Further Math P1 Topical Workbook (2012-2021)

Further Math P1 Topical 2012-2021 Compiled By A Fellow Student Roots Of Polonomial Rational Functions Summation Of Series Matrices 1 Polar Coordinates Vectors Proof By Induction 1-68 69-152 153-226 227-252 253-340 341-439 440-500 Assalam-o-Alaikum, So basically while doing AS I was unable to find any topical for FM that was not behind a paywall and easily accessible and due to that I was unable to practice much for the paper. Therefore, after my P1 paper I felt like I should compile this workbook so that others may have an easier time practicing and exploring this interesting/unique/difficult/depression inducing subject. Oh and as only a student compiled this workbook there might be some errors. I have tried my best to eliminate them but sorry if you find any. A fellow student, Craftear/AN Craftear Table Of Contents: 1 Craftear ROOTS OF POLYNOMIAL 2 Q1 3 6 The equation x 4 - 2x 3 - 1 = 0 has roots a , b , c , d . (a) Find a quartic equation whose roots are a 3 , b 3 , c 3 , d 3 and state the value of a 3 + b 3 + c 3 + d 3 . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 Craftear ............................................................................................................................................................ 3 (b) Find the value of 1 1 1 a b c 3+ 3+ 3+ 1 d3 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find the value of a 4 + b 4 + c 4 + d 4 . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 [Turn over Craftear ............................................................................................................................................................ 4 Q2 2 4 The cubic equation 2x 3 - 4x 2 + 3 = 0 has roots a , b , c . Let Sn = a n + b n + c n . (a) State the value of S1 and find the value of S2 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) (i) Express Sn +3 in terms of Sn + 2 and Sn . [1] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (ii) Hence, or otherwise, find the value of S4 . [2] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... © UCLES 2021 9231/13/M/J/21 Craftear .................................................................................................................................................... 5 (c) Use the substitution y = S 1 - x , where S1 is the numerical value found in part (a), to find and simplify an equation whose roots are a + b , b + c , c + a . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Find the value of 1 a+b + 1 b+c + 1 c+a . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 [Turn over Craftear ............................................................................................................................................................ 6 Q3 1 It is given that a + b + c = 3, a 2 + b 2 + c 2 = 5, a3 + b3 + c3 = 6. The cubic equation x 3 + bx 2 + cx + d = 0 has roots a , b , c . Find the values of b, c and d. [6] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... © UCLES 2021 9231/11/O/N/21 Craftear .................................................................................................................................................................... 7 Q4 4 6 The cubic equation x 3 + 2x 2 + 3x + 3 = 0 has roots a , b , c . (a) Find the value of a 2 + b 2 + c 2 . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Show that a 3 + b 3 + c 3 = 1. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 Craftear ............................................................................................................................................................ 8 Q5 3 6 The cubic equation x 3 + cx + 1 = 0 , where c is a constant, has roots a, b, c. (a) Find a cubic equation whose roots are a 3 , b 3 , c 3 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Show that a 6 + b 6 + c 6 = 3 - 2c 3 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 Craftear ............................................................................................................................................................ 9 f p 1 a3 b3 (c) Find the real value of c for which the matrix a 3 1 c 3 is singular. b3 c3 1 [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 [Turn over Craftear ............................................................................................................................................................ 10 Q6 1 The cubic equation x 3 + bx 2 + cx + d = 0 , where b, c and d are constants, has roots a, b, c. It is given that abc = -1. (a) State the value of d. [1] ............................................................................................................................................................ (b) Find a cubic equation, with coefficients in terms of b and c, whose roots are a + 1, b + 1, c + 1. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Given also that c + 1 = - a - 1, deduce that (c - 2b + 3) (b - 3) = b - c . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 [Turn over Craftear ............................................................................................................................................................ 11 Q7 2 4 The cubic equation 6x 3 + px 2 - 3x - 5 = 0 , where p is a constant, has roots a, b, c. (a) Find a cubic equation whose roots are a 2 , b 2 , c 2 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) It is given that a 2 + b 2 + c 2 = 2 (a + b + c) . (i) Find the value of p. [3] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... © UCLES 2020 9231/11/M/J/20 Craftear ............................................................................................................................................................ 12 .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (ii) Find the value of a 3 + b 3 + c 3 . [2] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... © UCLES 2020 9231/11/M/J/20 [Turn over Craftear .................................................................................................................................................... 13 Q8 1 2 The cubic equation 7x 3 + 3x 2 + 5x + 1 = 0 has roots a, b, c. (a) Find a cubic equation whose roots are a -1, b -1, c -1 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the value of a -2 + b -2 + c -2 . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find the value of a -3 + b -3 + c -3 . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 Craftear ............................................................................................................................................................ 14 Q9 7 The equation x3 + 2x2 + x + 7 = 0 has roots !, ", ' . (i) Use the relation x2 = −7y to show that the equation has roots ! " ' , , . "' '! !" 49y3 + 14y2 − 27y + 7 = 0 [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 Craftear ........................................................................................................................................................ 15 (ii) Show that !2 "2 '2 + + = 58 . "2 ' 2 ' 2 !2 !2 "2 49 [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find the exact value of "3 '3 !3 + + . "3 ' 3 ' 3 ! 3 !3 " 3 [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 [Turn over Craftear ........................................................................................................................................................ 16 Q10 6 The equation x3 − x + 1 = 0 has roots !, ", ' . (i) Use the relation x = y 3 to show that the equation 1 y3 + 3y2 + 2y + 1 = 0 has roots !3 , "3 , ' 3 . Hence write down the value of !3 + "3 + ' 3 . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 Craftear ........................................................................................................................................................ 17 Let Sn = !n + "n + ' n . (ii) Find the value of S−3 . [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Show that S6 = 5 and find the value of S9 . [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 [Turn over Craftear ........................................................................................................................................................ 18 Q11 9 A cubic equation x3 + bx2 + cx + d = 0 has real roots !, " and ' such that 1 1 1 + + =−5, 12 ! " ' !"' = −12, !3 + "3 + ' 3 = 90. (i) Find the values of c and d . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Express !2 + "2 + ' 2 in terms of b. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Show that b3 − 15b + 126 = 0. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 Craftear ........................................................................................................................................................ 19 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Given that 3 + iï 12 is a root of y3 − 15y + 126 = 0, deduce the value of b. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 [Turn over Craftear ........................................................................................................................................................ 20 Q12 4 It is given that the equation x3 − 21x2 + kx − 216 = 0, where k is a constant, has real roots a, ar and ar−1 . (i) Find the numerical values of the roots. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 Craftear ........................................................................................................................................................ 21 ........................................................................................................................................................ 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(ii) Deduce the value of k. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 [Turn over Craftear ........................................................................................................................................................ 22 Q13 6 The equation has roots !, ", ' . 9x3 − 9x2 + x − 2 = 0 (i) Use the substitution y = 3x − 1 to show that 3! − 1, 3" − 1, 3' − 1 are the roots of the equation y3 − 2y − 7 = 0. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The sum 3! − 1n + 3" − 1n + 3' − 1n is denoted by Sn . (ii) Find the value of S3 . [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 Craftear ........................................................................................................................................................ 23 (iii) Find the value of S−2 . [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 [Turn over Craftear ........................................................................................................................................................ 24 Q14 2 The roots of the equation x3 + px2 + qx + r = 0 are !, 2!, 4!, where p, q, r and ! are non-zero real constants. (i) Show that 2p! + q = 0. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Show that p3 r − q3 = 0. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 [Turn over Craftear ........................................................................................................................................................ 25 Q15 1 The roots of the cubic equation are !, ", ' . x3 − 5x2 + 13x − 4 = 0 (i) Find the value of !2 + "2 + ' 2 . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the value of !3 + "3 + ' 3 . [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 Craftear ........................................................................................................................................................ 26 Q16 7 By finding a cubic equation whose roots are !, " and ' , solve the set of simultaneous equations ! + " + ' = −1, ! + "2 + ' 2 = 29, 1 1 1 + + = −1. ! " ' 2 [8] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ 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................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 Craftear ................................................................................................................................................................ 27 ................................................................................................................................................................ 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................................................................................................................................................................ 28 Q17 1 The roots of the cubic equation x3 + 2x2 − 3 = 0 are !, " and ' . (i) By using the substitution y = 1 1 1 1 , find the cubic equation with roots 2 , 2 and 2 . 2 ! " ' x [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence find the value of 1 1 1 + 2 + 2. 2 ! " ' [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find also the value of 1 1 1 + 2 2 + 2 2. ! " " ' ' ! 2 2 [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 Craftear ........................................................................................................................................................ 29 Q18 4 The cubic equation 2x3 − 3x2 + 4x − 10 = 0 has roots !, " and ' . (i) Find the value of ! + 1 " + 1 ' + 1. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the value of " + ' ' + ! ! + ". [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 [Turn over Craftear ........................................................................................................................................................ Q19 1 , or x 1 1 1 otherwise, find the cubic equation whose roots are 1 + , 1 + and 1 + , giving your answer in the ! " ' form ay3 + by2 + cy + d = 0, where a, b, c and d are constants to be found. [4] The roots of the cubic equation 2x3 + x2 − 7 = 0 are !, " and ' . Using the substitution y = 1 + Craftear 1 30 © UCLES 2016 9231/11/M/J/16 [Turn over 31 Using the cartesian equation of the curve, find the area of this region. Q20 8 [3] The cubic equation z3 − z2 − z − 5 = 0 has roots !, " and ' . Show that the value of !3 + "3 + ' 3 is 19. [4] Find the value of !4 + "4 + ' 4 . [2] Show that the cubic equation with roots '−1 !−1 "−1 , and may be found using the substitution ! " ' Craftear 1 z= , and find this equation, giving your answer in the form px3 + qx2 + rx + s = 0, where p, q, r 1−x and s are constants to be determined. [4] © UCLES 2016 9231/13/M/J/16 [Turn over Q21 2 32 Find the cubic equation with roots !, " and ' such that ! + " + ' = 3, !2 + "2 + ' 2 = 1, !3 + "3 + ' 3 = −30, [6] Craftear giving your answer in the form x3 + px2 + qx + r = 0, where p, q and r are integers to be found. © UCLES 2016 9231/11/O/N/16 [Turn over Q22 4 33 The roots of the cubic equation x3 − 7x2 + 2x − 3 = 0 are !, " and ' . Find the values of (i) 1 , !" "' '! (ii) 1 1 1 + + , !" "' ' ! (iii) 1 1 1 . + 2 + ! "' !" ' !"' 2 2 [6] 1 1 1 , and . !" "' '! [2] Craftear Deduce a cubic equation, with integer coefficients, having roots © UCLES 2015 9231/11/M/J/15 [Turn over Q23 1 The quartic equation x4 − px2 + qx − r = 0, where p, q and r are real constants, has two pairs of equal roots. Show that p2 + 4r = 0 and state the value of q. [6] The curve C has polar equation r = e41 for 0 ≤ 1 ≤ !, where ! is measured in radians. The length of C is 2015. Find the value of !. [6] Craftear 2 34 © UCLES 2015 9231/13/M/J/15 [Turn over Q24 5 35 The cubic equation x3 + px2 + qx + r = 0, where p, q and r are integers, has roots !, " and ' , such that ! + " + ' = 15, ! + "2 + ' 2 = 83. 2 [3] Given that !, " and ' are all real and that !" + !' = 36, find ! and hence find the value of r. [5] Craftear Write down the value of p and find the value of q. © UCLES 2015 9231/11/O/N/15 [Turn over Q25 The equation x3 + px + q = 0, where p and q are constants, with q ≠ 0, has one root which is the [5] reciprocal of another root. Prove that p + q2 = 1. Craftear 1 36 © UCLES 2014 9231/11/M/J/14 [Turn over 37 Q26 11 4 Answer only one of the following two alternatives. EITHER The roots of the quartic equation x4 + 4x3 + 2x2 − 4x + 1 = 0 are , , and . Find the values of (i) + + + , [1] (ii) 2 + 2 + 2 + 2 , [2] 1 1 1 + + , [2] (iii) (iv) 1 + + + + . [2] Using the substitution y = x + 1, find a quartic equation in y. Solve this quartic equation and hence [7] find the roots of the equation x4 + 4x3 + 2x2 − 4x + 1 = 0. OR Craftear The square matrix A has as an eigenvalue with e as a corresponding eigenvector. Show that if A is non-singular then © UCLES 2014 9231/11/O/N/14 [Turn over Q27 3 38 The cubic equation x3 − 2x2 − 3x + 4 = 0 has roots !, ", ' . Given that c = ! + " + ' , state the value of c. [1] Use the substitution y = c − x to find a cubic equation whose roots are ! + ", " + ' , ' + !. Find a cubic equation whose roots are 1 1 1 + + . 2 2 ' + !2 ! + " " + ' [2] [2] Craftear Hence evaluate 1 1 1 , , . !+" "+' '+! [3] © UCLES 2013 9231/11/M/J/13 [Turn over Q28 2 39 The roots of the equation x4 − 4x2 + 3x − 2 = 0 are !, ", ' and $ ; the sum !n + "n + ' n + $ n is denoted by Sn . By using the relation y = x2 , or otherwise, show that !2 , "2 , ' 2 and $ 2 are the roots of the equation y4 − 8y3 + 12y2 + 7y + 4 = 0. 3 State the value of S2 and hence show that 3 Craftear S8 = 8S6 − 12S4 − 72. © UCLES 2013 9231/13/M/J/13 [Turn over Q29 2 40 The cubic equation x3 − px − q = 0, where p and q are constants, has roots !, ", ' . Show that (i) !2 + "2 + ' 2 = 2p, [1] (ii) !3 + "3 + ' 3 = 3q, [2] [3] Craftear (iii) 6 !5 + "5 + ' 5 = 5 !3 + "3 + ' 3 !2 + "2 + ' 2 . © UCLES 2013 9231/11/O/N/13 [Turn over Q30 5 41 The equation 8x3 + 36x2 + kx − 21 = 0, Craftear where k is a constant, has roots a − d, a, a + d. Find the numerical values of the roots and determine the value of k. [8] © UCLES 2013 9231/13/O/N/13 [Turn over Q31 1 42 The roots of the cubic equation x3 − 7x2 + 2x − 3 = 0 are α , β , γ . Find the values of (i) α 2 + β 2 + γ 2 , [2] (ii) α 3 + β 3 + γ 3 . Craftear [3] © UCLES 2012 9231/11/M/J/12 [Turn over Q32 43 11 EITHER The roots of the equation x4 − 3x2 + 5x − 2 = 0 are α , β , γ , δ , and α n + β n + γ n + δ n is denoted by Sn . Show that Sn+4 − 3Sn+2 + 5Sn+1 − 2Sn = 0. [2] Find the values of (i) S2 and S4 , [3] (ii) S3 and S5 . [6] Hence find the value of [3] Craftear α 2 (β 3 + γ 3 + δ 3 ) + β 2 (γ 3 + δ 3 + α 3 ) + γ 2 (δ 3 + α 3 + β 3 ) + δ 2 (α 3 + β 3 + γ 3 ). © UCLES 2012 9231/12/O/N/12 [Turn over Q33 7 44 A cubic equation has roots α , β and γ such that α + β + γ = 4, Find the value of αβ + βγ + γ α . α 2 + β 2 + γ 2 = 14, α 3 + β 3 + γ 3 = 34. [2] Show that the cubic equation is x3 − 4x2 + x + 6 = 0, [6] Craftear and solve this equation. © UCLES 2012 9231/13/O/N/12 [Turn over Q34 8 45 The cubic equation x3 − x2 − 3x − 10 = 0 has roots α , β , γ . (i) Let u = −α + β + γ . Show that u + 2α = 1, and hence find a cubic equation having roots −α + β + γ , α − β + γ, α + β − γ. [5] 1 1 1 , , . βγ γ α αβ [5] Craftear (ii) State the value of αβγ and hence find a cubic equation having roots © UCLES 2012 9231/13/M/J/12 [Turn over ROOTS OF POLYNOMIAL MS Craftear 46 Q1 3(c) 3(b) Question 3(a) Question Craftear α 3 β 3δ 3 + α 3 β 3γ 3 + β 3γ 3δ 3 + α 3γ 3δ 3 6 = −1 α 3 β 3γ 3δ 3 Guidance 3 A1 M1 1 1 1 1 A1 FT Relates α 3 + β 3 + γ 3 + δ 3 to coefficients. 2 A1 δ = 3 1 = 20 γ + 3 1 M1 Uses original equation. β + 3 1 Marks 4 α 4 + β 4 + γ 4 + δ 4 = 2(α 3 + β 3 + γ 3 + δ 3 ) + 4 −6 α + 3 1 α3 + β3 + γ 3 +δ 3 = 8 Answer A1 y 4 − 8 y 3 − 12 y 2 − 6 y − 1 = 0 B1 FT M1 Obtains an equation not involving radicals. 4 y 3 − 2 y −1 = 0  y 4 = ( 2 y + 1)3 = 8 y 3 + 12 y 2 + 6 y + 1 Guidance B1 Correct substitution. Marks y = x3 Answer 47 Q2 M1 Uses formula for sum of squares. A1 Correct answer implies M1A1. S2 = S12 − 2(0) =4 A1 =4 2(d) 2(c) M1 Uses their recursive formula from part (i) to find S4  S3 = 72  . S4 = 2S3 − 32 S1 = 2(2S2 − 23 S0 ) − 32 S1 2(b)(ii) = 8 3 Craftear OR Or use 2 S 2 − 8S1 + 8S0 − 3S −1 = 0 with substitution of their values α' 1 + 2 + 1 = α ' β '+ β ' γ '+ γ 'α ' . β' γ' α ' β 'γ ' 1 A1 FT FT from 2(c). M1 Uses A1 OE but must be an equation. 2 y3 − 8 y 2 + 8y − 3 = 0 8 2 −3 2 M1 Makes their substitution. 2(2 − y )3 − 4(2 − y )2 + 3 = 0 3 B1 SOI x =2− y 2 1 B1 CAO or as a single fraction. Sn+3 = 2Sn+ 2 − 32 Sn 3 B1 S1 = 2 2(b)(i) 2(a) 48 Q4 Q3 4(b) 4(a) Question 1 Question A1 −2 A1 AG 1 2 M1 Uses original equation or formula for sum of cubes. α 3 + β 3 + γ 3 = −2(−2) − 3(−2) − 3(3) 2 M1 Uses formula for sum of squares. Guidance (−2)2 − 2(3) Marks A1 [Equation is x 3 − 3 x 2 + 2 x + 1 = 0 ] d =1 Answer M1 Uses original equation or formula for sum of cubes. 6 − 3(5) + 2(3) + 3d = 0 6 A1 Craftear Guidance M1 A1 Uses formula for sum of squares. B1 Marks c=2 5 = 32 − 2(αβ + βγ + γα ) b = − (α + β + γ ) = −3 Answer 49 Q5 3(a) Question 4(c) n r =1 r =1 ) Craftear 3 A1 ( y 3 + 3 y 2 + 3 + c3 y + 1 = 0 Guidance M1 Correct attempt to eliminate cube root. 1 3 Marks y + cy 3 + 1 = 0  −c3 y = ( y + 1) = y 3 + 3 y 2 + 3 y + 1 Answer 6 M1 A1 Simplifies. M1 Applies formulae from MF19. B1 Substitutes. ) Expands. M1 A1 Collects like terms and uses results from parts (a) and (b). B1 y = x3  x = y 3 1 n + 14 n(n + 1) 3n 2 − 5n −16 ( n + 14 n(n + 1) ( −12 − 4(2n +1) + 3n(n +1) ) n − 3n(n + 1) − n( n + 1)(2n + 1) + 34 n 2 ( n + 1) 2 n − 6 ( 12 n( n + 1) ) − 6 ( 16 n( n + 1)( 2n + 1) ) + 34 n 2 ( n + 1) 2 3 3 3  ( (α + r ) + ( β + r ) + (γ + r ) ) =  (1 + 3(−2)r + 3(−2)r 2 + 3r 3 ) n (α + r )3 = α 3 + 3α 2 r + 3α r 2 + r 3 50 3(c) 3(b) Craftear A1 c=32 5 M1 Sets determinant equal to zero. 1 M1 A1 Evaluates determinant. B1 If using their answer to (a) FT 2c 3 − 4 = 0 β3 γ 3 1 α3 γ 3 = 1 − (α 6 + β 6 + γ 6 ) + 2α 3 β 3γ 3 = 2c 3 − 4 α3 β3 1 α 3 β 3γ 3 = −1 3 A1 AG α 6 + β 6 + γ 6 = (α 3 + β 3 + γ 3 )2 − 2(α 3 β 3 + β 3γ 3 + γ 3α 3 ) = 3 − 2c 3 2 M1 B1 FT Using their answer to (a). α 6 + β 6 + γ 6 = ( −3 ) − 2 ( 3 + c 3 ) α 3 + β 3 + γ 3 = −3 α 3 β 3 + β 3γ 3 + γ 3α 3 = 3 + c3 51 Q6 1(c) y = x +1 x = y −1 1(b) Answer A1 y 3 + (b − 3) y 2 + (c − 2b + 3) y + b − c = 0) B1 Uses sum of products in pairs. M1 Applies product of roots. A1 AG −(α +1)(α + 1) = c − 2b + 3 − (α + 1) ( β + 1) (α + 1) = − ( b − c )  ( c − 2b + 3) ( b − 3) = b − c 4 B1 Uses sum of roots. β + 1 = − ( b − 3) 3 M1 Using these relationships. (α + 1 + β + 1 + γ +1) = 3 − b, (α + 1)( β + 1)(γ +1) = c − b M1 A1 Substitutes and expands. B1 Craftear Guidance B1 Uses correct substitution. 1 B1 Marks (α + 1)( β +1) + (α + 1)(γ + 1) + ( β + 1)(γ + 1) = c − 2b + 3 Alternative method for question 1(b) y 3 + (b − 3) y 2 + (c − 2b + 3) y + b − c = 0 d =1 1(a) Question 52 Q8 Q7 1 1 Answer A1 p = −6 α −2 + β −2 + γ −2 = ( −5) 2 − 2 ( 3 ) = 19 α −3 + β −3 + γ −3 = −5(19) − 3(−5) − 21 = −101 1(c) 7 y −3 + 3 y −2 + 5y −1 +1 = 0  y 3 + 5 y 2 + 3y + 7 = 0 y = x−1 Craftear A1 α3 + β3 +γ 3 = 5 4 M1 A1 M1 A1 3 M1 A1 B1 Marks 2 M1 6(α 3 + β 3 + γ 3 ) = 6(α 2 + β 2 + γ 2 ) + 3(α + β + γ ) + 15 3 M1 p 2 + 36 2p =−  p 2 + 12 p + 36 = 0 36 6 2 B1 2 3 p 2 + 36 α + β +γ = 36 2 36 y 3 − ( p 2 + 36) y 2 + (10 p + 9) y − 25 = 0 y (6 y − 3) 2 = ( − py + 5) 2  y (36 y 2 − 36 y + 9) = p 2 y 2 −10 py + 25 A1 M1 3 6 y 2 + py − 3 y 2 − 5 = 0  y 2 ( 6 y − 3) = − py + 5 Marks B1 Answer y = x2 1(b) 1(a) Question 2(b)(ii) 2(b)(i) 2(a) Question 53 Q9 7(iii) 7(ii) 7(i) Question 2 x2 x2 = −7 αβγ ⇒ α3 β3 γ3 317 + + =− 3 3 3 3 3 3 βγ αγ α β 343  α3 β3 γ3   58   2 49  3 3 + 3 3 + 3 3  = −14   + 27  −  − 21 αγ α β   49   7 β γ α2 β2 γ2  2  27  58 + + =  −  − 2 −  = 2 2 2 2 2 2 β γ α γ α β  7  49  49 2 α β γ 2 1 1 1 27 + + =− , 2 + 2 + 2 =− 7 γ 49 βγ αγ αβ β α α2 α β2 β γ2 γ = , = , = So roots are αβγ βγ αβγ αγ αβγ αβ y= ⇒ 49 y 3 +14 y 2 − 27 y + 7 = 0 ⇒ −7 y ( −7 y + 1) = 14 y − 7 ⇒ −7 y ( −7 y + 1) = (14 y − 7 ) −7 y ( −7 y ) + 2 ( −7 y ) + −7 y + 7 = 0 Answer Craftear 2 Guidance AG 2 A1 M1 Uses 49α ' 3 = −14α ′ 2 + 27α ′ − 7 . 3 2 α ' 2 + β ' 2 + γ ' 2 = (α '+ β '+ γ ') − 2 (α ' β '+ α ' γ '+ β ' γ ') M1 A1 Uses B1 States sum of roots and α ' β '+ α ' γ '+ β ' γ ' . 4 A1 AG M1 Uses αβγ = −7 . A1 AG M1 Uses given substitution and eliminates radical. Marks 54 Q11 Q10 9(i) Question 6(iii) 6(ii) 6(i) Question + β 1 d = 12 α 1 + γ 1 = αβ + β γ + αγ ⇒ c = αβ + βγ + αγ = 5 αβγ Answer S9 = −3S6 − 2S3 − 3 = −3 ( 5 ) − 2 ( −3) − 3 = −12 2 S6 = ( −3) − 2 ( 2 ) = 5 α 3 β 3 + β 3γ 3 + α 3γ 3 2 = = −2 −1 α 3 β 3γ 3 Craftear ∑αβ . AG. 3 B1 M1 A1 Marks 4 Uses α 1 + β 1 + γ 1 = αβ + βγ + αγ . αβγ Guidance M1 A1 Sums y 3 − 3 y 2 + 2 y − 1 = 0 for y = α 3 , β 3 , γ 3 . α ≠β M1 A1 Uses ( ∑ α )2 = ∑ α 2 + 2 2 M1 A1 3 B1 S3 = −3 S −3 = Guidance M1 Obtains an equation in y not involving radicals. A1 AG 3 Marks y = y 3 + 3 y 2 + 3y + 1 ⇒ y 3 + 3y 2 + 2 y +1 = 0 y = ( y +1) Answer 55 9(iv) 9(iii) 9(ii) ) (−b = α + β + γ ) Real root is b = −6 Craftear M1 A1 Gives the real root for b . 4 M1 A1 Equates expressions. 5b − 126 = b2 − 10 ⇒ b3 −15b + 126 = 0 b 2 M1 A1 Finds sum of squares in terms of b. A1 (M1) Substitutes known values. M1 A1 Uses expansion of (α + β + γ )3 A1 M1 A1 Sums over roots and uses 9(ii) M1 Formulates equation. 2 M1 A1 Sums over roots α 2 + β 2 + γ 2 = (α + β + γ ) − 2 (αβ + βγ + γα ) = b 2 − 10 Alternative method 2 for 9(iii) b3 −15b + 126 = 0 −b3 = 90 −15b + 36 3 (α + β + γ ) = α 3 + β 3 + γ 3 + 3(α + β + γ ) (αβ + βγ + γα ) − 3αβγ Alternative method 1 for 9(iii) b3 −15b + 126 = 0 90 + b b 2 − 10 − 5b + 36 = 0 ( x 3 + bx 2 + 5 x + 12 = 0 90 + bS2 − 5b + 36 = 0 ⇒ S2 = Alternative method for 9(ii) 5b −126 b A1 (α + β + γ ) −10 = b2 −10 2 M1 2 α 2 + β 2 + γ 2 = (α + β + γ ) − 2 (αβ + βγ + γα ) 56 Q13 Q12 6(ii) 6(i) Question 4(ii) 4(i) Question ) M1 Uses y 3 = 2 y + 7 . Or uses formula for Σ (3α – 1)3 A1 S3 = 2 S1 + 7 × 3 = 21 Craftear Guidance M1 Accept substitution of y = 3x − 1 into given equation and derivation of equation in x . Marks 2 M1 A1 Or finds coefficient of x in ( x − 3)( x − 6 )( x − 12 ) . Or substitutes root into equation 6 A1 A1 AG. y +1 3 Guidance M1 A1 Substitutes for a and solves quadratic M1 Uses sum of roots M1 A1 Uses product of roots Marks Obtains the given result Substitutes x = Answer k = αβ + αγ + βγ = 6 (12 ) + 6 ( 3) + 12 ( 3) =126 Roots are 6, 12, 3 2 r 2 − 5r + 2 = 0 ⇒ r = 2 or r = 0.5 6 1 + r + r −1 = 21 ( a + ar + ar −1 = 21 αβγ = a3 = 216 ⇒ a = 6 Answer 57 Q14 2(i) Question 6(iii) 4 49 1 = ∑(3α −1) 2 (3β − 1) 2 Craftear A1 Verifies result (AG). ⇒ 2 pα + q = 0 4 M1 Combines equations. 14α 2 q =− p 7α Guidance B1 Sum of products in pairs. Marks 7z3 + 2z2 – 1 = 0 A1 Uses S2 = (S1)2 – 2xΣαβ M1 4 A1 = 49 1 , etc. M1 2 written down directly. 7 Alt method: Finds cubic with roots 2α 2 + 4α 2 + 8α 2 = q Answer 8 A1 M1 M1 A1 A1 B1 Sum of roots. 4 ( − 2)2 – 2(7)(0) = 2 49 7 Award M1A1 if S−1 = − M1 Uses 7 y −2 = y − 2 y −1 . M1 A1 α + 2α + 4α = − p = 2 ( 3α − 1) (3β − 1)2 (3γ − 1)2 (∑ ( 3α − 1)( 3β − 1))2 − 2 ( 3α −1) ( 3β −1)( 3γ −1) (∑ ( 3α −1)) s−2 = ∑ ( 3α − 1)( 3β −1) + ( 3α − 1)( 3γ − 1) + ( 3β − 1) ( 3γ − 1) = −2 . 7 ( 3α − 1)( 3β −1) ( 3γ − 1) 7 S −2 = S1 − 2 S −1 S −1 = 58 Q15 1(ii) 1(i) Question 2(ii) ) Or ( ∑α ) 3 – 3 ( ∑α ) ( ∑ αβ ) +3 αβγ Craftear Alt method: Use formula e.g. ∑ α 3 = ( ∑ α ) ∑ α 2 − ∑ αβ + 3αβγ 2 A1 = 5( −1) −13( 5) + 12 = −58 ( M1 Uses α 3 = 5α 2 − 13α + 4. α 3 + β 3 + γ 3 = 5 (α 2 + β 2 + γ 2 ) − 13 (α + β + γ ) + 12 3 www A1 = ‒1 Guidance M1 Uses ∑ α 2 = ( ∑ α )2 − 2 ( ∑ αβ ) Marks α 2 + β 2 + γ 2 = 52 − 2 (13) Answer 2 B1 Verifies result (AG). B1 Sum of roots and αβ + αγ + βγ . SOI q3 ⇒ p3r − q3 = 0 3 p B1 Product of roots. α + β + γ = 5 , αβ + αγ + βγ = 13 ⇒r= 8α 3 = −r 59 Q17 Q16 1(iii) 1(ii) 1(i) Question 7 Question ) 2 1 2 −3=0⇒ = 3− y y y y 1 2 2 + α β α 2 1 + β 2 1 β γ 2 2 2 1 γ 1 + + = γ α 2 1 2 = 12 4 or 9 3 4 9 ⇒ 9 y 3 − 12 y 2 + 4 y − 1 = 0 SR B1 for finding cubic by manipulating roots + 1 1 ⇒x= 2 x y y y 1 y= Craftear Answer ⇒Solution is –1, in ± 14 any order. Accept ±3.74 (awrt) SR B1 for correct roots without working ⇒ ( x + 1) x 2 − 14 ( ⇒ x 3 + x 2 − 14 x − 14 = 0 αβγ Total: Guidance 8 A1 Total: Total: Total: Guidance 1 B1FT 1 B1FT 3 A1 OE M1 Substituting and squaring M1 Rearranging to make x the subject Marks M1A1 Attempt to factorise cubic A1 M1A1 FT ∑αβ = −14 = −1 ⇒ αβγ = 14 αβγ M1A1 Marks ∑αβ = 1 − 29 ⇒ ∑αβ = −14 2 Answer 60 Q18 4(ii) 4(i) Question 3 2 αβ + βγ + γα = 2 1  2 27 9 3 3 − × + × 2 − 5 = −2 8 4 2 2 = Guidance M1, A1 A1 M1 4 M1A1 A1 M1 Alt methods: = (∑ α ) (∑ αβ ) − αβγ or ∑ α 2 ∑ α + 2αβγ − ∑ α 3 4 A1FT Alt method: Let x= y–1 Sub and expand 2y3 – 9y2 16y – 19 = 0 Product of roots = 19/2 M1A1 Multiply out and group for M1 B1 (Can be awarded in (ii) if not seen here) SOI Marks Craftear 27 9 3 − (α + β + γ ) + (αβ + βγ + γα ) − αβγ 8 4 2 1  2 = 1  2  ( β + γ )( γ + α ) (α + β ) = 1 − α  1 − β   1 − γ  1 1 = 5 + 2 +1 +1 = 9 2 2  αβγ = 5 + β + γ = (α + 1)( β + 1) (γ + 1) = αβγ + (αβ + βγ + γα ) + (α + β + γ ) + 1 3 αβ + βγ + γα = 2αβγ = 5 2 α + β +γ = Answer 61 62 y =1+ 1 1 ⇒x= x y −1 2 ( y − 1) ( 3 + 1 ( y −1) M1 3 2 − 7 = 0 ⇒ 2 + ( y − 1) − 7 ( y − 1) = 0 ) ⇒ 7 y 3 − 3 y 2 + 3y −1 − y + 1 − 2 = 0 ⇒ 7 y 3 − 21y 2 + 20 y − 8 = 0 Q20 8 A1 M1A1 [4] ALT METHOD: ∑α, ∑αβ, αβγM1 A1, ∑(1+1/α) etc M1 A1 Let Sn denote αn + βn +γn. S 2 = S12 − 2 ∑αβ = 12 − 2 × ( −1) = 3 S3 = S2 + S1 +15 = 3 + 1 + 15 = 19 (AG) Alt method: ∑ α = 1, Σαβ= -1, and αβγ = 5 M1, A1 S3 = S13- 3Σα Σαβ + 3αβγ =19 M1, A1 S 4 = S3 + S 2 + 5S1 = 19 + 3 + 5 = 27 z −1 1 1 1 =1− ⇒ =1− x ⇒ z = z z z 1− x 1 1 1 − − − 5 = 0 is the required equation. Hence 3 2 (1 − x ) (1 − x ) (1 − x ) x= 2 3 ⇒ 1 − (1− x ) − (1− x ) − 5 (1− x ) = 0 ⇒ … ⇒ 5 x 3 − 16 x 2 + 18 x − 6 = 0 M1A1 M1A1 [4] M1A1 [2] B1 M1 M1A1 [4] Craftear Q19 1 Q21 Q22 2 ( 2 4 B1 (iii) ⇒ x3 − Craftear M1A1√ (2) Total: 8 M1A1 (6) 1 1 1 1 1  αβ + βγ + γα  2 1 1 1 = + +  + + = = 2 2 αβγ  α β γ  αβγ  αβγ α βγ αβ γ αβγ  9 (ii) 7 2 2 1 x + x − = 0 ⇒ 9 x 3 − 21x 2 + 2 x −1 = 0 3 9 9 M1A1 1 1 1 α + β +γ 7 + + = = αβ βγ γα αβγ 3 (i) 2 B1 α + β + γ = 7 , αβ + βγ + γα = 2 , αβγ = 3 (Stated or implied by working.) 1 1 1 = = 2 (αβ )( β γ )(γα ) (αβγ ) 9 ALT METHOD: S3 – 3S2+4S1 + 3r = 0 M1 3r = 30 +3 × 1 – 4 × 3 A1 r=7 A1 3 A1 ⇒ αβγ = −7 Required cubic equation is x − 3x + 4 x + 7 = 0 must see final equation M1 A1 A1 3 ( ∑ α ) = ∑ α 3 + 3 ∑ α ∑ αβ − 3αβγ ) M1A1 Correct substitution in formula or Use of, e.g.: ∑ α 3 − 3αβγ = ∑ α ∑ α 2 − ∑ αβ 2 ∑ αβ = 9 − 1 ⇒ ∑ αβ = 4 [6] 63 64 Q23 2α + 2β = 0 ii α 2 + 4αβ + β 2 = − p iii 2α 2 β + 2αβ 2 = − q iv α 2 β 2 = −r 1 i All four correct (Any two correct) Use of β = −α in ii, iii or iv B2 (B1) M1 ⇒ p = 2α 2 and α 4 = − r ⇒ p 2 + 4r = 0 (AG) M1A1 B1 (6) Total 6 and q = 0 (CAO) Q24 α + β + γ = − p = 15 ⇒ p = −15 B1 ( ) 2(αβ + βγ + γα) = (α + β + γ ) 2 − α 2 + β 2 + γ 2 = 2q 1 ⇒ q = (225 − 83) = 71 2 M1 A1 [3] 36 = 15 − α (= [β + γ ]) α ⇒ a 2 − 15α + 36 = 0 ⇒ α = 3 , α ≠ 12 , e.g. since 122 > 83 or other reason βγ = 71− 36 = 35 ⇒ r = −αβγ = −3× 35 = −105 (extra answer penalised) M1 M1A1 B1 A1 [5] Total 8 Q25 1 Let roots be α, α–1 and β ⇒ α + α–1 + β = 0 (1) Any of three for M1 Product of roots ⇒ β = –q (2) A1 for another. Sum of products in pairs ⇒ 1+ β (α + α–1) = p (3) A1 for a third M1 A1 A1 From (1) and (3) 1 – β2 = p Using (2) 1 – q2 = p or p + q2 = 1 M1 A1 [5] Wrong sign in (2) scores M1A0A1M1A0 Craftear 5 Q26 11E (i) 65 α + β + γ + δ = −4 B1 (1) (ii) α 2 + β 2 + γ 2 + δ 2 = (− 4) − 2 × 2 = 12 (iii) 1 1 1 1 − (− 4) + + + = =4 α β γ δ 1 (iv) α β γ δ α 2 + β 2 + γ 2 + δ 2 12 + + + = = = 12 βγδ αγδ αβδ αβγ αβγδ 1 2 M1A1 (2) M1A1 (2) M1A1 (2) y = x +1 ⇒ x = y −1 ( y −1)4 + 4( y −1)3 = y 4 − 6 y 2 + 8y − 3 2 2( y − 1) − 4( y − 1) +1 = 2 y 2 − 8y + 7 4 3 2 4 M1A1 A1 2 ⇒ x + 4 x + 2 x − 4x +1 = y − 4 y + 4 = 0 A1 (y − 2) = 0 ⇒ y = ± 2 (twice). A1 2 2 ⇒ x = ± 2 −1 (twice). (Some indication of four roots for fina M1A1 (7) [14] Q27 Uses −b ∑α = a . Uses substitution c=2 B1 ( α + β = c − γ etc.) ⇒ y = c − x ⇒ x = c − y M1 M1 (2 − y ) 3 − 2(2 − y ) 2 − 3(2 − y) + 4 = 0 …(their c) to obtain required cubic equation. ⇒ y3 − 4y2 + y + 2 = 0 A1 Obtains equation whose roots are reciprocals of those in previous cubic equation. Uses z = y −1 to obtain 2 z 3 + z 2 − 4 z + 1 = 0 M1A M1A1 Uses ∑ α = (∑ α ) − 2∑ αβ 2 2 ∑ 1 Craftear 3 2 1 1 1 =   − 2(−2) = 4 2 4 (α + β ) 2 M1A1 2 [8] Q28 2 1 Makes substitution. Squares. Obtains result. y 2 − 4 y + 3y 2 − 2 = 0 4 2 2 3 ⇒ 9 y = 4 + y + 16 y − 4 y + 16 y − 8y (N.B. Must see both terms in y2.) ⇒ y 4 − 8y 3 + 12 y 2 + 7 y + 4 = 0 (AG) S 2 = 0 2 − 2 × (−4) = 8 S 8 = 8S 6 − 12 S 4 − 7 S 2 − 16 ⇒ S 8 = 8S 6 − 12 S 4 − 56 − 16 = 8S 6 − 12 S 4 − 72 (AG) Alternatively – for final two marks. S2 = 8 , S3 = –9 , S4 = 40 , S5 = –60 , S6 =203 , S7 = –378 S8 = 1072 (generated by substitution of roots in equations and summing.) Then 8S 6 −12S 4 − 72 = 1624 − 480 − 72 = 1072 = S 8 M1 requires a complete method, A1 if all correct. M1 M1 A1 3 B1 M1 A1 3 [6] 66 Q29 2 (i) α 2 + β 2 + γ 2 = (α + β + γ ) − 2(αβ + βγ + γα ) = 0 − 2(− p ) = 2 p (AG) B1 1 2 2 Finds S2. (ii) Finds S3. α 3 + β 3 + γ 3 = p ∑α + 3q = 0 + 3q = 3q (AG) M1A1 (iii) Finds S5. α5 + β5 +γ 5 = p ∑α + q∑α M1 3 2 = p.3q + q.2 p = 5 pq 5 3 ⇒ 6 ∑α = 30pq = 5∑α ∑α 2 A1 (AG) A1 [6] 3 Q30 36 9 3 Uses sum of roots ∑ α = 3a = − 8 = − 2 ⇒ a = − 2 Uses product of roots αβγ = a a 2 − d 2 = ) 218 M1 Substitutes for a, 9 2 21 7 −d2 = − × =− 4 3 8 4 M1 and solves ⇒ d 2 = 4 ⇒ d = ±2 A1 7 3 1 Roots are − , − , 2 2 2 A1 Uses sum of products in pairs. (Or expands (2x + 7)(2x + 3)(2x + 1) ( 21 7 3 k M1A1 ∑ αβ = 4 − 4 − 4 = 8 M1 ⇒ k = 22 A1 Craftear 5 (8) and equates coefficients) [8] Q31 1 States ∑ α and ∑ αβ Uses formula for correctly. Uses formula for to obtain result. ∑α 3 ∑ α = 7 ∑ αβ = 2 ∑ α = 7 − 2 × 2 = 45 2 2 ∑ α = 7 ∑ α − 2∑ α + 9 3 2 = 315-14+9 = 310 B1 B1 2 M1 A1A1 3 [5] 67 Q32 (i) (ii) ∑ α = (∑ α ) − 2∑ αβ 2 2 α is a root ⇒ α 4 − 3α 2 + 5α − 2 = 0 ⇒ α n+ 4 − 3α n+ 2 + 5α n+1 − 2α n = 0 Repeat for β , γ , δ and sum ⇒ S n+ 4 − 3S n+ 2 + 5S n+1 − 2 S n = 0 (AG) Finds S 4 from formula. S 2 = 0 − 2 × ( −3) = 6 S 4 = 3 × 6 − 5 × 0 + 2 × 4 = 26 ∑ αβγ S = S −1 = Uses −1 αβγδ Finds S 3 from formula. Finds S 5 from formula. 2 B1 M1A1 3 −5 5 = −2 2 M1A1 S3 = 3 × 0 − 5 × 4 + 2 × M1A1 5 = −15 2 S 5 = 3 × (−15) − 5 × 6 + 2 × 0 = −75 ∑α β = S S − S 2 3 2 3 5 = 6 × (−15) − (−75) = −15 Q33 7 M1 A1 (α + β + γ ) 2 − 2(αβ + βγ + γα ) = α 2 + β 2 + γ 2 ⇒ Either Required equation is x 3 − 4 x 2 + x + c = 0 ⇒ ∑ α 3 − 4∑ a 2 + 4 + 3c = 0 ⇒ 3c = 56 − 34 − 4 = 18 ⇒ c = 6 ∑ αβ = 1 (AG) Or α 3 + β 3 + γ 3 − 3αβγ = (α + β + γ )(α 2 + β 2 + γ 2 − αβ − βγ − γα ) (or some other appropriate identity, e.g. (α + β + γ ) 3 = α 3 + β 3 + γ 3 + 3(α + β + γ )(αβ + βγ + γα ) − 3αβγ ) ⇒ ... ⇒ αβγ = −6 (AG) ⇒ x3 − 4x 2 + x + 6 = 0 ⇒ (x + 1)(x − 2)(x − 3) = 0 ⇒ x = –1,2,3. M1A1 6 M1 M1A1 3 M1A1 [14] 2 M1 M1 A1 Craftear EITHER Substitute α into equation. Multiply by α n . Obtain result. 11 (M1) (M1A1) A1 M1A1 6 [8] Q34 8 (i) Deduces initial result. Substitutes into cubic equation. 68 u = −α + β + γ ⇒ u + 2α = α + β + γ = 1 1− u  ⇒α =   2  3 Deduces new cubic equation. (ii) Deduces initial result. Substitutes into cubic equation. Deduces new cubic equation. M1 2 1− u  1− u  1− u  ⇒  −  − 3  −10 = 0  2   2   2  ⇒ … ⇒ u 3 − u 2 − 13u + 93 = 0 αβγ = 10 ⇒v= M1A1 A1 A1 5 B1 1 v 1 1 ⇒ = = ⇒ α = 10v βγ α αβγ 10 M1A1 (10v )3 − (10v )2 − 3(10v ) − 10 = 0 M1 ⇒ 100v 3 − 10v 2 − 3v − 1 = 0 A1 5 [10] Alternatively: Award M1 for an attempt at formulae for all three coefficients. A1 for any two correct. A1 for completion (ii) For final 4 marks in (ii): Award M1 for an attempt at formulae for all three coefficients. A1 for any one correct. A1 for a second one correct. A1 for completion. –b = ∑α = 1 ⇒ b = –1 c = 4∑αβ – (∑α)2 = 4 × (–3) – 12 = –13 −d =4 ∑α ∑αβ − (∑α ) − 8αβγ 3 = 4 × 1 × (–3) – 13 – 8 × 10 = –93 So u3 – u2 – 13u + 93 = 0 (5) Let equation be v3 + bv2 + cv + d = 0. Craftear 8 (i) For final 3 marks in (i): Let equation be u3 + bu2 + cu + d = 0. Σα 1 1 = ⇒b=− αβγ 10 10 3 Σαβ −3 c= = =− 100 (αβγ ) 2 10 2 −b = 1 1 1 = ⇒d =− 100 (αβγ ) 2 10 2 1 3 1 v− =0 So v 3 − v 2 − 10 100 100 or 100v3 – 10v2 – 3v – 1 = 0. −d = (5) [10] 69 Craftear RATIONAL FUNCTIONS 70 Q1 7 The curve C has equation y = x2 + x + 9 . x+1 (a) Find the equations of the asymptotes of C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the coordinates of the stationary points on C. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 Craftear ............................................................................................................................................................ 71 (c) Sketch C, stating the coordinates of any intersections with the axes. [3] (d) Sketch the curve with equation y = x2 + x + 9 and find the set of values of x for which x+1 2 x 2 + x + 9 2 13 x + 1 . © UCLES 2021 [5] 9231/11/M/J/21 Craftear ............................................................................................................................................................ 72 Q22 7 x2 - x - 3 . 1 + x - x2 (a) Find the equations of the asymptotes of C. The curve C has equation y = [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 [Turn over Craftear (b) Find the coordinates of any stationary points on C. 73 [3] ............................................................................................................................................................ (d) Sketch the curve with equation y = which © UCLES 2021 x2 - x - 3 1 3. 1 + x - x2 x2 - x - 3 and find in exact form the set of values of x for 1 + x - x2 [6] 9231/13/M/J/21 Craftear (c) Sketch C, stating the coordinates of the intersections with the axes. 74 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 Craftear ............................................................................................................................................................ 75 Q3 7 The curve C has equation y = 4x + 5 . 4 - 4x 2 (a) Find the equations of the asymptotes of C. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the coordinates of any stationary points on C. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 Craftear ............................................................................................................................................................ 76 (d) Sketch the curve with equation y = which 4 4x + 5 2 5 4 - 4x 2 . [3] 4x + 5 and find in exact form the set of values of x for 4 - 4x 2 [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 [Turn over Craftear (c) Sketch C, stating the coordinates of the intersections with the axes. 77 ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 Craftear ............................................................................................................................................................ 78 Q4 6 The curve C has equation y = x2 . x-3 (a) Find the equations of the asymptotes of C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Show that there is no point on C for which 0 1 y 1 12 . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 Craftear ............................................................................................................................................................ 79 (c) Sketch C. x2 and y = x - 3 on a single diagram, stating the coordinates x-3 of the intersections with the axes. [4] (d) (i) Sketch the graphs of y = (ii) Use your sketch to find the set of values of c for which x2 G x + c has no solution. [1] x-3 .................................................................................................................................................... .................................................................................................................................................... © UCLES 2021 9231/12/O/N/21 [Turn over Craftear [2] 80 Q55 6 The curve C has equation y = x2 + x - 1 . x-1 (a) Find the equations of the asymptotes of C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Show that there is no point on C for which 1 1 y 1 5. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 Craftear ............................................................................................................................................................ 81 (c) Find the coordinates of the intersections of C with the axes, and sketch C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ Craftear ............................................................................................................................................................ (d) Sketch the curve with equation y = © UCLES 2020 x2 + x - 1 . x-1 9231/11/O/N/20 [2] [Turn over 82 Q66 6 Let a be a positive constant. (a) The curve C1 has equation y = x-a . x - 2a [2] Sketch C1 . x-a 2 l . The curve C3 has equation y = x - a . x - 2a x - 2a (b) (i) Find the coordinates of any stationary points of C2 . [3] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... © UCLES 2020 9231/12/O/N/20 Craftear The curve C2 has equation y = b 83 (ii) Find also the coordinates of any points of intersection of C2 and C3 . [3] .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (c) Sketch C2 and C3 on a single diagram, clearly identifying each curve. Hence find the set of values of x for which b © UCLES 2020 x-a 2 l G x-a . x - 2a x - 2a [5] 9231/12/O/N/20 [Turn over Craftear .................................................................................................................................................... 84 Q7 1 Let a be a positive constant. ax . x+7 [2] Craftear (a) Sketch the curve with equation y = © UCLES 2020 9231/11/M/J/20 85 (b) Sketch the curve with equation y = ax ax a and find the set of values of x for which 2 . x+7 x+7 2 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 [Turn over Craftear [4] 86 Q88 3 The curve C has equation y = x2 . 2x + 1 (a) Find the equations of the asymptotes of C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the coordinates of the stationary points on C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 Craftear ............................................................................................................................................................ 87 (c) Sketch C. Craftear [3] © UCLES 2020 9231/11/M/J/20 [Turn over 88 Q9 6 The curve C has equation y = 10 + x - 2x 2 . 2x - 3 (a) Find the equations of the asymptotes of C. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Show that C has no turning points. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 [Turn over Craftear ............................................................................................................................................................ 89 [3] Craftear (c) Sketch C, stating the coordinates of the intersections with the axes. © UCLES 2020 9231/13/M/J/20 90 which 10 + x - 2x 2 1 4. 2x - 3 10 + x - 2x 2 and find in exact form the set of values of x for 2x - 3 [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 [Turn over Craftear (d) Sketch the curve with equation y = 91 Q10 4 The line y = 2x + 1 is an asymptote of the curve C with equation y= x2 + 1 . ax + b (i) Find the values of the constants a and b. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ [1] ........................................................................................................................................................ ........................................................................................................................................................ (iii) Sketch C. [Your sketch should indicate the coordinates of any points of intersection with the y-axis. You do not need to find the coordinates of any stationary points.] [3] © UCLES 2019 9231/11/O/N/19 [Turn over Craftear (ii) State the equation of the other asymptote of C. 92 Q11 10 The curves C1 and C2 have equations y= ax x+5 and y= respectively, where a is a constant and a > 2. x2 + a + 10x + 5a + 26 x+5 (i) Find the equations of the asymptotes of C1 . [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the equation of the oblique asymptote of C2 . [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Show that C1 and C2 do not intersect. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 Craftear ........................................................................................................................................................ 93 (iv) Find the coordinates of the stationary points of C2 . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (v) Sketch C1 and C2 on a single diagram. [You do not need to calculate the coordinates of any [3] points where C2 crosses the axes.] © UCLES 2019 9231/11/M/J/19 [Turn over Craftear ........................................................................................................................................................ 94 Q12 6 The curve C has equation y= where k is a positive constant. x2 , kx − 1 (i) Obtain the equations of the asymptotes of C. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the coordinates of the stationary points of C. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 Craftear ........................................................................................................................................................ 95 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Sketch C. © UCLES 2019 [3] 9231/13/M/J/19 [Turn over Craftear ........................................................................................................................................................ 96 Q13 6 The curve C has equation y= where b is a positive constant. x2 + b , x+b (i) Find the equations of the asymptotes of C. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Show that C does not intersect the x-axis. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 Craftear ........................................................................................................................................................ 97 (iii) Justifying your answer, find the number of stationary points on C. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Sketch C. Your sketch should indicate the coordinates of any points of intersection with the y-axis. You do not need to find the coordinates of any stationary points. [3] © UCLES 2018 9231/11/M/J/18 [Turn over Craftear ........................................................................................................................................................ 98 Q14 4 The curve C has equation y= x2 + 7x + 6 . x−2 (i) Find the coordinates of the points of intersection of C with the axes. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 Craftear (ii) Find the equation of each of the asymptotes of C. 99 (iii) Sketch C. Craftear [3] © UCLES 2018 9231/13/M/J/18 [Turn over 100 Q15 6 The curve C has equation where a is constant and a > 1. y= x2 + ax − 1 , x+1 (i) Find the equations of the asymptotes of C. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Show that C intersects the x-axis twice. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 Craftear ........................................................................................................................................................ 101 (iii) Justifying your answer, find the number of stationary points on C. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Sketch C, stating the coordinates of its point of intersection with the y-axis. © UCLES 2018 9231/11/O/N/18 [3] [Turn over Craftear ........................................................................................................................................................ 102 Q16 9 16 The curve C has equation y= 5x2 + 5x + 1 . x2 + x + 1 (i) Find the equation of the asymptote of C. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Show that, for all real values of x, − 13 ≤ y < 5. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 Craftear ........................................................................................................................................................ 103 17 (iii) Find the coordinates of any stationary points of C. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Sketch C, stating the coordinates of any intersections with the y-axis. © UCLES 2018 9231/12/O/N/18 [2] [Turn over Craftear ........................................................................................................................................................ 104 Q17 9 The curve C has equation y = x2 − 3x + 6 . 1−x (i) Find the equations of the asymptotes of C. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 Craftear (ii) Find the coordinates of the turning points of C. 105 (iii) Find the coordinates of any intersections with the coordinate axes. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Sketch C. Craftear [3] © UCLES 2017 9231/11/M/J/17 [Turn over 106 Q18 9 The curve C has equation y= 3x − 9 . x − 2 x + 1 (i) Find the equations of the asymptotes of C. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Show that there is no point on C for which 13 < y < 3. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 Craftear ........................................................................................................................................................ 107 (iii) Find the coordinates of the turning points of C. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Sketch C. © UCLES 2017 [3] 9231/11/O/N/17 [Turn over Craftear ........................................................................................................................................................ Q19 A curve C has equation y = x2 . Find the equations of the asymptotes of C. x−2 [3] Show that there are no points on C for which 0 < y < 8. [4] Sketch C, giving the coordinates of the turning points. [3] Craftear 7 108 © UCLES 2016 9231/11/M/J/16 [Turn over Q20 The curve C has equation y = dy x+2 . Show that < 0 at all points on C. 2 dx x −9 [3] State the equations of the asymptotes of C. [2] Sketch C, showing the coordinates of any points of intersection with the coordinate axes. [3] Craftear 5 109 © UCLES 2016 9231/13/M/J/16 [Turn over Q21 10 110 4x2 − 3x . Verify that the equation of C may be written in the form x2 + 1 1 3x − 12 9 x + 32 y=− + and also in the form y = − . [3] 2 2 x2 + 1 2 2 x2 + 1 The curve C has equation y = Hence show that − 12 ≤ y ≤ 92 . [2] Without differentiating, write down the coordinates of the turning points of C. [2] State the equation of the asymptote of C. [1] Craftear Sketch the graph of C, stating the coordinates of the intersections with the coordinate axes and the asymptote. [3] © UCLES 2015 9231/13/M/J/15 [Turn over Q22 8 The curve C has equation y = C has no stationary points. 111 2x2 + kx , where k is a constant. Find the set of values of k for which x+1 [5] Craftear For the case k = 4, find the equations of the asymptotes of C and sketch C, indicating the coordinates of the points where C intersects the coordinate axes. [6] © UCLES 2015 9231/11/O/N/15 [Turn over Q23 112 12 OR The curve C has equation ax2 + bx + c , x+d where a, b, c and d are constants. The curve cuts the y-axis at 0, −2 and has asymptotes x = 2 and y = x + 1. y= [1] (ii) Determine the values of a, b and c. [6] (iii) Show that, at all points on C, either y ≤ 3 − 26 or y ≥ 3 + 26. [7] Craftear (i) Write down the value of d. © UCLES 2014 9231/11/M/J/14 [Turn over Q24 113 11: EITHER Express 2x2 − x + 5 A B in the form 2 + + , where A and B are integers to be found. 2 x−1 x+1 x −1 The curve C has equation y = dy = 0. dx [3] 2x2 − x + 5 . Show that there are two distinct values of x for which x2 − 1 [4] Craftear Sketch C, stating the equations of the asymptotes and giving the coordinates of any points of intersection with the coordinate axes and with the asymptotes. You do not need to find the coordinates of the turning points. [7] © UCLES 2014 9231/13/M/J/14 [Turn over Q25 4 A curve C has equation y = 114 2x2 + x − 1 . Find the equations of the asymptotes of C. x−1 [3] Show that there is no point on C for which 1 < y < 9. Craftear [4] © UCLES 2014 9231/11/O/N/14 [Turn over Q26 10 The curve C has equation y = 115 2x2 − 3x − 2 . State the equations of the asymptotes of C. x2 − 2x + 1 [2] at all points of C. Show that y ≤ 25 12 [4] Find the coordinates of any stationary points of C. [3] Craftear Sketch C, stating the coordinates of any intersections of C with the coordinate axes and the asymptotes. [4] © UCLES 2013 9231/11/M/J/13 [Turn over Q27 10 116 The curve C has equation y= where p is a positive constant and p ≠ 3. px2 + 4x + 1 , x+1 (i) Obtain the equations of the asymptotes of C. [3] (ii) Find the value of p for which the x-axis is a tangent to C, and sketch C in this case. [4] Craftear (iii) For the case p = 1, show that C has no turning points, and sketch C, giving the exact coordinates of the points of intersection of C with the x-axis. [5] © UCLES 2013 9231/11/O/N/13 [Turn over Q28 7 117 The curve C has equation y= 2x2 + 5x − 1 . x+2 Find the equations of the asymptotes of C. Show that [3] dy > 2 at all points on C. dx [3] Sketch C. 8 [3] The points A, B, C have position vectors 4i + 5j + 6k, 5i + 7j + 8k, 2i + 6j + 4k, [4] Craftear respectively, relative to the origin O. Find a cartesian equation of the plane ABC. © UCLES 2013 9231/13/O/N/13 [Turn over Q29 9 118 The curve C has equation y= Show that, for all x, 1 ≤ y ≤ 52 . 2x2 + 2x + 3 . x2 + 2 [4] Find the coordinates of the turning points on C. [3] Find the equation of the asymptote of C. [2] Craftear Sketch the graph of C, stating the coordinates of any intersections with the y-axis and the asymptote. [2] © UCLES 2012 9231/11/M/J/12 [Turn over Q30 7 119 The curve C has equation x , x−2 where λ is a non-zero constant. Find the equations of the asymptotes of C. y = λx + [3] [3] Sketch C in the case λ = −1, stating the coordinates of the intersections with the axes. [3] Craftear Show that C has no turning points if λ < 0. © UCLES 2012 9231/12/O/N/12 [Turn over Q31 The curve C has equation y = x2 − 3x + 3 . Find the equations of the asymptotes of C . x−2 [3] Show that there are no points on C for which −1 < y < 3. [4] Find the coordinates of the turning points of C. [3] Sketch C. [2] Craftear 9 120 © UCLES 2012 9231/13/O/N/12 [Turn over Q32 The curve C has equation y = x2 . Find the equations of the asymptotes of C. x−2 [3] Find the coordinates of the turning points on C. [3] Draw a sketch of C. [3] Craftear 6 121 © UCLES 2012 9231/13/M/J/12 [Turn over RATIONAL FUNCTIONS MS Craftear 122 Q1 7(c) 7(b) 7(a) Question x( x + 1) + 9 9 =x+ x +1 x +1 9 x Craftear 3 B1 Lower branch correct and good approach to asymptotes throughout, no extra branches. B1 Upper branch with (0, 9) stated or shown on diagram. B1 Axes labelled and correct asymptotes drawn. 4 A1 ( −4, − 7 ) M1 A1 Differentiates and sets derivative equal to 0. 3 A1 M1 Finds oblique asymptote. A1 y Guidance B1 States vertical asymptote. Marks ( 2, 5) dy = 1 − 9( x +1)−2 = 0  ( x + 1)2 = 9 dx y=x y= x = −1 Answer 123 7(d) Question Craftear 5 A1 + 152 x + 312 = 0 x < 12 and x > 5. or x 2 solutions since 7 > 132 . M1 M1 Finds critical points, award M1 for each case. May state that x 2 + x + 9 = − 132 (x + 1) has no real A1 − 112 x + 52 = 0 13 13 (x +1) or x 2 + x + 9 = − (x + 1) 2 2 x Guidance B1 FT FT from sketch in (c) with asymptotes shown. Marks x = 12 , 5 x 2 x2 + x + 9 = 9 y Answer 124 Q2 7(c) 7(b) 7(a) Question ) 1 2 ( + 1 2 13,0 , ) ( − 1 2 1 2 13,0 , (0, −3) ) x Craftear dy equal to 0 and forms equation. dx 3 B1 States exact coordinates of intersections with axes. B1 Correct shape and position, with all asymptotic behaviour clear. B1 Both axes labelled and correct asymptotes shown. 3 A1 WWW dy . dx Sets their Finds ( 12 , − 135 ) M1 2 B1 Horizontal asymptote. M1 y Guidance B1 Vertical asymptotes. Must be exact. Marks (2 x − 1)(−2) = 0 dy (1 + x − x 2 )(2 x −1) − ( x 2 − x − 3)(1 − 2 x) = 2 dx 1 + x − x2 ( 1− 5 1+ 5 , x= 2 2 y = −1 x= Answer 125 Q3 7(b) 7(a) Question 7(d) ( ) ( −2, 14 ) , ( − 12 ,1) 16 x 2 + 40 x + 16 = 0 dy (4 − 4 x 2 )(4) − (4x + 5)(−8 x) = 2 dx 4 − 4 x2 Craftear dy . dx Guidance 4 A1 A1 M1 Sets equal to 0 and forms equation. M1 Finds B1 Horizontal asymptote. y=0 2 B1 Vertical asymptotes. Marks x = 1, x = −1 Answer 6 A1 FT Must be three distinct regions and strict inequalities. A1 Must be exact. x = 12 + 12 7, x = 12 − 12 7 , x = 0, x = 1 x < 12 − 12 7, 0 < x < 1, x > 12 + 12 7 M2 Finds critical points, award M1 for each case. x2 − x − 3 x2 − x − 3 or = 3 = −3 1 + x − x2 1 + x − x2 4 x 2 − 4 x − 6 = 0 or −2 x 2 + 2 x = 0 B1 Correct shape as x tends to infinity and intersections with x axis. B1 FT FT from sketch in (c). 126 7(d) 4x൅5 4-4x ଶ x B1 Correct shape at infinity. B1 FT FT from sketch in part (c). 3 B1 States coordinates of intersections with axes. 2 − 53 5 6 < x < − 54 , 0 < x < 52 + 53 6 , x ≠ ±1 Craftear 6 A1 FT Condone exclusion of x = ±1 from the range. A1 A0 if –1.07, 1.87 4x൅5 ฬ yൌ ฬ 4-4x ଶ yൌ x B1 Correct shape and position. B1 Axes and asymptotes. x = − 54 , x = 0 or x = 52 − 53 6, x = 52 + 53 6 xൌ-1 M2 Finds critical points, award M1 for each case. ( 0, 54 ) xൌ-1 4x + 5 5 4x + 5 5 = or =− 2 2 4 4 4 − 4x 4 − 4x 2 2 5 x + 4 x = 0 or 5 x − 4 x − 1 0 = 0 (− 54 ,0), xൌ1 xൌ1 7(c) 127 6(c) 6(b) 6(a) Question 9 leading to y = x + 3 x −3 Answer 𝑦 0 < y < 12 𝑥 𝑥 3 y 2 − 4(3 y) < 0 leading to y 2 − 12 y < 0 yx − 3y = x2 leading to x2 − yx + 3 y = 0 y = x +3+ x=3 3 𝑥 Q4 Craftear Guidance 2 B1 Branches correct. B1 Axes and asymptotes. 4 A1 AG M1 Uses that discriminant is negative. M1 A1 Forms quadratic in x. 3 M1 A1 Finds oblique asymptote. B1 States vertical asymptote. Marks 128 Q5 6(b) 6(a) Question 6(d)(ii) Answer B1 States vertical asymptote. Guidance M1 A1 Forms quadratic in x. 3 M1 A1 Finds oblique asymptote. Marks 1 B1 4 4 A1 AG Craftear x B1 Correct intercepts with axes (may be seen on graph). B1 Correct shape of y = x − 3. B1 Correct shape at infinity. 1< y < 5 3 B1 FT FT from sketch in (c). M1 Uses that discriminant is negative if there are no values of x. 𝑥 𝑥 (1 − y )2 − 4( y −1) < 0  y 2 − 6 y + 5 < 0 yx − y = x 2 + x − 1 x 2 + (1− y)x + y −1 = 0 x 2 + x −1 = ( x −1)(x + 2) +1 y = x + 2 x =1 c ⩽ –3 𝑦 3 𝑥 6(d)(i) 129 6(d) 6(c) (0,1), 1 2 1 2 1 2 1 2 ( − + √ 5,0 ) , ( − − √ 5,0 ) y x Craftear x 2 B1 Correct shape at extremities. B1 FT FT from sketch in (c) with both branches. 3 B1 States coordinates of intersections with axes, can be labelled on graph. Accept ( 0.618,0 ) and ( −1.62,0 ) . B1 Branches correct. B1 Axes and asymptotes. 130 Q6 6(b)(ii) 6(b)(i) 6(a) Question Craftear A1 Since x ≠ 2a, stationary point is ( a,0 ) . A1 (a, 0) and ( 32 a , 1) 3 A1 Both x values or one correct point. x = a and 32 a or (a, 0) or ( 32 a , 1) dy = 0. dx M1 Finds a critical point. Sets ( x − a )2 = ( x − a)( x − 2a) or ( x − a )2 = −( x − a)( x − 2a) 3 M1 2 Solves for x. x M1 2a Guidance B1 Correct position and shape. Not too truncated. B1 Asymptotes labelled. Marks 2a ( x − a ) dy =− =0 3 dx ( x − 2a ) 1 y Answer 131 Q7 1(a) C2 1 y C3 C2 2a C3 C3 C2 x Answer Craftear (B1 for axes and asymptotes correct, B1 for branches correct) x ⩽ 32 a Question 6(c) 5 B1 Accept algebraic method. B1 Relative positions and intersections correct. Marks 2 B1 B1 C3 FT from their sketch in part (a). Clearly identified. B1 Correct shape of C2 . Clearly identified. B1 FT B1 Axes and asymptotes. 132 Q8 3(b) 3(a) Question 1(b) 7 3 Craftear A1 (0,0),(−1, −1) 3 A1 3 A1 M1 B1 x = 0, −1 1 4( 2 x + 1) Marks M1 = 12 x − 14 + Answer 4 A1 A1 M1 dy 1 1 = − = 0  (2 x + 1) 2 = 1 dx 2 2 ( 2x +1)2 2x + 1 ( 2 x + 1) ( 12 x − 14 ) + 14 y = 12 x − 14 y= x = − 12 7 x < −7 , −7 < x < − , x > 7 3 x = 7 and x = − ax ax a a = or =− x+7 2 x+7 2 (B1 FT from sketch in part (a)) B1 FT 133 Q9 6(b) 6(a) Question 3(c) )  dy  14  or  = −1 − 2  dx ( 2 x − 3)   2 Craftear 122 − 4(4)(23) = −224 < 0 (or y ' < 0 )  No turning points. 4 x 2 − 12 x + 23 = 0 dy ( 2 x − 3)(1 − 4 x ) − 2 10 + x − 2x = dx ( 2 x − 3) 2 ( −2 x 2 + x + 10 = ( 2 x − 3 ) ( − x − 1) + 7  y = − x − 1 x = 32 Answer (B1 for axes and asymptotes correct, B1 for upper branch correct, B1 for lower branch correct) 3 A1 A1 M1 3 M1 A1 B1 Marks 3 B1 B1 B1 134 6(c) (0, − 103 ), ( −2,0 ) ,( 52 ,0) Craftear (B1 for axes and asymptotes correct, B1 for branches correct) 3 B1 B1 B1 135 4(ii) 4(i) Q10 Question 6(d) 1 2 ( Answer − 112 < x < 14 9 − √ 65 and 2 < x < 14 9 + √ 65 ) ) Marks Guidance Craftear 1 B1 FT 3 A1 A1 M1 Uses that 2 x + 1 is the quotient. 6 A1 FT A1 x = − 112 , 2 or x = 14 ( 9 ± √ 65 ) ( M2 B1 B1 FT Marks 10 + x − 2 x 2 10 + x − 2 x 2 = −4 = 4 or 2x − 3 2x − 3 2 x 2 + 7 x − 22 = 0 or 2 x 2 − 9 x + 2 = 0 1 1 ⇒a= , b=− 2 4 x= Answer (B1 FT for their sketch in (c), B1 for correct shape at infinity) x 2 + 1 = ( ax + b )( 2x + 1) + c Question 136 Q11 M1 By inspection or long division. A1 oblique asymptote is y = x + a + 5 Guidance x 2 + ( a + 10 ) x + 5a + 26 = ( x + 5 )( x + a + 5 ) + 1 2 2 B1 B1 Marks 3 Deduct at most one mark for poor forms at infinity. B1 FT Right branch correct. B1 Left branch correct. 10(ii) Craftear Guidance B1 Intersection (0,-4) given and asymptotes drawn. Marks x = −5 and y = a Answer Answer 10(i) Question 4(iii) Question 137 10(v) 10(iv) 10(iii) Craftear A1 Correct discriminant and conclusion. 102 − 4 ( 5a + 26 ) = −4 − 20a < 0 so no intersection point A1 A1 Must have both points. x 2 + 10 x + 24 = 0 Stationary points are ( −4,a + 2 ) and ( −6,a − 2 ) C2 correct. B1 3 C1 correct. B1 B1 Asymptotes drawn, intersection correct. 3 M1 Differentiates and forms quadratic equation. ( x + 5 )( 2 x + a + 10 ) − x 2 − ax − 10 x − 5a − 26 = 0 2 M1 Puts y –values equal and forms quadratic equation. x 2 + 10 x + 5a + 26 = 0 138 Q12 6(iii) 6(ii) 6(i) Question kx − 1 Craftear 3 B1 Lower branch correct. Deduct at most 1 mark for poor forms at infinity. B1 Upper branch correct. B1 Axes and asymptotes correct. 3 A1 Finds y -coordinates M1 Differentiates and equates to 0. ( 0,0 ) , ( 2k −1 , 4k −2 ) 2 = 0 ⇒ kx 2 − 2 x = 0 A1 Finds x -coordinates. ( kx − 1) 2 x ( kx − 1) − kx 2 x = 0, 2k −1 y′ = 3 A1 M1 Finds oblique asymptote. ( kx − 1) ( k −1 x + k −2 ) + k −2 Guidance B1 States vertical asymptote. Marks 1 k Oblique asymptote is y = k −1 x + k −2 y= x= Answer 139 Q13 6(iii) 6(ii) 6(i) Question 2 ) ( x + b )2 b2 + b =0. Craftear dy b2 + b = 0 gives and setting dx x+b b2 + b > 0 Therefore there are two stationary points on C 1− Or differentiating y = x − b + dy 2 x ( x + b ) − x + b = = 0 ⇒ x 2 + 2bx − b = 0 2 dx ( x + b) ( If y = 0 then x 2 + b = 0 which has no real root. Thus the oblique asymptote is y = x − b x−b x + b x2 + 0x + b Find dy and set = 0 dx 2 A1 Use discriminant or (x + b)2 to show two stationary points M1 1 B1 Must refer to b > 0 OE 3 A1 M1 By inspection or long division. x 2 + b = ( x + b ) ( x − b ) + b 2 + b or Guidance B1 Marks Vertical asymptote is x = −b . Answer 140 Q14 4(iii) 4(ii) 4(i) Question 6(iv) 24 ⇒ other asymptote is y = x + 9 . x−2 Craftear B1 Lower branch correctly located and orientated. Penalise at most 1 mark for poor forms at infinity 8 B1 Upper branch correctly located and orientated. B1 Sketches axes and asymptotes, labelled or to scale M1 A1 By inspection or long division. A0 if error in division B1 One asymptote is x = 2 . y = x+9+ B1 States y -intercept ( 0, −3) Guidance B1 States points of intersection with x -axis. Marks ( −6,0 ) , ( −1,0 ) Answer 3 B1 B1 Each branch correct Penalise at most one mark for poor forms at infinitty 1) g ven and asym mptotes drawn B1 Intersection (0,1 141 Q15 Craftear A1 Therefore there are no stationary points on C . 6(iv) M1 ( x + 1)( 2 x + a ) − ( x 2 + ax − 1) = 0 ⇒ x2 + 2 x + a + 1 = 0 2 ( x + 1) Discriminant = 4 − 4 ( a + 1) < 0 6(iii) 3 B1 B1 Each branch correect 2 pt and asymptotess drawn. B1 Correct y-intercep 1 B1 3 A1 a2 + 4 > 0 Thus the oblique asymptote is y = x + a −1 2 x + a −1 or x +1 x + ax − 1 M1 By inspection or long division. x 2 + ax − 1 = ( x + 1)( x + a − 1) − a Guidance B1 Marks Vertical asymptote is x = −1 . Answer 6(ii) 6(i) Question 142 Q16 9(iii) 9(ii) 9(i) Question 4 x + x +1 2 2 A1 ) 1 1 ⇒ 4 ( 2 x + 1) = 0 ⇒ x = − , y = − 2 3 ( M1 Differentiates and equates to 0. ) y ′ = 0 ⇒ x 2 + x + 1 (10 x + 5 ) − 5 x 2 + 5 x + 1 ( 2 x + 1) = 0 ( A1 Explaining strict upper inequality (AG) 1 ⇒ − - y < 5, because y = 5 is an asymptote (www) 3 4 M1 Factorising 2 ⇒ ( y − 5)( 3 y + 1) - 0 B1 Forms quadratic equation in x . 2 A1 M1 Uses discriminant Craftear Guidance M1 Alt method: Finding limit Marks For real x , ( y − 5) − 4 ( y − 5) ( y − 1) . 0 (condone >) ⇒ ( y − 5) x2 + ( y − 5) x + ( y − 1) = 0 yx2 + yx + y = 5x2 + 5x + 1 As x → ±∞ , y → 5 ∴ y = 5 CAO y =5− Answer 143 Q17 9(iii) 9(ii) 9(i) Question 9(iv) Total: Craftear 2 B1 y = 0 ⇒ x 2 − 3x + 6 ; ∆ = 9 − 24 ⇒ No intersection with x-axis Total: B1 (0 , 6) 3 A1 M1 Turning points are ( −1,5) and ( 3,−3) =0 3 A1 −2 Total: M1A1 B1 Marks ⇒ x = −1,3 y ' = −1+ 4 (1 − x ) y =2− x x =1 Answer 2 Guidance orrect asymptote and completely corrrect B1 Co graaph. B1FT Possitive y-intercept at (0,1), FT dep on min nimum point from (iii). 144 Q18 9(ii) 9(i) Question 9(iv) = 0 is 1 < y<3 3 Craftear 4 A1 AG A1 ⇒ 9 y 2 − 30 y + 9 < 0 ⇒ 3 y 2 − 10 y + 3 < 0 ⇒ ( 3 y −1)( y − 3) < 0 ⇒ M1 No points on C if ( y + 3) − 4 y ( 9 − 2 y ) < 0 2 M1 2 B1 B1 Marks yx 2 − ( y + 3) x + 9 − 2 y = 0 ( x + 1)( x − 2 ) = 0 ⇒ x = −1 and ⇒ x = 2 are vertical asymptotes. Degree of numerator < degree of denominator ⇒ horizontal asymptote. Answer Total: 3 Guidance B1B1 Each brancch. B1 1 FT Asymptotees correct. 145 9(iv) 9(iii) Craftear RH branch ; Other two branchhes Axes, asympptotes and points on axes (0, 4.5) (3,0). 3 B1 1B1 B1 3 B1  1 ⇒ Turning points are (1,3) and  5,  .  3 B1 B1 ) ⇒ … ⇒ ( x − 1) ( x − 5) = 0 ( dy = 0 ⇒ 3 x 2 − x − 2 − ( 3 x − 9 )( 2 x −1) = 0 dx 146 147 Q19 7 B1 Vertical asymptote is x = 2. 4 y = x+2+ ⇒ Oblique asymptote is y = x + 2 . x−2 M1A1 [3] x2 ⇒ x 2 − yx + 2 y = 0 x−2 Quadratic has no real roots (i.e. no points on C) if ∆ < 0 ⇒ y 2 − 8 y < 0 y= B1 M1 ⇒ y ( y − 8) < 0 ⇒ 0 < y < 8 . (AG) Correct inequality M1A1 [4] B1 B1B1 [3] Axes and asymptotes. Each branch, showing (0,0) and (4,8). (Deduct at most 1 mark for poor forms at infinity and/or missing coordinates.) Q20 ( ) x2 − 9 − 2x ( x + 2) dy = 2 dx x2 − 9 = ( − x2 − 4 x − 9 ( x − 9) 2 2 B1 ) 2 = − ( x + 2) − 5 ( x − 9) 2 2 ⇒ dy <0 dx M1A1 [3] Asymptotes: x = ±3 ; y = 0 B1B1 [2] Sketch: Axes and asymptotes ; Outside branches 2  Middle branch, showing  0, −  and ( −2,0 ) . 9  (Deduct at most 1 mark for poor forms at infinity.) B1B1 B1 [3] 1 (3 x − 1) 2 − x 2 − 1+ 9 x 2 − 6x +1 8 x 2 − 6 x 4 x 2 − 3 x + = = = 2 2 2( x 2 + 1) x +1 2( x 2 + 1) 2( x 2 + 1) M1A1 Craftear 5 Q21 10 − 9 ( x + 3) 2 9 x 2 + 9 − x 2 − 6 x − 9 8 x 2 − 6 x 4 x 2 −3x − = = = 2 2 2( x 2 + 1) 2( x 2 + 1) 2( 1) x +1 A1 (3) (3 x − 1) 2 1 >0⇒ y>− 2 2 2( x + 1) B1 (x + 3) 2 9 1 9 >0⇒ y< ⇒ − < y< (Not hence gets B1 only.) 2 2 2 2 2( x + 1)   Turning points occur at equality, so  −3, B1 (2) 9 1 1  and  , −  2 2 3 B1B1 (2) Asymptote is y = 4 . B1 (1) 3 4    4  3   Intersection with axes at (0, 0) and  , 0  , Intersection with asymptote at  − , 4  B1B1 Correct graph. B1 (3) Total 11 Q22 ( ) y′ = 0 ⇒ (x + 1)(4 x + k) − 2x 2 + kx ×1 = 0 2 2 2 M1 ⇒ 4 x + (4 + k ) x + k − 2 x − kx = 0 ⇒ 2 x + 4 x + k = 0 A1 B 2 − 4 AC < 0 ⇒ no stationary points ⇒ 16 − 8k < 0 ⇒ k > 2 for no stationary points. M1A1 A1 [5] When k = 4: Vertical asymptote: x = –1 Oblique asymptote: y = 2 x + 2 − 2 ⇒ y = 2x + 2 x +1 Axes and asymptotes Each branch. B1 M1A1 B1 B1B1 [6] Total 11 Q23 12O (i) d = –2 c ⇒c=4 d 4 + 4a + 2b y = ax + (b + 2a) + x−2 Oblique asymptote y = x + 1 ⇒ a = 1, b + 2a = 1 ⇒ b = –1 (ii) x = 0 ⇒ y = –2 = (iii) x2 − x + 4 ⇒ x 2 − ( y +1) x + 2 y + 4 = 0 x−2 For real x B2 – 4AC > 0 ⇒ (y + 1)2 – 4(2y + 4) > 0 ⇒ y2 – y – 15 > 0 y= B1 [1] M1A1 M1A1 A1A1 [6] M1 M1 A1 Craftear 8 148 ⇒ (y – 3)2 > 24 (or for solving y2 – 6y – 15 = 0) ⇒ y – 3 < – 2 6 or y – 3 > – 2 6 (or for thumbnail sketch) M1A1 M1 ⇒ y < 3 – 2 6 or y > 3 + 2 6 (AG) (Note: if done by differentiation, max and min must be demonstrated for full marks.) A1 [7] EITH A B 2( x 2 − 1) + A( x +1) + B( x −1) 2 x 2 + ( A + B ) x + A − B − 2 2 x 2 − x + 5 2+ + = = = x − 1 x +1 x2 −1 x2 −1 x2 −1 ⇒ A = 3 , B = –4 M1 Q24 11 1 y′ = 0 ⇒ (x2 – 1)(4x – 1) – (2x2 – x + 5).2x = 0 ⇒ x2 – 14x + 1 = 0 B2 – 4AC = (–14)2 – 4 × 1 × 1 = 192 > 0 ⇒ two distinct turning points. Asymptotes are x = 1 , x = –1 : y = 2 y = 2 ⇒ 2x2 – x + 5 = 2x2 – 2 ⇒ x = 7 ⇒ (7, 2) (Accept if labelled on graph.) Sketch: Middle branch crossing y-axis at (0, –5) and left branch. Right branch. Working to show no intersections with x-axis. A1A1 [3] M1A1 M1A1 [4] B1B1 M1A1 B1 B1 B1 [7] Q25 4 149 Vertical asymptote is x = 1 −1 y = 2 x + 3 + 2( x − 1) ⇒ y = 2 x + 3 is the oblique asymptote. B1 M1A1 (3) 2 x 2 + (1 − y )x + ( y −1) = 0 has real roots ⇔ (1 − y ) − 8( y − 1) > 0 2 ⇔ y 2 − 10 y + 9 > 0 ⇔ ( y −1)( y − 9 ) > 0 Hence ( y − 1)( y − 9 ) < 0 ⇒ no real roots. i.e. 1 < y < 9 ⇒ no points on C. (AG) Thumbnail sketch, or similar, required. M1A1 M1 A1 (4) [7] Q26 States asymptotes. Vertical: x = 1 and Horizontal: y = 2 B1B1 Obtains quadratic form in x. yx 2 − 2 yx + y = 2x 2 − 3x − 2 M1A1 2 ⇒ ( y − 2) x 2 − (2 y − 3) x + ( y + 2) = 0 Uses B 2 − 4 AC ≥ 0 for real roots. Finds condition for y ′ = 0 . For real x (2 y − 3) 2 − 4( y − 2)( y + 2) ≥ 0 25 . ⇒ 12 y ≤ 25 ⇒ y ≤ 12 M1 y′ = 0 ⇒ M1 A1 4 ( x 2 − 2 x + 1)(4 x − 3) − (2 x 2 − 3x − 2)(2 x − 2) = 0 Solves ⇒ x 2 − 8 x + 7 = 0 ⇒ ( x − 7)( x − 1) = 0 ⇒ x = 7 , (since x = 1 is vertical asymptote). A1 Obtains stationary point.  25  Stationary point is  7 ,   12  A1 Sketch showing: Axes and asymptotes (-0.5,0) , (2,0) , (0,–2) and (4,2) Left hand branch. Right hand branch. B1 B1 B1 B1 Craftear 10 4 [13] Q27 10 (i) (ii) (iii) Vertical asymptote. Oblique asymptote. Asymptotes : x = –1 −1 y = px + 4 − p + ( p − 3)( x +1) ⇒ y = px + 4 − p B1 M1A1 3 Obtains value of p. Sketches curve. p = 4 ⇒ x-axis is a tangent Correct location of turning points and asymptotes. Each branch. B1 B1 B1B1 4 Proves required result. Sketches graph. p = 1 ⇒ y = x + 3 − 2( x + 1) ⇒ y′ = 1 + 2( x + 1) −1 ( Intersections on x-axis at − 2 ± 3 , 0 Each branch. −2 ) ([ 1) M1A1 B1 B1B1 5 [12] 150 Q28 7 States vertical asymptote Vertical asymptote is x = −2 B1 Expresses y in a suitable form y = 2x + 1 − 3(x + 2) M1 States oblique asymptote Oblique asymptote is y = 2x + 1 A1 Differentiates wrt x y ′ = 2 + 3(x + 2) M1A1 and explains result (x + 2)−2 > 0 ⇒ y ′ > 2 (AG) A1 Sketches graph. Axes and asymptotes B1 (Deduct 1 mark for poor Each branch, showing intersection with axes. forms at infinity) B1B1 −1 −2 (3) (3) (3) [9] Q29 Possible approach for first two parts together. 2x2 + 2x + 3 (x + 1) 2 = + 1 x2 + 2 x2 + 2 (x +1) 2 States 2 ≥ 0 ⇒ y ≥1 x +2 From this it is clear that (–1, 1) is a turning point. Writes y = Writes y = States 2 x 2 + 2 x + 3 5 (x − 2) 2 = − 2 2( x 2 + 2) x2 + 2 (B1) (M1A1) (B1) 5 (x − 2) 2 ≥0⇒ y ≤ 2 2 2( x + 2) From this it is clear that (2, 2 (i) can come after finding turning points: Continuous function (implied by graph) ⇒ (2,2.5) Max and (–1,1) Min 5 (AG) ⇒1≤ y ≤ 2 N.B. Award B1 if Max and Min assumed without proof. i.e. 1/4 . (B1) Craftear 9(i) and (ii) 1 ) is the other turning point. 2 (B1) (A1) (7) (M1) (M1A1) (A1) (4) 151 OR Forms quadratic equation in x. yx 2 + 2 y = 2x 2 + 2x + 3 ⇒ ( y − 2) x 2 − 2 x + (2 y − 3) = 0 M1 A1 Uses discriminant to obtain condition for real roots. For real x 4 – 4(y – 2)(2y – 3) ≥ 0 M1 5 (AG) 2 A1 Differentiates and equates to zero. Solves equation. ⇒ (2 y − 5)( y −1) ≤ 0 ⇒1≤ y ≤ y′ = 0 ⇒ (x 2 + 2)(4x + 2) − 2x(2x 2 + 2x + 3) = 0 Expresses y in an appropriate form. (May alternatively divide numerator and denominator by x2. Finds y-intercept and intersection with y = 2. Completes graph. M1 ⇒ (x –2)(x + 1) = 0 ⇒ x = –1 or x = 2 (Or substitutes y =1 and States coordinates of turning points. 4 5 in equation of C.) 2 1  Turning points are (–1, 1) and  2, 2  2  y = 2+ 2x − 1 x2 + 2 A1A1 3 M1 As x → ±∞ y → 2 ∴ y = 2 A1 1  1  Shows  0, 1  and  , 2  2  2  B1 Completely correct graph. B1 2 2 Craftear 9 [11] Q30 7 Finds asymptotes to C. Differentiates and equates to 0. Sketch of graph. Deduct 1 mark for poor forms at infinity. Deduct 1 mark if intersections with axes not shown. Vertical asymptote x = 2. 2 y = λx + 1 + ⇒ y = λx + 1 is oblique asymptote. x−2 y ′ = λ − 2( x − 2) −2 = 0 for turning points. M1A1 2 > 0 ⇒ no turning points if λ < 0 . (x − 2) 2 A1 λ= B1 3 M1A1 Or y ′ = 0 ⇒ λx 2 − 4λx + 4λ − 2 = 0 Uses discriminant to show 8λ < 0 ⇒ no T.P.s. (M1A1) (A1) Axes and asymptotes. LH branch. RH branch. (Indicating intersections with axes at (0,0) and (3,0).) B1 B1B1 3 3 [9] 152 9 States vertical asymptote. Vertical asymptote is x = 2 . ⇒ oblique asymptote is y = x − 1 . Divides and states oblique asymptote. Rearranges as quadratic in x. xy − 2 y = x 2 − 3x + 3 ⇒ x 2 − ( y + 3) x + (3 + 2y) = 0 Uses discriminant to obtain condition stated. For real x, B 2 − 4AC ≥ 0 ∴( y + 3) 2 − 4(3 + 2 y) ≥ 0 ⇒ ... ⇒ ( y − 3)( y + 1)) ≥ 0 ⇒ y ≤ −1 or y ≥ 3 ∴ no points for −1 < y < 3 (AG) B1 M1 A1 3 B1 M1 A1 A1 4 Differentiates, puts = 0 and obtains x-values. y′ =1 − (x − 2) −2 = 0 ⇒ x = 1 or 3 (Or uses y = –1 and 3 to obtain x-values.) M1 States stationary points. Stationary points are (1,–1) and (3,3) A1A1 3 Sketch. One mark for each branch correctly placed. B1B1 2 Vertical asymptote is x = 2. B1 y = x+2+ M1 A1 Q32 6 States vertical asymptote. Finds oblique asymptote. Oblique asymptote is y = x + 2. y′ = 1 − Differentiates and equates to zero. Finds x coordinates. 4 x−2 4 = 0 ⇒ (x − 2) 2 = 4 2 (x − 2) 3 [12] Craftear Q31 M1 x = 0, 4. A1 Turning points are (0,0) and (4,8) A1 3 Axes and both asymptotes correct. Upper branch correct. Lower branch correct. B1 B1 B1 3 States coordinates of turning points. Deduct at most 1 mark for poor forms at infinity. [9] 153 Craftear SUMMATION OF SERIES 154 Q1 2 n / `1 - r - r 2j in terms of n, (a) Use standard results from the List of formulae (MF19) to find simplifying your answer. [3] r =1 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 Craftear ............................................................................................................................................................ 155 (b) Show that r+1 r 1 - r - r2 = 2 2 2 2 `r + 2r + 2j`r + 1j `r + 1j + 1 r + 1 and hence use the method of differences to find n / `r + 12r-+r2-j`rr + 1j . r =1 2 2 2 [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 3 (c) Deduce the value of / 1 - r - r2 . 2 2 ` j ` j + + + r 2 r 2 r 1 r =1 [1] ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 [Turn over Craftear ............................................................................................................................................................ 156 Q2 1 (a) Show that tan (r + 1) - tan r = sin 1 . cos (r + 1) cos r [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ Let ur = 1 . cos (r + 1) cos r (b) Use the method of differences to find n /u . r =1 r [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 Craftear ............................................................................................................................................................ 157 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Explain why the infinite series u1 + u2 + u3 + f does not converge. [1] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 [Turn over Craftear ............................................................................................................................................................ 158 .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... 2 Q3 (a) Use standard results from the list of formulae (MF19) to find n / r (r + 1) (r + 2) in terms of n, r =1 fully factorising your answer. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 [Turn over Craftear ............................................................................................................................................................ 159 (b) Express 1 in partial fractions and hence use the method of differences to find r (r + 1) (r + 2) n / r (r + 11) (r + 2) . [5] r =1 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ / r (r + 11) (r + 2) . 3 (c) Deduce the value of [1] r =1 ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 Craftear ............................................................................................................................................................ 160 Q4 n 3 Let Sn = / ln r(r(r++12) ) . r =1 2 (a) Using the method of differences, or otherwise, show that S n = ln n+2 . 2 (n + 1) [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ 161 / ln r(r(r++12) ) . 3 Let S = r =1 2 (b) Find the least value of n such that Sn - S 1 0.01. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ 162 (c) Use standard results from the list of formulae (MF19) to show that n /`(a + r) + (b + r) + (c + r) j = n + n (n + 1)àn + bn + cj, 3 3 3 r =1 where a, b and c are constants to be determined. 1 4 2 [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 [Turn over Craftear ............................................................................................................................................................ 163 Q5 2 4 (a) Use standard results from the List of Formulae (MF19) to show that n / (7r + 1) (7r + 8) = an 3 + bn 2 + cn , r =1 where a, b and c are constants to be determined. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 Craftear ............................................................................................................................................................ 164 n (b) Use the method of differences to find / (7r + 1)1(7r + 8) in terms of n. [4] r=1 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ / (7r + 1)1(7r + 8) . 3 (c) Deduce the value of [1] r=1 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 [Turn over Craftear ............................................................................................................................................................ 165 Q6 3 (a) By simplifying `x n - x 2n + 1j`x n + x 2n + 1j , show that 1 = - x n - x 2n + 1 . 2n x - x +1 n [1] ............................................................................................................................................................ ............................................................................................................................................................ Let un = x n+1 + x 2n+2 + 1 + 1 . x - x 2n + 1 n N (b) Use the method of differences to find / un in terms of N and x. n =1 [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Deduce the set of values of x for which the infinite series u1 + u2 + u3 + f is convergent and give the sum to infinity when this exists. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 [Turn over Craftear ............................................................................................................................................................ 166 Q7 4 8 (a) By first expressing 1 in partial fractions, show that r -1 2 n / r 1- 1 = 34 - 2nan(n++b1) , r =2 2 where a and b are integers to be found. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 Craftear ............................................................................................................................................................ 167 3 (b) Deduce the value of / r 1- 1 . r =2 [1] 2 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 2n / r n- 1 . r = n +1 2 [4] ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 [Turn over Craftear (c) Find the limit, as n " 3, of 168 Q8 3 Let Sn = 2 2 + 6 2 + 10 2 + f + `4n - 2j . 2 (a) Use standard results from the List of Formulae (MF19) to show that S n = 43 n (4n 2 - 1) . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 Craftear ............................................................................................................................................................ 169 n (b) Express in partial fractions and find Sn N / Sn in terms of N. n =1 n [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 3 (c) Deduce the value of / Sn . n =1 [1] n ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 [Turn over Craftear ............................................................................................................................................................ 170 Q9 Let SN = Ð 5r + 1 5r + 6 and TN = N 5 r=1 Ð 5r + 1 5r + 6 . N 1 r=1 (i) Use standard results from the List of Formulae (MF10) to show that SN = 13 N 25N 2 + 90N + 83. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Use the method of differences to express TN in terms of N . [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 Craftear ........................................................................................................................................................ 171 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find lim N −3 SN TN . [2] N →∞ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 [Turn over Craftear ........................................................................................................................................................ 172 Q10 2 Let un = 4 sin n − 12 sin 12 cos 2n − 1 + cos 1 . (i) Using the formulae for cos P ± cos Q given in the List of Formulae MF10, show that un = 1 1 − . cos n cos n − 1 [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Use the method of differences to find Ð un . N [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Explain why the infinite series u1 + u2 + u3 + à does not converge. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 [Turn over Craftear n=1 173 Q11 4 Ð 3r + 1 3r − 2 = 3 − 3 3N + 1 . N (i) Use the method of differences to show that 1 1 1 [4] r=1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 Craftear ........................................................................................................................................................ 174 (ii) Find the limit, as N → ∞, of Ð 3r + 1 3r − 2 . N2 N [4] r=N +1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 [Turn over Craftear ........................................................................................................................................................ 175 Q12 Let Sn = Ð −1r−1 r2 . n 5 r=1 (i) Use the standard result for Ð r 2 given in the List of Formulae (MF10) to show that n r=1 S2n = −n 2n + 1. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 Craftear ........................................................................................................................................................ 176 S2n S and find lim 2n2+1 . 2 n→∞ n n→∞ n (ii) State the value of lim [4] ........................................................................................................................................................ 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........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 [Turn over Craftear ........................................................................................................................................................ 177 Q13 2 (i) Verify that n e − 1 + e 1 1 . = n− n+1 ne n n + 1e n + 1en+1 [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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Ð n n + 1e n=1 n e − 1 + e n+1 . (ii) Express SN in terms of N and e. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 Craftear Let SN = N 178 Let S = lim SN . N →∞ (iii) Find the least value of N such that N + 1 S − SN < 10−3 . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 [Turn over Craftear ........................................................................................................................................................ 179 Q14 11 20 Answer only one of the following two alternatives. EITHER (i) By considering 2r + 12 − 2r − 12 , use the method of differences to prove that Ð r = 12 n n + 1. n r=1 [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 Craftear ........................................................................................................................................................ 180 (ii) By considering 2r + 14 − 2r − 14 , use the method of differences and the result given in part (i) to prove that Ð r3 = 14 n2 n + 12 . n r=1 [5] ........................................................................................................................................................ 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........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 [Turn over Craftear ........................................................................................................................................................ 181 The sums S and T are defined as follows: S = 13 + 23 + 33 + 43 + à + 2N 3 + 2N + 13 , T = 13 + 33 + 53 + 73 + à + 2N − 13 + 2N + 13 . (iii) Use the result given in part (ii) to show that S = 2N + 12 N + 12 . [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Hence, or otherwise, find an expression in terms of N for T , factorising your answer as far as possible. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (v) Deduce the value of S as N → ∞. T [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 Craftear ........................................................................................................................................................ 182 Q15 7 Let SN = Ð 3r + 1 3r + 4 N r=1 and TN = Ð 3r + 1 3r + 4 . N 1 r=1 (i) Use standard results from the List of Formulae (MF10) to show that SN = N 3N 2 + 12N + 13. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Use the method of differences to show that TN = 1 1 − . 12 3 3N + 4 [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 Craftear ........................................................................................................................................................ 183 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Deduce that SN is an integer. TN [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ SN . N →∞ N 3 T N (iv) Find lim [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 [Turn over Craftear ........................................................................................................................................................ 184 Q16 It is given that Ð ur = n2 2n + 3, where n is a positive integer. n 1 r=1 Ð ur . 2n (i) Find [2] r=n+1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find ur . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 [Turn over Craftear ........................................................................................................................................................ 185 Q17 2 (i) Verify that 2r + 1 1 = r r + 1 r + 2 2 D E 2r + 1 2r + 3 2r − 1 2r + 1 . − r + 1 r + 2 r r + 1 [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Ð n (ii) Hence show that r=1 2r + 1 1 = r r + 1 r + 2 2 D E 2n + 1 2n + 3 3 . − n + 1 n + 2 2 [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ∞ (iii) Deduce the value of Ð r r + 1 r + 2 . 2r + 1 [2] r=1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 [Turn over Craftear ........................................................................................................................................................ 186 Q18 Find Ð 4r − 3 4r + 1, giving your answer in its simplest form. n 1 r=1 [4] ................................................................................................................................................................ ................................................................................................................................................................ 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................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 Craftear ................................................................................................................................................................ 187 Q19 2 4 Express in partial fractions and hence find r r + 1 r + 2 ∞ Deduce the value of Ð r r + 1 r + 2 . 4 Ð r r + 1 r + 2 . n 4 [5] r =1 [1] Craftear r =1 © UCLES 2016 9231/11/M/J/16 [Turn over 188 Q20 1 Verify that 1 1 1 1 − = . 3r + 1 3r + 4 3 3r + 1 3r + 4 Ð N Let SN denote r=1 1 and let S denote 3r + 1 3r + 4 ∞ Ð 3r + 1 3r + 4 . Find the least value of N r=1 1 [5] Craftear such that S − SN < 10 1000 . [1] © UCLES 2016 9231/13/M/J/16 [Turn over 189 Q21 1 Use the method of differences to find ∞ Deduce the value of 1 . 2r2 − 1 Ð 2r − 1 . [1] 2 Craftear r =1 1 [4] © UCLES 2016 9231/11/O/N/16 [Turn over Q22 1 190 Use the List of Formulae (MF10) to show that Ð 3r2 − 5r + 1 and Ð r3 − 1 have the same 9 r=1 r=0 [4] Craftear numerical value. 13 © UCLES 2015 9231/11/M/J/15 [Turn over Q23 4 191 The sequence a1 , a2 , a3 , … is such that, for all positive integers n, an = n+5 n+6 − 2 . n − n + 1 n + n + 1 2 The sum Ð an is denoted by SN . Find N n=1 [3] (ii) the least value of N for which SN > 4.9. [4] Craftear (i) the value of S30 correct to 3 decimal places, © UCLES 2015 9231/11/O/N/15 [Turn over Q24 2 192 Expand and simplify r + 14 − r4 . [1] n n r=1 r=1 Use the method of differences together with the standard results for r and r2 to show that n r3 = 14 n2 n + 12 . 4 Craftear r=1 © UCLES 2014 9231/11/M/J/14 [Turn over Show that a, b, c is a basis for >3 . 193 [3] Express d in terms of a, b and c. Q25 [2] 2 Show that the difference between the squares of consecutive integers is an odd integer. [1] Find the sum to n terms of the series 5 7 2r + 1 3 + 2 + 2 +à+ 2 +à 2 2 2 1 ×2 2 ×3 3 ×4 r r + 12 2 [5] Craftear and deduce the sum to infinity of the series. © UCLES 2014 9231/13/M/J/14 [Turn over Q26 1 194 Given that uk = 1 1 , − 2k − 1 2k + 1 n express uk in terms of n. [4] k=13 ∞ Deduce the value of uk . [1] Craftear k=13 © UCLES 2014 9231/11/O/N/14 [Turn over Q27 5 195 Ð 2r + 1 2r + 3 = 6 − 2 2N + 3 . N Use the method of differences to show that 1 1 1 [5] r=1 Ð 2r + 1 2r + 3 < 8N . 2N Deduce that 1 1 [4] Craftear r=N +1 © UCLES 2013 9231/11/M/J/13 [Turn over Q28 Let f r = r! r − 1. Simplify f r + 1 − f r and hence find Ð r! r2 + 1. 2n r=n+1 [5] Craftear 1 196 © UCLES 2013 9231/13/M/J/13 [Turn over Q29 3 197 It is given that Sn = Ð ur = 2n2 + n. n r=1 Write down the values of S1 , S2 , S3 , S4 . Express ur in terms of r, justifying your answer. [4] Find Ð ur . 3 9231/11/O/N/13 [Turn over 2n Craftear r=n+1 © UCLES 2013 Q30 1 Express 198 1 in partial fractions. r r + 1 r − 1 Find [1] Ð r r + 1 r − 1 . 4 Ð r r + 1 r − 1 . 1 n 1 r=2 State the value of ∞ 1 Craftear r=2 © UCLES 2013 9231/13/O/N/13 [Turn over Q31 3 1 Given that f (r) = , show that (r + 1)(r + 2) 199 f (r − 1) − f (r) = n Hence find 2 . r(r + 1)(r + 2) 1 ∑ r(r + 1)(r + 2) . [2] [3] r=1 ∞ Deduce the value of 1 ∑ r(r + 1)(r + 2) . [1] Craftear r=1 © UCLES 2012 9231/11/M/J/12 [Turn over 200 Q32 4 Let f (r) = r(r + 1)(r + 2). Show that f (r) − f (r − 1) = 3r(r + 1). [1] n Hence show that ∑ r(r + 1) = 13 n(n + 1)(n + 2). r=1 n n r =1 r=1 Using the standard result for ∑ r, deduce that ∑ r2 = 16 n(n + 1)(2n + 1). [2] [2] Find the sum of the series 12 + 2 × 22 + 32 + 2 × 42 + 52 + 2 × 62 + . . . + 2(n − 1)2 + n2 , where n is odd. Craftear [3] © UCLES 2012 9231/12/O/N/12 [Turn over Q33 201 2n Show that ∑ r2 = 16 n(2n + 1)(7n + 1). [4] r=n+1 Craftear 1 © UCLES 2012 9231/13/O/N/12 [Turn over Q34 1 202 Find the sum of the first n terms of the series 1 1 1 + + + ... 1×3 2×4 3×5 [5] Craftear and deduce the sum to infinity. © UCLES 2012 9231/13/M/J/12 [Turn over SUMMATION OF SERIES MS Craftear 203 2(c) 2(b) 2(a) Question Q1 ( ( ) ( )( ) ) ( )( ) 2 1− r − r2 2 = n  r +1 2 r  r =1 n +1 n 2 1 3 2 4 3 − + − + − + + − 2 2 5 2 10 5 17 10 (n + 1) +1 n +1 − 12 1 n +1 =− + 2 ( n + 1)2 +1 = r =1 2  ( r + 2r + 2)( r + 1)   (r + 1) +1 − r +1  n Craftear (r + 1) r 2 + 1 − r r 2 + 2r + 2 r +1 r 1− r − r2 = − = (r + 1) 2 +1 r 2 +1 r 2 + 2r + 2 r 2 + 1 r 2 + 2r + 2 r 2 + 1 1 n − n 2 − 13 n 3 3 n − 12 n(n + 1) − 16 n(n + 1)(2n + 1) Answer Guidance 1 B1 FT FT from their answer to part (b). 5 A1 ISW M1 A1 Shows at least three complete terms including first and last. Cancellation may be implicit. M1 A1 Puts over a common denominator and expands, AG. 3 A1 Simplifies by collecting terms. M1A1 Substitutes correct formulae from MF19. Expanding brackets correctly. Marks 204 1(c) 1(b) 1(a) Question Q2 sin1 cos(r +1) cos r = ur = tan(n + 1) − tan1 sin1 r =1 r  u = tan 2 − tan1 + tan 3 − tan 2 + ... + tan(n + 1) − tan n n Craftear tan(n +1) oscillates as n → ∞ so u1 + u2 + u3 + does not converge. r =1  n [sin1] tan r + tan1 tan r + tan1 − tan r + tan 2 r tan 1 − tan r = 1 − tan r tan1 1 − tan r tan1 2 tan1sec r tan1 sin1 = = = 1 − tan r tan1 cos r(cos r − sin r tan1) cos r (cos r cos1 − sin r sin1) sin1 = cos r cos(r +1) sin r cos r cos1+ cos 2 r sin1 − sin r cos r cos1 + sin 2 r sin1 cos(r +1)cos r = sin( r + 1) sin r sin(r + 1) cos r − cos( r + 1) sin r sin(r + 1 − r) − = = cos(r + 1) cos r cos(r + 1) cos r cos(r +1) cos r sin( r + 1) sin r sin r cos1 + cos r sin1 sin r OR = − − cos(r + 1) cos r cos( r + 1) cos r Answer Guidance 1 B1 States ‘oscillates’ or refers to diverging values of tan(n +1) , or states that tan(n +1) does not tend to a limit. 3 A1 OE and ISW. M1 A1 Shows enough terms for cancellation to be clear. There must be three complete terms including first and last. Cancelling may be implied. 2 A1 SC B2 for this route completely correct to AG. M1 Applies all relevant correct addition formulae from MF19 for the route chosen and combines into a single fraction. Marks 205 2(c) 2(b) 2(a) Question Q3 3 2 1 4 2 2 1 2 ( ) r =1 1 f (1) − f (2) + 12 f (3) 2 1 4 = 1 1 1 − + 4 2 ( n + 1) 2 ( n + 2 ) + 12 f (n) − f (n + 1) + 12 f (n + 2) + 12 f (n − 1) − f (n) + 12 f (n +1)  + 12 f (3) − f (4) + 12 f (5) + 12 f (2) − f (3) + 12 f (4) 1 1 r 1 B1 FT FT from their answer to part (b). 5 A1 ISW Where f ( x) = M1 A1 Shows enough terms for cancelation to be clear. n  r (r + 1)(r + 2) = 3 A1 M1 A1 Finds partial fractions. Craftear Guidance M1 A1 Expands and substitutes formulae. At least two formulae used for M1. Marks 1 1 1 1 + = − r (r + 1)(r + 2) 2r r + 1 2(r + 2) = 14 n( n + 1) n 2 + n + 4n + 2 + 4 = 14 n( n + 1)( n + 2)( n + 3) r =1  ( r + 3r + 2r ) = n (n + 1) + n(n + 1)(2n + 1) + n(n + 1) n Answer 206 3(b) 3(a) Question Q4 Craftear A1 CAO Least value of n is 99 3 M1 Forms inequality n+2 0.01 S n − S = ln   < 0.01 leading to n + 2 < e (n +1)  n +1  4 A1 AG M1 A1 Shows enough terms to make cancellation clear. B1 States sum to infinity. AEF n+2 . 2(n + 1) Guidance B1 Separates logarithms into correct form using a difference. Or as logarithm of product. Marks S = − ln 2 [ln1] − ln 2 − ln( n + 1) + ln( n + 2) = ln ln n − 2ln(n + 1) + ln(n + 2) ln(n − 1) − 2ln n + ln(n +1) ln 3 − 2 ln 4 + ln 5  ln 2 − 2ln 3 + ln 4 ln1 − 2ln 2 + ln 3 r =1  ( ln r − 2ln(r + 1) + ln(r + 2) ) n Answer 207 2(c) 2(b) 2(a) Question Q5 2 1 56 11 1 1 1 1  Craftear 1 B1 FT 4 A1 1 1  11 =  −  7  8 7n + 8  r =1 n M1 Writes at least three correct terms, including first and last. 1 M1 A1 Finds partial fractions. 2  r and  r .  (7r + 1)(7r + 8) = 7  8 − 15 + 15 − 22 +  + 7n + 1 − 7n + 8  1 1 1 1  =  −  (7 r + 1)(7 r + 8) 7  7r + 1 7r + 8  3 A1 M1 Expands. = 493 n3 + 56n 2 + 143 n 3 r =1  49r + 63r + 8 n Guidance M1 Substitutes formulae for (7 r + 1)(7 r + 8) = Marks = 49 ( 16 n(n + 1)(2n + 1) ) + 63 ( 12 n(n + 1) ) + 8n r =1  n Answer 208 3(c) 3(b) 3(a) Question Q6 n 2n n 2n ( ) ( ) 2n n 2 4 N 2 2N 3 6 2 4 2 N +2 ) N +1 ( ( ) ( ) x N +1 → 0 as n → ∞  u1 + u2 + u3 +  = 1 − x − √ x 2 + 1 −1 < x < 1 = x N +1 + √ x 2 N + 2 + 1 − x − √ x 2 + 1 n =1 Craftear  u = x + √ ( x + 1) − x − √ ( x + 1) + x + √ ( x + 1) − x − √ ( x + 1) + + x + √(x + 1) − x − √ ( x + 1) N un = x n +1 + √ x 2 n + 2 + 1 − x n − √ x 2 n + 1 2n ( x − √ ( x + 1)) ( x + √ ( x + 1)) = x − ( x + 1) = −1 Answer Guidance 3 M1 A1 Finds sum to infinity. B1 3 A1 (Don’t allow in terms of n .) M1 Writes at least three complete correct terms, including the first and last. B1 Uses the identity given in part (a). 1 B1 Uses the difference of two squares and obtains the given answer. AG Marks 209 2 1 1 1 1 1 1 1 1 1 1 1 1 Answer 1    → 12 as n → ∞ Craftear 3 3 4n + 1  2n + 1  n ( 2n +1) n ( 4n + 1) − = n  −  − n  −  =  4 2n ( n + 1)  2n ( n + 1) 4n ( 2n +1)  4 4n ( 2n + 1)   2n n n n n = − 2 2 2 r = n +1 r − 1 r =2 r − 1 r =2 r −1 2n 4(c) 3 2n + 1 − 4 2n ( n + 1) 3 4 = 1 1 1 1  = 1+ − −  2  2 n n +1 r =2  r − 1 = 2  1 − 3 + 2 − 4 + 3 − 5 + 4 − 6 +  + n − 2 − n + n − 1 − n + 1  n 1 1 1 1 1  = =  −  r − 1 ( r −1)( r + 1) 2  r − 1 r + 1  2 4(b) 4(a) Question Q7 4 A1 M1A1 M1 1 B1 5 A1 A1 M1 M1 A1 Marks 210 3(c) 3(b) 3(a) Question Q8 r =1  n (4r − 2) 2 = 16 r =1  n r 2 − 16 r =1 ( 4n − 1) AG n n n =1 3 1 1 1 1 1 1 1  3 8 3 1  = 1−  8  2N + 1   S = 8 1 − 3 + 3 − 5 + 5 − 7  + 2 N − 1 − 2 N + 1  N n 3 3 1 1  = =  −  S n 4 ( 2n − 1) (2n + 1) 8  2n − 1 2n + 1  4 n(4n 2 + 6n + 2 − 6n − 6 + 3) = 34 n 3 2  r + 4n n = 83 n(n + 1)(2n + 1) − 8n(n + 1) + 4n Sn = Craftear Answer 1 B1 4 A1 M1 M1 A1 4 A1 M1 M1 A1 Marks 211 5(iii) 5(ii) 5(i) Question Q9 2 ) SN 25 1 5 T → × = 3 N 3 30 18 N Craftear 2 M1 A1 Divides S by N 3 and takes limits as N → ∞ N 4 A1 At least 3 terms including last. 11 1  1 1 −  − = 5  6 5 N + 6  30 5 ( 5 N + 6 ) M1 A1 Finds partial fractions. 3 M1 Expresses terms as differences. ( 1 1 1 1 1 1 1  TN =  − + − + " + −  5  6 11 11 16 5 N +1 5 N + 6  1 1 1  =  − ( 5r + 1)( 5r + 6 ) 5  5r + 1 5r + 6  1 ) A1 Simplifies to the given answer (AG). ( 35 35  25  1 = N 2 N 2 + 3N + 1 + N + + 6  = N 25 N 2 + 90 N + 83 2 2  6  3 r =1 M1 Expands. Guidance M1 Substitutes formulae for ∑ r and ∑ r 2 . r =1 r =1 N Marks 1  1  25  N ( N +1)( 2 N + 1)  + 35  N ( N + 1)  + 6 N 6  2  N N ∑ ( 5r + 1)( 5r + 6 ) = 25∑r + 35∑r + 6N Answer 212 2(iii) 2(ii) 2(i) Question Q10 ∑ Craftear cos N oscillates as N → ∞ so u1 + u 2 + u 3 + … does not converge. 1 cos N 1 1 1 1 1 1 − + − − + …+ cos1 1 cos 2 cos1 cos N cos ( N − 1) −1 + = ∑ 1 1  4sin  n −  sin N 1 1 2 2  = − cos ( 2n − 1) + cos1 n =1 cos n cos ( n − 1) n =1 N 1 B1 States “oscillates” or refers to diverging values of cos N . 2 A1 M1 Applies (i), shows enough terms and cancelation. 2 A1 AG 1 1 − cos n cos ( n −1) Guidance M1 Uses formulae for cos P ± cos Q. Marks 1 1  4sin  n −  sin 2  2 2 ( cos ( n − 1) − cos n )  = 2cos n cos ( n −1) cos ( 2n −1) + cos1 Answer 213 4(ii) 4(i) Question Q11 = N2 N − → 1 as N → ∞ 9 ( ) ( N r =1 ) N N N3 − N2 − = 3 ( 3N +1) 3 3 N 2 + 1 ( 3N +1) 3 N 2 + 1 ) = ( N  N N N −  − −  2 3 3 3N + 1  3 3 ( 3N + 1)  r =1 = N N Craftear ∑ (3r + 1)( 3r − 2 ) ∑ ( 3r + 1)( 3r − 2) ∑ ( 3r +1)( 3r − 2 ) r = N +1 N2 1 Uses r =1 − r =1 N 4 B1 A1 Allow simplification to common denominator. = N2 ∑ ∑ ∑ r = N +1 N2 M1 Applies (i) M1 4 A1 AG ∑ N 1 1  = 1 −  3  3N + 1  Guidance M1 At least 3 term including final term. M1 A1 Finds partial fractions. Marks 11 1 1 1 1 1  =  − + − +… −  3r + 1)( 3r − 2 ) 3  1 4 4 7 3N − 2 3 N +1  r =1 ( 1  1 1 =  − ( 3r +1)( 3r − 2 ) 3  3r − 2 3r + 1  1 Answer 214 5(ii) 5( 5(i) Question Q12 2 2n 2 r =1 n 2 r =1 Thus lim n →∞ S2 n+1 =2 n2 Craftear n 1 2 AG 2 correct sign 4 S 2n and n2 M1 Uses the result given in (i) or using lim n→∞ A1 n 1 A1 Alt: Find limit from previous line directly M1 So, S2n+1 = −n ( 2n + 1) + ( 2n + 1) = ( 2n + 1) ( n + 1) 2 M1 ( 2n + 1)2 S2 n +1 = S2n + ( −1) 2n B1 4 n 1 = −n ( 2n + 1) 2 A1 n 1 ∑ 4 r − 4∑ ( r ) + n − 4∑ ( r ) n 1 ∑ ( 2r − 1) − ∑ ( 2r ) =A1 M1 A1 Alt method: Use 2 Guidance M1 Uses correct difference. Marks S2 n = −2 n →∞ n 2 lim 1 Factorising, S 2n = n ( 2n + 1) ( 4n + 1 − 4n − 4 ) = − n ( 2n +1) 3 1 8 ( 2n ) ( 2n + 1) ( 4n + 1) − n ( n +1)( 2n +1) 6 6 Thus S 2n = n r =1 2n 2 r =1 so S 2 n = ∑ r − 2∑ ( 2r ) = ∑ r − 8∑ r 2 2 2 2 2n = 1 – 2 + 3 –4 ….. Answer 215 11E(i) Question Q14 2(iii) 2(ii) 2(i) Question Q13 1 e  1 1 ⇒ r =1 1 r = −12 + ( 2n + 1) r =1 ∑r = 2 n ( n + 1) n ⇒ 8 ∑ n 2 ( 2r + 1) − ( 2r −1) = 8r 2 ⇒ e N +1 > 103 ⇒ least such N is 6 . Answer 1 < 10−3 ⇒ N +1 < 10−3 e ( N +1) ( S − S N ) S= N ∑  ne − ( n + 1) e Craftear Marks Guidance 3 A1 AG. M1 Sums both sides and uses method of differences. B1 6 A1 M1 Attempts to find difference between sum and sum to infinity. B1 Finds S A1 SN = 1 1 − . e ( N +1) e N +1 Guidance M1 Uses difference method to sum. B1 Verifies result (AG). Marks  1 1 1 1 =( − + + −  n n +1  2 2 3 e 2e 2e 3e n =1   1 1 − …………. ) SOI = N Ne ( N + 1) e N +1 ( n + 1) e − n = n ( e − 1) + e 1 1 − = n n +1 n ( n + 1) en +1 n ( n 1) e ne ( n + 1) e Answer 216 11E(v) 11E(iv) 11E(iii) 11E(ii) ( ) ( r3 + 2 M1 Sums both sides and uses method of differences. A1 M1 Uses difference of squares or expands. 4 4 2 Craftear 2 A1 Accept ( N + 1)2 ( 2N ( N + 2 ) + 1) . M1 Eliminates even terms from S . 2 2 A1 2 Converges to 2 as N → ∞ . 2 M1 Writes fraction as quadratic in N divided by quadratic in N . 2 1 B1 Uses formula for ∑ r 3 . AG. 5 ( 2 N +1) = 4 N 2 + 4 N + 1 S = T 2N 2 + 4N + 1 2N 2 + 4N + 1 2 ( N + 1) ( 2 N 2 + 4 N + 1) r =1 3 ∑ ( 2r ) = ( 2 N +1) ( N +1) − 2 N ( N + 1) N 1 2 2 2 2 ( 2 N + 1) ( 2 N + 2 ) = ( 2 N + 1) ( N + 1) 4 3 T =S− S= ∑ n A1 AG 4 2 2 2 2 2  1   1    1 1  1 2 =   n +  −      n +  +    − n ( n +1) = n 2 ( n + 1)     2   2    2 2  2  ∑ 4 1  1  r = − 1 −  +  n +  2  2  r =1 n 2 M1 Uses formula for ∑ r r =1 ∑ n ) 2 1 1 1  4 r =  n +  −   − n ( n +1) 2 2 2  r =1 ⇒4 = 8r 2 + 2 ( 8r ) = 16 4r 3 + r 4 ( 2r + 1) − ( 2r − 1) = ( ( 2r + 1) + ( 2r − 1) )( ( 2r + 1) − ( 2r − 1) ) 217 7(iii) 7(ii) 7(i) Question Q15 N So SN is an integer because all terms are integers TN SN N ⇒ = 4 ( 3 N + 4 ) 3N 2 + 12 N + 13 4 ( 3N + 4 ) TN ( ) Craftear SN as a polynomial TN 2 A1 Justifies expression being integer M1 Writes A1 Cancels to given answer (AG). 1 1 1  1 1  − = − 3  4 3N + 4  12 3 ( 3N + 4 ) TN = M1 Expresses terms as differences. 1 1 1 1 1 1 1  TN =  − + − + " + −  3  4 7 7 10 3 ( N − 1) + 1 3 N + 4  3 B1 Finds partial fractions. 1 1 1 1  =  − ( 3r + 1) ( 3r + 4 ) 3  3r + 1 3r + 4  2 3 A1 Shows simplification to the given answer (AG). ) = N ( 3 N + 12N + 13) ( 15 15 9  = N  2 N 2 + 3N + 1 + N + + 4  6 2 2   r =1 Guidance M1 Substitutes formulae for ∑ r and ∑ r 2 . r =1 r =1 M1 Expands Marks 1  1  9  N ( N + 1)( 2 N + 1)  + 15  N ( N + 1)  + 4 N 6  2  N N ∑ ( 3r + 1) ( 3r + 4) = 9∑r 2 +15∑r + 4N Answer 218 2(i) Question Q17 1(ii) 1(i) Question Q16 7(iv) ( 2 2n n +1 ur = 2 Answer ) ( ) Total: Craftear 2 A1 AG 2 Marks 3 2 3 2 1  4r + 8r + 3r − 4r + 8r − r − 2  1  ( 4r + 2 )  ( 2r +1) =   =  = r ( r + 1) ( r + 2 ) 2  2  r ( r +1) ( r + 2 )  r ( r +1) ( r + 2 )  ( 3 Guidance A1 SR: CAO B1 without wrong working M1A1 Method mark for using Sr – Sr–1 OE 2 A1 M1 2 Answer Total: Total: Guidance M1 Method mark for using S2n – Sn Marks 2 A1 M1 Divides expression in (iii) by N 3 and takes limit 1  4r + 8r + 3r − 4r −1 ( r + 2 )  RHS =   2 r ( r +1) ( r + 2 )    3 ur = r 2 ( 2r + 3) − ( r − 1) ( 2r +1) = 6r 2 − 1 ) ( 2n )2 ( 4n + 3) − n 2 ( 2n + 3) = 14n3 + 9n 2 ∑ → 4 ( 3)( 3) = 36 4 ( 3 N + 4 ) 3 N + 16 N + 9 SN = N3 N 3TN 219 1 Question Q18 2(iii) 2(ii 2(ii) r =1 ∑ n r =1 n =… = ( ) 2 − 3n n 16n 2 +12n − 13 (3 terms) 3 6 n ( n + 1) Answer −8 ∑ r − 3n r2 − 8 n ( n + 1)( 2n + 1) r =1 n ∑ = 16 ur = 16 1 3 1 S∞ = × 4 − = 1 . 2 4 4 1  ( 2n +1)( 2n + 3) 3  =  −  2  2  ( n +1)( n + 2 ) Craftear Guidance 4 M1A1 A1 AG M1 4 A1 OE M1 For using formulae correctly in their expression M1A1 M1 for split into 3 parts Marks Total:  ( 2n − 1)( 2n +1) ( 2n − 3)( 2n − 1)   ( 2n + 1)( 2n + 3) ( 2n − 1)( 2n +1)   1  3.5 1.3   5.7 3.5  − − + − +…+  −  +    n ( n − 1) n ( n + 1) n ( n +1) 2   2.3 1.2   3.4 2.3       ( n +1)( n + 2 ) to n terms is: 220 221 Q19 2 2 4 2 − + (Award B2 if written down by cover up rule.) r r +1 r + 2 2  4 1 4 2  2 4 2    2 − + +  2 − 2 +  + 1 − +  + … +  + −  + 3 3 2 n − 1 n n + 1 n n + 1 n 2        2 2 + (AEF) = 1− n +1 n + 2 Sum to infinity = 1 M1A1 M1A1 A1 [5] B1 [1] Q20 1 1 1  1  ( 3r + 4 ) − ( 3r + 1)  1 AG −  =   =  3  3r + 1 3r + 4  3  ( 3r + 1)( 3r + 4 )  ( 3r + 1)( 3r + 4 )  1  1  1 1   1 1  1  1  1 1  S N =  −  +  −  + … +    =  −  − 3  4 7   7 10    3N +1  3N + 4   3  4 3N + 4  1 ⇒S = 12 1 1 < ⇒ S – SN = 3 ( 3N + 4 ) 10000 ⇒ 3N + 4 > 10000 7 ⇒ N > 1109 . Thus least N is 1110. 3 9 B1 [1] M1 A1 A1 M1 A1 [5] Craftear 1 Q21 1 1 1 1  1 1  1 1  M1A1 Craftear B1 [1] 1 ∞ n 1 1  1 1 1 − = ⇒ = ∑   2 2  2n + 1  2n +1 2 r =1 ( 2r ) − 1 r =1 2 [4] 1 M1A1 ∑ ( 2r ) − 1 = 2  1 − 3  +  3 − 5  +…+  2n − 1 − 2n +1  = 2 1 − 2n + 1  ( OE ) n 1 1 1  =  − ( 2r − 1) ( 2r + 1) 2  2r − 1 2r + 1  1 222 223 Q22 3 × 1 13 × 14 × 27 13 × 14 − 5× +13 = 2015 6 2 M1A1 2  9 × 10   2  −10 = 2015 (Award M1 for subtracting 9 or 10 here.)   M1A1 (4) Total: 4 Q23 (i) 7   7 8  36  36  35  6 = 4.820 − −  = 6 − −  + K +    +  3  3 7 931  931  871  1 (ii) 6− n+6 2 > 4.9 ⇒ 0.21n 2 − 10.79n − 34.79(> 0) n + n +1 ⇒ n > 54.42 K so 55 terms required. M1A1 A1 [3] M1A1 dM1A1 [4] Total 7 Q24 (r + 1)4 – r4 = 4r3 + 6r2 + 4r + 1 2 (n + 1)4 – 14 = 4Σ nr=1 r3 + 6Σ nr=1 r2 + 4Σ nr=1 r + n B1 [1] M1 n4 +4n3 + 6n2 + 4n = 4Σ nr=1 r3 + n(2n2 + 3n + 1) + 2n2 + 2n + n A1A1 ⇒ … ⇒ Σ nr=1 r3 = 1 n2(n + 1)2. 4 A1 [4] (AG) Craftear 4 Q25 (n + 1)2 – n2 = n2 + 2n + 1 – n2 = 2n + 1 ⇒ odd. 3 5 7 2n +1 2 2 − 12 32 − 2 2 4 2 − 32 (n +1) 2 − n 2 + + + K = + + + K 12.2 2 2 2.32 32.4 2 n 2 (n +1) 2 12.2 2 2 2.32 32.4 2 n 2 (n +1) 2 1 1 1 1 1 1 1 = 1− 2 + 2 − 2 + 2 − 2 +K+ 2 − 2 2 3 3 4 (n +1) 2 n 1 =1− (n +1) 2 Sum to infinity =1. 2 Q26 1 ( 1 − 25 1 27 )+ ( ∑ n 1 − 27 u = 15 − r =13 k ∑ ∞ u = 15 r =13 k 1 2 n+1 1 29 )+ ... + ( 1 − 2 n −1 1 2 n +1 ) B1 [1] M1A1 M1 A1 A1 [5] M1A1 M1A1 (4) B1 (1) [5] 224 Q27 5 1 1 1 1  =  − (`2r + 1)(2r + 3) 2  2r + 1 2r + 3  Finds partial fractions. N M1A1 1 ∑ (2r + 1)(2r + 3) r =1 Expresses terms as differences. Shows cancellation. 2N Uses ∑ N +1 = 2N N 1 1 1 1 1 1 1  −   −  + ... +  23 5 2  2N + 1 2N + 3  = 1 1 (AG) − 6 2(2 N + 3) M1A1 A A11 2N ∑ ∑ − = 1  1  1 1  −  −  =  − 2(4 N + 3)   6 2(2 N + 3)  N +1  6 M1 ∑ . 1 Applies result and simplifies. Deduces inequality. = 1 1 1  −   2  2N + 3 4N + 3  A1 = N (2 N + 3)(4 N + 3) M1 < N 1 (AG) = 2 N.4N 8 N A1 [9] 1 Simplifies. f(r + 1) – f(r) = r(r + 1)!−(r −1)r! 2 2 = r!(r + r − r + 1) = r!(r + 1) Uses difference method. Obtains result. n ∑ = f(2) – f(1) + f(3) – f(2) +…f(n + 1) – f(n) 1 ∴ Craftear Q28 M1 A1 M1 A1 = n(n + 1)!−0 = n(n + 1)! 2n ∑ = 2n(2n + 1)!−n(n + 1)! A1 [5] 5 n+1 2n (Or directly using ∑ = f (2n + 1) − f (n + 1) from the n +1 method of differences.) Q29 3 Writes first four sums. S1…S4 ~ 3, 10, 21, 36 Deduces first four terms, conjectures and justifies result. u1…u4 ~ 3, 7, 11, 15 ⇒ ur = 4r ‒ 1 n since S n = {6 + 4(n −1)} = 2n 2 + n as given. 2 B1B1 Or ur = S r − S r −1 = 2r 2 + r − 2(r −1) − (r −1) = 4r – 1 B1B1 B1 B1 2 Obtains required sum. ∑ 2n n+1 (4r − 1) = 4. 2n(2n +1)  n(n +1)  − 2n −  4. − n 2 2   ( ) = 8n 2 + 2n − 2n 2 + n = 6n 2 + n Or Sum of AP = n (4n + 3 + 8n − 1) = 6n 2 + n 2 B1 4 M1A1 A1 3 [7] 225 Q30 1 Finds partial fractions. 1 1 1 1 = − + r (r -1) (r + 1) 2(r − 1) r 2(r + 1) Expresses each term in fractions 1 1  1 1 1 1 1 1  1  − +  − +  +  − + ...  2 2 6   4 3 8   2(n − 1) n 2(n + 1)  Cancels terms and sums = Find sums to infinity S∞ = 1 1 1 − + 4 2n 2(n + 1) (OE) 1 4 B1 (1) M1A1 M1A1 (4) B1 (1) [6] Q31 Proves initial result. 1 1 − r (r + 1) (r +1)(r + 2) r +2−r 2 = = (AG) r (r +1)(r + 2) r (r +1)(r + 2) f(r – 1) – f(r) = 1 n Sets up method of differences. Shows cancellation to get result. States sum to infinity. ‘Non hence’ method for last two parts i.e. penalty of 1 mar i. 1 1 1  ∑ r (r + 1)(r + 2) = 2 1 × 2 − 2 × 3  …… 1 +  1 1 1 −   2  n(n + 1) (n + 1)(n + 2)  =  1 1 1 −   (OE) 4 2  (n + 1)(n + 2)  1 1 ∴∑ = 4 1 r ( r + 1)( r + 2) ∞ 1 1 1 1 = − + r (r + 1)(r − 2) 2r (r +1) 2(r + 2) ⇒ ⋅⋅⋅ ⇒ 1 1 1 1 1 1 − + ... + − + 2 2 4 2(n + 1) (n +1) 2(n + 2) = M1 A1 2 M1A1 A1 3 A1√ 1 (M1) (A1)  1 1 1 −   (OE) 4 2  (n +1)(n + 2)  (A1) (3) 1 1 = 4 1 r ( r + 1)( r + 2) (A1√) (1) ∞ ∴∑ Craftear 3 [6] 226 Q32 4 r(r + 1)(r + 2) − (r − 1)r(r + 1) = r(r + 1)(r + 2 − r + 1) = 3r(r + 1) (AG) Verifies given result. n ∑ Uses method of differences to sum first series. 1 r =1 1 {[f (n) − f (n −1)] + [f (n −1) − f (n − 2)] + ... + [f (1) − f (0]} 3 1 = n(n + 1)(n + 2) (AG) (Award B1 if ‘not 3 hence’.) n n Subtracts B1 r ( r + 1) = ∑ ∑r r =1 r =1 to obtain sum of second series. n ∑ r ( r + 1) − ∑r = (12 + 2 2 + ... + n 2 ) + (2 2 + 4 2 + ... + (n −1) 2 ) = Splits series into two series. Applies sum of squares formula to obtain result. A1 2 n n(n + 1)(n + 2) n(n + 1) − 3 2 r =1 r =1 1 1 = n(n + 1)(2n + 4 − 3) = n(n + 1)(2n + 1) (AG) 6 6 r2 = M1  n − 1  n + 1  4  n 1 n(n + 1)(2n + 1) 2  2   =…= n 2 (n + 1) + 2 6 6 M1 A1 2 M1 M1A1 3 [8] 4 [4] 5 [5] 1 Use of: Use of: Obtains result. 2n 2n n n+1 n 1 1 ∑ = ∑ −∑ Craftear Q33 M1 n( n + 1)(2n + 1) 6 1 2n(2n + 1)(4n + 1) n( n +1)(2n + 1) − 6 6 1 1 = n(2n + 1)(8n + 2 − n − 1) = n(2n + 1)(7 n + 1) (AG) 6 6 ∑r = M1 2 A1 A1 Q34 1 Finds partial fractions. 1 1 1 1  =  −  r (r + 2) 2  r r + 2  n 1 = ∑ r =1 r ( r + 2) Use method of differences. 1  1 1   1 1   1 1   1  + − + ... +  −  + 1 −    −   2  n n + 2   n − 1 n + 1  2 4   3 Obtains results. = 3 1 3 1 1  −  −  (acf) ⇒ S ∞ = 4 2 2 n + 1 n + 2 M1A1 M1 A1A1√ 227 Craftear MATRICES 1 228 Q1 4 8 The matrix M represents the sequence of two transformations in the x-y plane given by a rotation of 60° anticlockwise about the origin followed by a one-way stretch in the x-direction, scale factor d (d ! 0) . (a) Find M in terms of d. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) The unit square in the x-y plane is transformed by M onto a parallelogram of area 12 d 2 units2. Show that d = 2 . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 Craftear ............................................................................................................................................................ 229 J1 1N The matrix N is such that MN = KK1 1OO . L2 2P (c) Find N. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by MN. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 [Turn over Craftear ............................................................................................................................................................ 230 Q2 4 The matrices A, B and C are given by 2 k k A = f5 - 1 3p, 1 0 1 1 0 B = f0 1p and 1 0 0 1 1 o, C =e -1 2 0 where k is a real constant. (a) Find CAB. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Given that A is singular, find the value of k. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 Craftear ............................................................................................................................................................ 231 (c) Using the value of k from part (b), find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by CAB. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 [Turn over Craftear ............................................................................................................................................................ 232 Q3 4 cos i - sin i 3 0 oe o. The matrix M is given by M = e sin i cos i 0 1 (a) The matrix M represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the values of i, for 0 G i G r , for which the transformation represented by M has exactly one invariant line through the origin, giving your answers in terms of r . 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............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 Craftear ............................................................................................................................................................ 233 ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 [Turn over Craftear ............................................................................................................................................................ 234 Q4 1 (a) Give full details of the geometrical transformation in the x-y plane represented by the matrix 6 0 e o. [1] 0 6 ............................................................................................................................................................ ............................................................................................................................................................ 3 4 o. Let A = e 2 2 (b) The triangle DEF in the x-y plane is transformed by A onto triangle PQR. Given that the area of triangle DEF is 13 cm2, find the area of triangle PQR. [2] ............................................................................................................................................................ ............................................................................................................................................................ 6 0 o. (c) Find the matrix B such that AB = e 0 6 [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Show that the origin is the only invariant point of the transformation in the x-y plane represented by A. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 Craftear ............................................................................................................................................................ 235 Q5 1 2 1 b a 0 oe o, where a and b are positive constants. The matrix M is given by M = e 0 1 0 1 (a) The matrix M represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The unit square in the x-y plane is transformed by M onto parallelogram OPQR. (b) Find, in terms of a and b, the matrix which transforms parallelogram OPQR onto the unit square. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 Craftear ............................................................................................................................................................ 236 It is given that the area of OPQR is 2 cm 2 and that the line x + 3y = 0 is invariant under the transformation represented by M. (c) Find the values of a and b. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 [Turn over Craftear ............................................................................................................................................................ 237 Q6 4 The matrices A and B are given by 1 1 -2 3 0 1 2 o and B = f 1 A =e p. 1 1 0 3 2 2 (a) Give full details of the geometrical transformation in the x-y plane represented by A. [1] ............................................................................................................................................................ (b) Give full details of the geometrical transformation in the x-y plane represented by B. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The triangle DEF in the x-y plane is transformed by AB onto triangle PQR. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 Craftear (c) Show that the triangles DEF and PQR have the same area. 238 (d) Find the matrix which transforms triangle PQR onto triangle DEF. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (e) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by AB. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 [Turn over Craftear ............................................................................................................................................................ 239 Q7 6 2 0 o. Let A = e 1 1 (a) The transformation in the x-y plane represented by A -1 transforms a triangle of area 30 cm 2 into a triangle of area d cm 2 . Find the value of d. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Prove by mathematical induction that, for all positive integers n, 2n 0 o. An = e n 2 -1 1 [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 Craftear ............................................................................................................................................................ 240 (c) The line y = 2x is invariant under the transformation in the x-y plane represented by A n B , where 1 0 o. B=e 33 0 Find the value of n. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 [Turn over Craftear ............................................................................................................................................................ 241 Q8 4 The matrix A is given by k 0 2 A = f0 - 1 - 1p, 1 1 -k where k is a real constant. (a) Show that A is non-singular. 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Craftear The matrices B and C are given by 0 -3 -3 -1 1 o. B = f- 1 3p and C = e 1 1 2 0 0 - 2 - 32 It is given that CAB = f p. - 1 - 32 (b) Find the value of k. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 242 (c) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by CAB. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 [Turn over Craftear ............................................................................................................................................................ MATRICES 1 MS Craftear 243 Q1 4(c) 4(b) 4(a) Question 1 1 =  2 2− 3  2 3  1  1 1+ 3 1+ 3  =   4  1 − 3 1− 3  1 1 1  1 N = M −1  1 1  =  2 3  2 2  2  − 2 M −1 3  1 1  1 1  1   2 2  Craftear A1 CAO M1 Multiplies on the left by the inverse of their M. B1 FT Inverse of their M. 2 A1 AG 4 M1 A1 Correct order. B1 One-way stretch in the x-direction, scale factor d. d ≠0 d =2 − d 23   1  2  Guidance B1 Rotation 60° anticlockwise about the origin. M1 Uses value of det M . − 23   d2 = 1   3 2   2 − 23   1  2  Marks d = 12 d 2  d 0   12 M=   0 1   23  d 0    0 1  cos 60 −sin 60   12  =  sin 60 cos 60   23 Answer 244 Q2 4(b) 4(a) Question 4(d) −1 3 5 3 5 −1 −k +k = 0 leading to −2 − 2k + k = 0 0 1 1 1 1 0 k = −2 2 −1   10    −k +14 −k − 2  Craftear  2 k k  1 0  2 + k k   0 1 1    0 1 1   −1   5 −1 3  0 1  =    8  −1 2 0   1 0    −1 2 0   2 0     1 0  1 0  2 k k  1 0  4  −1  0 1 1  6   Or    5 −1 3  0 1  =   0 1 k k 1 2 0 8 − − − 2 − + 6    1 0 1  1 0    1 0      Answer Guidance 3 A1 M1 A1 Sets determinant equal to zero and forms linear equation. 3 A1 M1 A1 Multiplies two matrices correctly. Marks 5 A1 WWW y = 12 x and y = −x M1 A1 Uses y = mx and Y = mX .  x X Transforms   to   .  y Y  A1 ( x + mx ) B1 1 + m = 2m + 2 m 2  2m 2 + m − 1 = 0 1 x + 12 mx = m 2  1 1 x   x + y   1 1   =  1 1   2 2  y   2 x + 2 y  245 Q3 4(a) Question 4(c) Craftear One-way stretch [scale factor 3 parallel to the x-axis] followed by a rotation, [anticlockwise centred at the origin, through an angle θ . ] Answer [m2 − 10m + 16 = 0] y = 2 x and y = 8 x 16 = 10m − m2 10x − mx = X and 16 x = mX 10 −1  x  10 x − y    =    16 0   y   16 x  x Transforms   .  y 1  Allow q   where q is x, t or a nonzero number.  m Guidance 2 B2 Award B1 if given in the wrong order. Marks 5 A1 A1 OE Expect 16x = m (10 x − mx ) . M1 A1 Uses y = mx and Y = mX . M1 246 4(b) 1 or 3 tan θ = ± θ = 16 π and θ = 56 π cosθ = ± 23 −sin θ   x   3x cosθ − y sin θ    =   cosθ  y   3x sin θ + y cosθ  −sin θ   cosθ   3cosθ   3sin θ  3cosθ M =  3sin θ Craftear x X Transforms   to   .  y Y  9 A1 Allow 30° and 150° M1 Uses an appropriate trigonometric identity. M1 A1 Sets discriminant equal to 0. A1 M1 A1 Uses y = mx and Y = mX . B1 B1 247 Q4 Craftear A1 Area = 2 × 13 = 26 cm2 1(d) 1(c) M1 Finds det A. det A = 6 − 8 = −2 A1 AEF Could be solved by equations.  2 −4   −6 12  B = −3  =   −2 3   6 −9  4 M1 A1  2 4 Solves equations or states that det   ≠ 0 , AG. 2 1  x  X  Uses   =   to form simultaneous equations.  y  Y  M1 3x + 4 y = x  2 x + 4 y = 0  leading to  2x + 2 y = y 2 x + y = 0 y = −2 x leading to −6 x = 0 leading to x = 0, y = 0 X Finds   Y  B1  3 4   x   3x + 4 y     =    2 2  y   2 x + 2 y  2 M1 Finds A −1 . Guidance  2 −4  A −1 = − 12    −2 3  2 1 1(b) B1 Marks Enlargement, scale factor 6. Answer 1(a) Question 248 Q5 1(c) 1(b) 1(a) Question A1 b=3 Craftear M1 Uses that line is invariant (or X + 3Y = 0 ). x = ax − 13 bx  1 = a − 13 b 5 M1 Uses x + 3 y = 0 .  ax − 13 bx  =   −1 x  3   x X Transforms   to   .  y Y  B1  a b  x   ax + by     =    0 1  y   y  2 M1 A1 2 B2 Both named correctly. Award B1 if given in the wrong order. Guidance B1 1  1 −b    a 0 a  Marks a=2 M −1 = One-way stretch followed by a shear. Answer 249 Q6 4(e) 4(d) 4(c) B1 Or 60° 1 π 3 1  − 12   12 √ 3 2 =    1 1 1   √ 3  2 − 2 √ 3  2 x 1 X x Transforms   to   . Accept t   instead of   . m Y   y  y A1 OE y = (2 − √ 3) x and y + (2 + √ 3) x = 0 5 A1 M1A1 Uses Y = mX . OE. Allow working with y = mx + c as long as c = 0 is stated explicitly. B1 2 M1 A1 or using (AB)–1 = B–1A–1 1 − m √ 3 = m √ 3 + m 2  m 2 + 2m √ 3 −1 = 0  m = ±2 − √ 3 1 − 12 m √ 3 = 12 m √ 3 + 12 m 2 2 1  12 √ 3   x   12 x √ 3 + 12 y  2   1   =  − 12 √ 3   y   12 x − 12 y √ 3   2 ( AB ) − 1 √3 = − 2 1  − 2  A1 Area of PQR = |-1| Area of DEF −1 B1 det AB = −1 3 M1 Finds AB or uses product of determinants. Full marks here may be obtained by arguing that both reflection and rotation preserve [absolute] value of area and so also does their combination. 2 1  1 √3  2 AB =  2 1  or det AB = det A det B 1  − 2 √ 3   2 anticlockwise about the origin. B1 Writes ‘rotation’ or ‘rotate’ (and no other transformation) Rotation 1 4(b) Craftear Guidance B1 Only mention reflection (and no other transformation). Marks Reflection in the line y = x. Answer 4(a) Question 250 Q7 6(c) 6(b) 6(a) Question ( ) 2k +1 0   2k +1  2k 0 2 0  0   = k = =      2 − 1 1   1 1   2 2k − 1 +1 1   2k +1 − 1 1        (2n + 32) x = 2n+1 x  2n = 32  n = 5  2n 0   x   2n x   n    =  n  2 + 32 0    y   (2 + 32) x   2n 0   1 0   2n 0 AnB =  n =  2 − 1 1   33 0   2n + 32 0      Craftear So, it is also true for n = k +1 . Hence, by induction, true for all positive integers. Then A k +1 5 M1 A1 B1 M1A1 5 A1 M1A1 B1  2k 0 Assume that it is true for n = k , so A =  k .  2 − 1 1    k B1 3 A1 M1 A1 Marks 1 0  2 0  2 A= =   so true when n = 1.   1  1 1   2 − 1 1  d = 15 −1 det A −1 = ( det A ) = 12 Answer 251 Q8 4(c) 4(b) 4(a) Question −1 −1 0 −1 = 0  k2 + k + 2 0 +2 1 −k 1 1 Craftear A1 y = −x and 3 y − 2 x = 0 5 A1 M1 A1 x + 32 mx = m ( 2x + 32 mx ) 1 + 32 m = 2 m + 32 m 2  3m 2 + m − 2 = 0 B1 3 A1 M1 A1 3 A1 M1 A1 Marks 3  2 x + 32 y   x 2   =  3    3 y 2    x + 2 y  2  1  −2 9k + 3   −2 − 32  =  k = − 12 = 3 − 1 − 1 −3 3 − − k    2  k 0 2  0 −3   0 −3k   −3 −1 1     −3 −1 1        0 −1 −1   −1 3  =    1 −3  1 1 2 1 1 2    1 1 −k  0 0     −1 0       1 − 4(2) = −7 < 0  Non-singular k Answer 252 253 Craftear POLAR COORDINATES 254 Q1 5 The curve C has polar equation r = a cot b13 r - il, where a is a positive constant and 0 G i G 16 r . It is given that the greatest distance of a point on C from the pole is 2 3. (a) Sketch C and show that a = 2 . [3] ............................................................................................................................................................ (b) Find the exact value of the area of the region bounded by C, the initial line and the half-line 1 [4] i = 6 r. ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 Craftear ............................................................................................................................................................ 255 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Show that C has Cartesian equation 2 `x + y 3j = `x 3 - yj x 2 + y 2 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 [Turn over Craftear ............................................................................................................................................................ 256 Q2 The curve C has polar equation r = 1 r-i - 1 r , where 0 G i G 12 r . (a) Sketch C. [3] (b) Show that the area of the region bounded by the half-line i = 12 r and C is 3 - 4 ln 2 . 4r [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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The curve C has polar equation r = 2 cos i (1 + sin i) , for 0 G i G 12 r . (a) Find the polar coordinates of the point on C that is furthest from the pole. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 Craftear ............................................................................................................................................................ 259 (b) Sketch C. [2] (c) Find the area of the region bounded by C and the initial line, giving your answer in exact form. [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 [Turn over Craftear ............................................................................................................................................................ 260 Q4 The curve C has polar equation r = 3 + 2 sin i , for - r 1 i G r . (a) The diagram shows part of C. Sketch the rest of C on the diagram. O [1] i=0 The straight line l has polar equation r sin i = 2 . (b) Add l to the diagram in part (a) and find the polar coordinates of the points of intersection of C and l. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 Craftear 5 8 261 (c) The region R is enclosed by C and l, and contains the pole. Find the area of R, giving your answer in exact form. [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 [Turn over Craftear ............................................................................................................................................................ 262 Q5 7 14 (a) Show that the curve with Cartesian equation 5 (x 2 + y 2) 2 = 4xy (x 2 - y 2) has polar equation r = sin 4i . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 Craftear ............................................................................................................................................................ 263 The curve C has polar equation r = sin 4i , for 0 G i G 14 r . (b) Sketch C and state the equation of the line of symmetry. [3] ............................................................................................................................................................ ............................................................................................................................................................ [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 [Turn over Craftear (c) Find the exact value of the area of the region enclosed by C. 264 (d) Using the identity sin 4i / 4 sin i cos 3 i - 4 sin 3 i cos i , find the maximum distance of C from the [6] line i = 12 r . Give your answer correct to 2 decimal places. ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 Craftear ............................................................................................................................................................ 265 Q6 5 8 The curve C has polar equation r = ln (1 + r - i) , for 0 G i G r . (a) Sketch C and state the polar coordinates of the point of C furthest from the pole. [3] ............................................................................................................................................................ ............................................................................................................................................................ 1 ` j 2 (1 + r) ln (1 + r) ln (1 + r) - 2 + r . [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 Craftear (b) Using the substitution u = 1 + r - i , or otherwise, show that the area of the region enclosed by C and the initial line is 266 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Show that, at the point of C furthest from the initial line, (1 + r - i) ln (1 + r - i) - tan i = 0 and verify that this equation has a root between 1.2 and 1.3. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 [Turn over Craftear ............................................................................................................................................................ 267 Q7 7 The curve C1 has polar equation r = i cos i , for 0 G i G 12 r . (a) The point on C1 furthest from the line i = 12 r is denoted by P. Show that, at P, 2i tan i - 1 = 0 and verify that this equation has a root between 0.6 and 0.7. 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The curve C2 has polar equation r = i sin i , for 0 G i G 12 r . The curves C1 and C2 intersect at the pole, denoted by O, and at another point Q. (b) Find the polar coordinates of Q, giving your answers in exact form. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 Craftear ............................................................................................................................................................ 268 (c) Sketch C1 and C2 on the same diagram. [3] (d) Find, in terms of r, the area of the region bounded by the arc OQ of C1 and the arc OQ of C2 . [7] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 Craftear ............................................................................................................................................................ 269 Q8 The curve C has polar equation r = a tan i , where a is a positive constant and 0 G i G 14 r . (a) Sketch C and state the greatest distance of a point on C from the pole. [2] (b) Find the exact value of the area of the region bounded by C and the half-line i = 14 r . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 Craftear 5 8 270 x2 . [3] a2 - x2 ............................................................................................................................................................ (c) Show that C has Cartesian equation y = ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 1 2a 2 x2 dx. [2] a2 - x2 0 ............................................................................................................................................................ (d) Using your answer to part (b), deduce the exact value of y ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 [Turn over Craftear ............................................................................................................................................................ 271 Q9 OR The curves C1 and C2 have polar equations, for 0 ≤ 1 ≤ 12 0, as follows: C1 : r = 2 e1 + e−1 , C2 : r = e21 − e−21 . The curves intersect at the point P where 1 = !. (i) Show that e2! − 2e! − 1 = 0. Hence find the exact value of ! and show that the value of r at P is 4ï2. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 Craftear ........................................................................................................................................................ 272 (ii) Sketch C1 and C2 on the same diagram. [3] (iii) Find the area of the region enclosed by C1 , C2 and the initial line, giving your answer correct to [5] 3 significant figures. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 Craftear ........................................................................................................................................................ 273 Q10 11 20 Answer only one of the following two alternatives. EITHER The curve C1 has polar equation r 2 = 21, for 0 ≤ 1 ≤ 12 0. (i) The point on C1 furthest from the line 1 = 12 0 is denoted by P. Show that, at P, 21 tan 1 = 1 and verify that this equation has a root between 0.6 and 0.7. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 Craftear ........................................................................................................................................................ 274 The curve C2 has polar equation r 2 = 1 sec2 1, for 0 ≤ 1 < 12 0. The curves C1 and C2 intersect at the pole, denoted by O, and at another point Q. (ii) Find the exact value of 1 at Q. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) The diagram below shows the curve C2 . Sketch C1 on this diagram. [2] 1 = 21 0 C2 1=0 O © UCLES 2019 9231/11/M/J/19 [Turn over Craftear ........................................................................................................................................................ 275 (iv) Find, in exact form, the area of the region OPQ enclosed by C1 and C2 . [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 Craftear ........................................................................................................................................................ 276 Q11 The curve C has polar equation r2 = ln 1 + 1, for 0 ≤ 1 ≤ 20. (i) Sketch C. [2] (ii) Using the substitution u = 1 + 1, or otherwise, find the area of the region bounded by C and the initial line, leaving your answer in an exact form. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 Craftear 2 277 Q12 The curve C has polar equation r = cos 21, for − 14 0 ≤ 1 ≤ 14 0. (i) Sketch C. [2] (ii) Find the area of the region enclosed by C, showing full working. [3] Craftear 3 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 278 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find a cartesian equation of C. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 [Turn over Craftear [3] 279 Q13 8 The curves C1 and C2 have polar equations, for 0 ≤ 1 ≤ 0, as follows: C1 : r = a, C2 : r = 2a cos 1 , where a is a positive constant. The curves intersect at the points P1 and P2 . (i) Find the polar coordinates of P1 and P2 . [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) In a single diagram, sketch C1 , C2 and their line of symmetry. © UCLES 2018 9231/13/M/J/18 [3] Craftear ........................................................................................................................................................ 280 (iii) The region R enclosed by C1 and C2 is bounded by the arcs OP1 , P1 P2 and P2 O, where O is the pole. Find the area of R, giving your answer in exact form. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 [Turn over Craftear ........................................................................................................................................................ 281 Q14 9 The curve C has polar equation where 0.01 ≤ 1 ≤ 12 0. r = 5 cot 1, (i) Find the area of the finite region bounded by C and the line 1 = 0.01, showing full working. Give your answer correct to 1 decimal place. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Let P be the point on C where 1 = 0.01. (ii) Find the distance of P from the initial line, giving your answer correct to 1 decimal place. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 Craftear ........................................................................................................................................................ 282 (iii) Find the maximum distance of C from the initial line. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iv) Sketch C. © UCLES 2018 [2] 9231/11/O/N/18 [Turn over Craftear ........................................................................................................................................................ 283 Q15 The curve C has polar equation r = a cos 31, for − 16 0 ≤ 1 ≤ 16 0 , where a is a positive constant. (i) Sketch C. [2] (ii) Find the area of the region enclosed by C, showing full working. [3] Craftear 3 4 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 284 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 [Turn over Craftear (iii) Using the identity cos 31 4 cos3 1 − 3 cos 1, find a cartesian equation of C. 285 Q16 11 The curve C has polar equation r = a 1 + sin 1 for −0 < 1 ≤ 0, where a is a positive constant. (i) Sketch C. [2] (ii) Find the area of the region enclosed by C. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 Craftear ........................................................................................................................................................ 286 (iii) Show that the length of the arc of C from the pole to the point furthest from the pole is given by s = ï2a Ô 10 2 − 12 0 1 + sin 1 d1. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 [Turn over Craftear ........................................................................................................................................................ 287 2 (iv) Show that the substitution u = 1 + sin 1 reduces this integral for s to ï2aÔ 0 evaluate s. 1 du. Hence 2 − u [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 Craftear ........................................................................................................................................................ 288 Q17 11 19 Answer only one of the following two alternatives. EITHER A curve C has polar equation r = 2a cos 21 + 12 0 for 0 ≤ 1 < 20, where a is a positive constant. (i) Show that r = −2a sin 21 and sketch C. [4] ........................................................................................................................................................ ........................................................................................................................................................ Craftear ........................................................................................................................................................ (ii) Deduce that the cartesian equation of C is x2 + y2 2 = −4axy. 3 [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 [Turn over 289 (iii) Find the area of one loop of C. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 Craftear ........................................................................................................................................................ 290 (iv) Show that, at the points (other than the pole) at which a tangent to C is parallel to the initial line, 2 tan 1 = − tan 21. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 [Turn over Craftear ........................................................................................................................................................ 291 Q18 OR The polar equation of a curve C is r = a 1 + cos 1 for 0 ≤ 1 < 20, where a is a positive constant. (i) Sketch C. (ii) Show that the cartesian equation of C is x2 + y2 = a x + x2 + y2 . [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 [Turn over Craftear [2] 292 (iii) Find the area of the sector of C between 1 = 0 and 1 = 13 0 . [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 Craftear ........................................................................................................................................................ 293 (iv) Find the arc length of C between the point where 1 = 0 and the point where 1 = 13 0 . [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 Craftear ........................................................................................................................................................ Q19 4 294 A curve C has polar equation r2 = 8 cosec 21 for 0 < 1 < 12 0. Find a cartesian equation of C. [3] Sketch C. [2] Determine the exact area of the sector bounded by the arc of C between 1 = 16 0 and 1 = 13 0, the half-line 1 = 16 0 and the half-line 1 = 13 0. [3] Craftear [It is given that Ó cosec x dx = ln tan 12 x + c.] © UCLES 2016 9231/11/M/J/16 [Turn over Q20 7 A curve has polar equation r = equation of the curve. 295 1 , for 0 < 1 < 20. Find, in the form y2 = f x, the cartesian 1 − cos 1 [3] Hence sketch the curve, and shade the region whose area is given by 1 Ô 2 Using the cartesian equation of the curve, find the area of this region. 8 30 2 10 2 1 d1. 1 − cos 12 [3] [3] The cubic equation z3 − z2 − z − 5 = 0 [4] Find the value of !4 + "4 + ' 4 . [2] Craftear has roots !, " and ' . Show that the value of !3 + "3 + ' 3 is 19. © UCLES 2016 9231/13/M/J/16 [Turn over Q21 5 296 The curves C1 and C2 have polar equations C1 : C2 : 1 , for 0 ≤ 1 < 20, ï2 r = sin 12 1 , for 0 ≤ 1 ≤ 0. r= [2] Sketch C1 and C2 on the same diagram. [3] Find the exact value of the area of the region enclosed by C1 , C2 and the half-line 1 = 0. [4] Craftear Find the polar coordinates of the point of intersection of C1 and C2 . © UCLES 2015 9231/11/M/J/15 [Turn over 1 297 p, q and r are real constants, has two pairs of equal The quartic equation x4 − px2 + qx − r = 0, where 2 roots. Show that p + 4r = 0 and state the value of q. [6] Q22 The curve C has polar equation r = e41 for 0 ≤ 1 ≤ !, where ! is measured in radians. The length of C is 2015. Find the value of !. [6] Craftear 2 © UCLES 2015 9231/13/M/J/15 [Turn over Q23 298 11 OR The curve C has polar equation r = a 1 − cos 1 for 0 ≤ 1 < 20. Sketch C. [2] Find the area of the region enclosed by the arc of C for which 21 0 ≤ 1 ≤ 32 0, the half-line 1 = 12 0 and the half-line 1 = 32 0. [5] Show that @ A ds 2 = 4a2 sin2 12 1 , d1 [7] Craftear where s denotes arc length, and find the length of the arc of C for which 21 0 ≤ 1 ≤ 32 0. © UCLES 2015 9231/11/O/N/15 [Turn over Q24 5 299 The curve C has polar equation r = a1 + sin , where a is a positive constant and 0 ≤ < 2. Draw a sketch of C. [2] Find the exact value of the area of the region enclosed by C and the half-lines = 13 and = 23 . [4] The linear transformation T : 4 → 4 is represented by the matrix M, where Craftear 6 © UCLES 2014 9231/11/M/J/14 [Turn over Q25 The curve C has cartesian equation x2 + y2 polar equation of C is r2 = a2 sin 21. 2 = 2a2 xy, where a is a positive constant. Show that the [3] Sketch C for −0 < 1 ≤ 0. [2] Find the area enclosed by one loop of C. [2] Craftear 4 300 © UCLES 2014 9231/13/M/J/14 [Turn over Q26 A circle has polar equation r = a, for 0 ≤ < 2, and a cardioid has polar equation r = a1 − cos , for 0 ≤ < 2, where a is a positive constant. Draw sketches of the circle and the cardioid on the same diagram. [3] Write down the polar coordinates of the points of intersection of the circle and the cardioid. [2] Show that the area of the region that is both inside the circle and inside the cardioid is 5 − 2 a2 . 4 6 Craftear 8 301 © UCLES 2014 9231/11/O/N/14 [Turn over Q27 Find the area of the region enclosed by the curve with polar equation r = 2 1 + cos 1, for 0 ≤ 1 < 20. [4] Craftear 1 302 © UCLES 2013 9231/11/M/J/13 [Turn over Q28 10 303 Use the identity 2 sin P cos Q sin P + Q + sin P − Q to show that 2 sin 1 cos 1 − 14 0 cos 21 − 34 0 + 1 . ï2 3 A curve has polar equation r = 2 sin 1 cos 1 − 14 0, for 0 ≤ 1 ≤ 43 0. Sketch the curve and state the polar equation of its line of symmetry, justifying your answer. [3] [6] Craftear Show that the area of the region enclosed by the curve is 83 0 + 1. © UCLES 2013 9231/13/M/J/13 [Turn over Q29 The curve C has polar equation r = 2e1 , for 61 0 ≤ 1 ≤ 12 0. Find (i) the area of the region bounded by the half-lines 1 = 16 0, 1 = 12 0 and C, [2] (ii) the length of C. [3] Craftear 1 304 © UCLES 2013 9231/11/O/N/13 [Turn over Q30 10 305 The curve C has polar equation r = 2 sin 1 1 − cos 1, for 0 ≤ 1 ≤ 0. Find coordinates of the point of C that is furthest from the pole. dr and hence find the polar d1 [5] [2] Find the exact area of the sector from 1 = 0 to 1 = 14 0. [6] Craftear Sketch C. © UCLES 2013 9231/13/O/N/13 [Turn over Q31 4 306 The curve C has polar equation r = 2 + 2 cos θ , for 0 ≤ θ ≤ π . Sketch the graph of C. [2] Find the area of the region R enclosed by C and the initial line. [4] Craftear The half-line θ = 15 π divides R into two parts. Find the area of each part, correct to 3 decimal places. [3] © UCLES 2012 9231/11/M/J/12 [Turn over 307 Q32 1 √ Find the cartesian equation corresponding to the polar equation r = ( 2) sec(θ − 14 π ). [3] Craftear √ Sketch the the graph of r = ( 2) sec(θ − 14 π ), for − 14 π < θ < 34 π , indicating clearly the polar coordinates of the intersection with the initial line. [2] © UCLES 2012 9231/12/O/N/12 [Turn over Q33 The curve C has polar equation r = 1 + 2 cos θ . Sketch the curve for − 23 π ≤ θ < 23 π . [2] Find the area bounded by C and the half-lines θ = − 13 π , θ = 13 π . [4] Craftear 5 308 © UCLES 2012 9231/13/O/N/12 [Turn over Q34 309 11 EITHER The curve C has cartesian equation (x2 + y2 ) = a2 (x2 − y2 ), 2 where a is a positive constant. Show that C has polar equation r2 = a2 cos 2θ . [2] [2] Find the area of the sector between θ = − 14 π and θ = 14 π . [3] Find the polar coordinates of all points of C where the tangent is parallel to the initial line. [7] Craftear Sketch C for −π < θ ≤ π . © UCLES 2012 9231/13/M/J/12 [Turn over POLAR COORDINATES MS Craftear 310 Q1 5(c) 5(b) 5(a) Question 1 3 ) 3 = 34 3 − 13 π ) ( 2 ) x +y r cos θ + 3r sin θ 3r cos θ − r sin θ 2 x+ y 3 = x 3− y ( 1 3 cos π3 cos θ + sin π3 sin θ sin π3 cos θ − cos π3 sin θ 1 6 x2 + y 2 = 2 r=2 0 2 ( 3− π− 1 3 2 2  cosec ( π − θ ) −1 dθ = 2 cot ( π − θ ) − θ  1π 6 4cot 2 ( 13 π − θ ) dθ 2  1π 1 6 2 0 a cot 16 π = 2 3  a = 2 θ =0 Answer 0 1π 6 Craftear Guidance  3 A1 AG. M1 Applies r = x 2 + y 2 , x = r cos θ and y = r sin θ . M1 Uses the identities for cos( A − B) and sin( A − B). Or identity for tan (A – B). 4 A1 OE, must be exact. M1 A1 Uses cot 2 ( 1 π − θ ) = cosec2 ( 1 π − θ ) − 1 and integrates. 3 3 M1 Uses 1 r 2 dθ with correct limits. 2 3 B1 Substitutes θ = 1 π . AG. 6 B1 B1 Initial line with correct position and shape of curve. SCB1 if correct shape with wrong starting position. Pole position must be clear. Marks 311 Q2 5(b) 5(a) Question 1π 1 2 2 2 1  1 −  dθ   π −θ π  1 1π 2 θ = 0] = 3 − 4ln 2 4π 1 2 2 π 1 1 2  1  3 2 1  −  + ln π   =  + ln   + ln + 2  π π 2 2π  π π   2  2π π 2  2 1 1 θ 2 + ln( + θ π − ) 2  π − θ π π 2  0 1 2 2 0  ( π − θ ) − π ( π − θ ) + π dθ  1π 1 2 2 0 [O] Answer Craftear B1 Approximately correct curve passing through the pole, O, with correct domain. Guidance  6 A1 AG M1 Substitute limits into an expression of the correct form and simplify the log terms. M1 A1 Integrates all terms to obtain correct form. M1 Expands. M1 Uses 1 r 2 dθ with correct limits. 2 3 B1 Correct form at O. Approx tangential to initial line at O. B1 r strictly increasing over the domain 0 to π . 2 Marks 312 Q3 6(c) Question 6(b) 6(a) 6 ) 4c 2 (1 + s)2 dθ = 2 2 2 0 2 1 4 2 2 2  c + 2sc + s c dθ 1π 2 Answer θ =0 1π 2 0 2 2 1 4 1 4  ( cos 2θ + 1) + 2 sin θ cos θ + sin 2θ dθ 1π 2 1 1π 3, 16 π 1π 5 π + 43 8 3 1 4 1 5 2  4 θ + 2 sin 2θ − 3 cos θ − 16 sin 4θ  0 0  cos 2θ + 1 + 4 sin θ cos θ + − cos 4θ dθ 2  3 2 1 2 2 0 ( Craftear Finds dr and sets equal to 0. dθ increasing to π then decreasing. 6 Correct shape. Tangential to θ = π at pole, r strictly 2 Guidance 6 A1 M1 Integrates. A1 Obtains integrable form. M1 Uses double angle formulae on first and last terms. M1 A1 Uses 1 r 2 dθ with correct limits. 2 Marks 2 B1 Correct position. All in first quadrant. B1 5 A1 Allow ( 2.60, 1 π ) 3sf 6 A1 1 − s − 2s 2 = 0 sin θ = 12 leading to θ M1 2 ( c(c) − (1 + s)s ) = 0 313 Q4 Answer Line l parallel to initial line and correct side of pole. 5 A1 A1 SC1 For finding both angles correctly Forms quadratic in sin θ . Or in r r = 3 + ( 4, 16 π ) ( 4, 56 π ) M1 B1 1 M1 Solves for sin θ . Craftear Guidance B1 Correct symmetrical shape, closed loop. Marks sin θ = 12 2 = 3sin θ + 2sin 2 θ 5(b) 5(a) Question 4 r 314 Q5 7(a) Question 5(c) − π2 2  ( 3 + 2 sin θ ) dθ π 6 π 6 π 2 ( ( Craftear 4 A1 Applies both double angle formulae, AG. r = sin 4θ B1 Uses x = r cos θ and y = r sin θ . B1 Uses r 2 = x 2 + y 2 . Guidance M1 Applies at least one double angle formula. Marks 6 A1 M1 Adds area of triangle. A1 A1 M1 Uses double angle formula and integrates. M1 Finds the part of the required area enclosed by the curved outer edge and two line segments from the pole. Limits must be correct. r = 2sin 2θ cos 2θ ) Answer r 5 = 4r 4 cos θ sin θ cos 2 θ − sin 2 θ ) 3 22 π − 52 3 r 5 = 4 xy x 2 − y 2 3 + (4cos 6π ) × 2 22 π − 132 3 [11θ − 12cosθ − sin 2θ ]− = 223 π − 132 3 − π2  9 +12sin θ + 2 (1 − cos 2θ ) dθ π 6 2 × 12 315 7(d) 7(c) 7(b) 2  = 14 0 Craftear A1 Substituting θ = ±0.369 gives maximum value of | x | . θ = 0.369  x = 0.93 (or θ = −0.369  | x |= 0.93 ) 6 B1 Allow use of decimals. ) tan 2 θ = 14 7 ± √ 41  θ = ±0.369, ± 1.071 ( M1 Forms quadratic in tan 2 θ . Must see consideration of cos θ = 0. cos θ = 0 or 2 tan 4 θ − 7 tan 2 θ + 1 = 0 ) A1 ( cos θ 2sin 4 θ − 7 sin 2 θ cos 2 θ + cos 4 θ = 0 B1 Uses x = r cos θ . 4 A1 M1 A1 Applies double angle formula and integrates. M1 Differentiates and sets equal to 0. )  M1 Uses 1 r 2 dθ with correct limits. 2 3 B1 States the equation of the line of symmetry. −4sin 2 θ cos3 θ + cos5 θ + 2sin 4 θ cos θ − 3sin 2 θ cos3 θ = 0 x = 4 sin θ cos 4 θ − sin 3 θ cos 2 θ ( 1π 1 − cos8θ dθ = 14 θ − 18 sin 8θ  04 1π 4 = 161 π 0  sin 4θ dθ 1 2 1π 4 θ = 18 π B1 Correct shape at extremities. B1 Initial line drawn and one loop in the first quadrant. 316 Q6 5(c) 5(b) 5(a) Question ln u du 1+π − 1 1+ π 1 (1 + π)ln(1 + π)(ln(1 + π) − 2) + π 2 Craftear 5 B1 Shows sign change. Values correct to 2sf should be shown. (1 + π −1.2) ln(1 + π −1.2) − tan1.2 = 0.602 and (1 + π −1.3) ln(1 + π −1.3) − tan1.3 = −0.634 M1 A1 Sets derivative equal to zero. B1 Uses y = r sinθ 6 A1 AG A1 M1 Integrates by parts again (or uses known result for integral of ln u) M1 A1 Integrates by parts once. M1 Uses correct formula with correct limits. 3 B1 May be seen on their diagram. Allow (1.42, 0) B1 Section 12 π ≤ θ ≤ π correct, becoming tangential at θ = π. A1 AG = Guidance B1 Correct shape, r decreasing. Marks cos θ ≠ 0  (1 + π − θ ) ln(1+ π − θ ) − tan θ = 0 dy sin θ = ln(1 + π − θ ) cosθ − =0 dθ 1+ π − θ y = ln(1 + π −θ )sinθ   1du  ln 2 u du 1 (1 + π) ln 2 (1 + π) − (1 + π) ln(1 + π) + π 2 1 =  12 u ln 2 u − u ln u + u  1 1+π −  [u ln u ]1  1 1+π =  12 u ln 2 u  1  − 1+π 1+π 1 θ )dθ = 12  1+π =  12 u ln 2 u   π 1 ln 2 (1 + π − 2 0 ( ln(1+ π),0 ) θ =0 Answer 317 Q7 7(c) 7(b) 7(a) Question B1 B1 2(0.6) tan 0.6 − 1 = −0.179 and 2(0.7) tan 0.7 − 1 = 0.179 Craftear (B1 for initial line drawn and C1 correct, B1 for C2 correct, B1 for intersection correct) 3 B1 B1 B1 2 A1 1 4 ( π √ 2, π ) 1 8 M1 θ cos θ = θ sin θ  tan θ = 1 5 A1 M1 A1 Marks cos θ ≠ 0  2θ tan θ − 1 = 0 dx = −2θ cosθ sin θ + cos 2 θ = 0 dθ x = θ cos 2 θ Answer 318 Q8 5(a) Question 7(d) 0 2  θ cos 2θ dθ 1π 4 2 2   Maximum distance of C from the pole is a. = 641 π 2 − 18 0 4 = 14 θ 2 sin 2θ + θ cos 2θ − 12 sin 2θ  1π 1π 1π 1π 4   4 cos 2θ dθ  = 12   12 θ 2 sin 2θ  −  − 12 θ cos 2θ  04 − 12 0 0   1π 1π 4   4 = 12   12 θ 2 sin 2θ  − θ sin 2θ dθ  0 0   = 12 2  θ ( cos θ − sin θ ) dθ 1π 1 4 2 0 Craftear Answer 2 B1 B1 Marks 7 A1 A1 M1 M1 A1 M1 M1 319 5(d) 5(c) 5(b) 0 2 2 y x ( ) 1π 4 0 ( ) AG ( a cos 14 π ) ( a sin 14 π ) − 12 a 2 (1 − 14 π ) = 14 a 2 ( 12 π − 1) √ a2 − x2 x2 (A1 FT their (b)) 1 2 y= x2 x2 + y 2 = a2 y 2  y 2 a 2 − x2 = x4 ) x2 + y 2 = a ( 1 2  sec θ −1dθ = a [ tan θ − θ ] 1π 4 = 12 a 2 (1 − 14 π ) = 12 a 2 2  tan θ dθ 1π 1 2 4 a 2 0 Craftear 11 M1 A1 FT 3 A1 M1 B1 4 A1 M1 A1 M1 320 Q9 11O(ii) 11O(i) Question ( ) ) ) r = 2 1 + 2 + 2 −1 = 4 2 ( α = ln 1 + 2 ( e 2α − 2eα − 1 = 0 ⇒ eα = 1 + 2 e 2α − e −2α = 2 eα + e −α ⇒ eα − e −α = 2 Answer Craftear Guidance C2 has correct shape. B1 3 B1 Intersection points positioned correctly. C1 has correct shape. B1 6 M1 A1 Substitutes to find r . A1 Must be exact. M1 A1 Forms quadratic in eα , AG. M1 Sets equations equal and divides by eα + e −α . Marks 321 Q10 11E(i) Question 11O(iii) Question ) −θ 2θ e +e θ ) 2 0 −2θ ( ) 0 ∫ ( ln 1+√2 e −e 2θ ( θ Answer −2 ) dθ 4 −4 Uses 5 B1 Shows sign change. 2 ( 0.6 ) tan ( 0.6 ) −1 = −0.179 and 2 ( 0.7 ) tan ( 0.7 ) −1 = 0.179 1 A1 AG 1 Guidance M1 A1 Sets derivative of r cosθ equal to zero. M1 Uses x = r cosθ . Marks 5 A1 Guidance 1 2 ∫ r dθ to formulate correct area. 2 M1 A1 Expands and integrates. M1 A1 Marks 1 − −θ 2 sin θ + θ 2 cos θ = 0 ⇒ cos θ = 2θ sin θ ⇒ 2θ tan θ = 1 2 Craftear ) − (1 + 2 ) − 81  (1 + 2 ) − (1 + 2 )  = 5.82 2 ( ln 1+√2 ) 2 1 1   1  d  1 −  2θ 2 cosθ  = 2  −θ 2 sin θ + θ 2 cosθ  = 0 dθ  2    x = 2θ 1 2 cos = 5ln(1 + √ 2) + 1 + 2 −2θ 1 1 − e 4θ − e −4θ dθ 2 2 1 dθ − 2 ∫ 5 + 2e + 2e ln 1+√2 ( 0 ∫ ( ) 1 1   = 5θ + e 2θ − e−2θ − e4θ + e−4θ  8 8  0 = 2 ( ln 1+√2 Answer 322 11E(iv) 11E(iii) 11E(ii) ∫ ∫ π 4 1 2 ∫( π 4 0 ) ( secθ − 2 ) = 0 ⇒ θ = π4 Craftear 5 A1 AEF, must be exact. 2  1 π  π  1  π  1 ππ  1 − − + ln 2  = ln 2 +  − 1 .       4 8  2  2   4  2 84   π A1 ∫ ( 2θ − tan θ ) dθ M1 A1 Integrates by parts. M1 Forms correct integral. 2 B1 Intersection correct. B1 Correct shape. 2 M1 A1 Finds value of θ . π 1 1 θ ( 2θ − tan θ )  04 − θ 2 + ln cos θ  4 0 2 2 1 θ ( 2θ − tan θ ) 2 4 π 1  04 − 20 π 1 1 1 2θ θ − θ sec 2 θ dθ = θ 2 − sec 2 θ dθ 20 20 20 π 4 2θ = θ sec θ ⇒ θ 2 323 Q11 2(ii) 2(i) Question ( ∫ 2π ) 1 ∫ 2π+1 1 ln (1 + θ ) dθ 20 2π+1 2 π +1 − [u ]1 1  A =  π +  ln ( 2π + 1) − π 2  1 1 ∫ ln u du = [u ln u ] 2π+1 0 2π A= Answer Craftear θ =0 Guidance States 1 2 ∫ r dθ with correct expression and limits. 2 5 A1 AEF, must be exact. M1 A1 Integrates ln u (by parts or otherwise). M1 Applies given substitution correctly, changes their limits. M1 2 B1 The initial line is tangential to C at the pole. B1 Correct shape, domain and orientation. Marks 324 Q12 3(iii) 3(ii) 3(i) Question π ∫ π 4 1 4 cos 4θ + 1dθ = 4 − π4 ) 3 Thus x 2 + y 2 2 = x 2 − y 2 ( r = cos 2 θ − sin 2 θ OE π 1 1 4 = 0.393 sin 4θ + θ  =  π 8 4 4 − = ∫ π 1 4 cos 2 2θ dθ 2 − π4 Answer 1 Craftear Guidance 2 B1 Single correct loop. B1 Correct position including label of 1 on initial line, and symmetric about initial 3 M1 A1 Uses x = r cos θ or y = r sin θ or both , AEF B1 Uses trig identity 3 A1 M1 Correct use of double angle formula M1 Correct integral Marks 325 Q13 8(iii) 8(ii) 8(i) Qu uestion 3 π ∫ cos 2θ + 1dθ 2 π π 1 2 Area = π π a2 a2 3 3 − a 2  − +  = − 6 2  6 2 3 π a2 2 1 Craftear  π  3 π  π 3 2 +   = a 2  − = 2a  sin 2θ + θ  = 2a 2  −    2 4 3   3 2  2 π    3 = 2a 2 3 π 4a 2 cos 2 θ dθ ∫ 2 π  π  2π    a,  and  a,  3   3 a = 2a cos θ ⇒ cos θ = ± Answ wer Guidance Uses cos 2 θ = 1 ( cos 2θ + 1) 2 10 M1 π a2 A1FT M1 for subtracting ‘their’ OP1P2 from 6 A1 Integrates correctly. M1 M1 Finds area of segment OP1 and OP2 of C2 2 and line of symm metry. B1 Other half of C2 uding r = 2a. B1 Half of C2 inclu B1 Semicircle for C1 including r = a.. ded for A1. A1 Both points need M1 Eliminates r . Marks 326 9(iii) 9(ii) 9(i) Q14 Question 2 π π 1 ⇒y= 5 2 2 ( = 3.54 ) 1 2 5 1 sin 2 2θ − dy 5 sin 2 2θ cos 2θ = 0 or max ( sin 2θ ) = 1 = dθ 2 θ = 0.01 ⇒ y ≈ 0.5 1 y = 5cos 2 θ sin 2 θ = 25 ln sin 0.01 ≈ 57.6 2 25 [ln sin θ ]0.201 2 =− = ∫ 25 cot θ dθ 2 0.01 Answer Craftear Uses 1 2 ∫ r dθ . 2 3 A1 M1 A1 2 A1 Sets Guidance dy = 0 or considers max (AEF). dθ M1 Uses y = r sin θ . 3 A1 A1 M1 Marks 327 Q15 3(i) Qu uestion 9(iv) Answer Craftear a 2 Guiidance 2 orrect position including (a, 0) labellled B1 Co or in table. mities B1 Jusst one loop, correcct shape at extrem Marks B1 Correct shape. B1 Intersecting the initial line only when x = 0 and y = 0. 328 Q16 11(i) Question 3(iii) π 6 − 6 π ∫π ( cos 6θ +1) dθ 6 ) ( ( ⇒ x2 + y 2 ( 2 2 2 ) = ax ( x − 3y ) ⇒ r 4 = ax 4 x 2 − 3r 2 ⇒ x 2 + y 2 ) ( 2 Answer 2 2 2 Craftear ) = ax ( 4 x − 3 ( x + y ) ) 2   x   x  r = a cos θ 4cos 2 θ − 3 ⇒ r = a    4   − 3    r    r   a2  1 π a2 6 sin 6 θ θ + = =  π 4  6 12 − a 4 2 6 − ∫ 1 a 2 cos 2 3θ dθ 2 π 6 π Total: Guidance 2 B1 Correct oriientation and labeelled B1 Sketch of a cardioid Marks 3 A1 Any equivalent cartesian form without fractions. M1 For eliminating θ B1 Uses x = r cosθ and x2 + y 2 = r 2 . 3 A1 M1 Using double angle formula correctly 329 11(iv) 11(iii) 11(ii) ∫ ( ) a2 π 1 + 2sin θ + sin 2 θ dθ 2 −π 2 2 ∫−π π ( ) , when r=2a θ = 2 π 2 1 ∫0 2 −u 2 du AG 1 2 = 2a  −2 ( 2 − u ) 2  = 4a  0 so s = 2a Total: 1 2 − 12 Total: Total: π 1 + sin θ dθ π dr = a cosθ dθ Craftear ∫ and that a 2 1 + 2sin θ + a 2 sin 2 θ + a 2 cos 2 θ dθ = 2a 2 π u = 1 + sin θ ⇒ dduθ = cos θ = 1 − ( u −1) = 2u − u 2 s= Show that when r=0, θ = − 2  a 2   3θ 1  π 3π a =    − 2cos θ − sin 2θ  = 4 2  −π  2  2 ∫  a2  π  1 1  1 + 2sin θ + − cos 2θ  dθ =   − π 2 2 2     A= 4 M1A1 A1 Including limits M1 3 M1A1 Uses correct formula for arc length; AG B1 4 M1A1 M1 M1 330 Q17 11E E(iii) 11E E(ii) 11E E(i) Quesstion 2 1 π 2 π ∫ (1 − cos 4θ ) dθ π 2 1  = a θ − sin 4θ  4   1π = a2 Area of one loop is π π ) 1 = π a2 2 π 2 π 2 ∫ ∫ 1 4a 2sin 2 2θ dθ = a 2 2sin 2 2θ dθ 21 1 ( 3 Craftear y x r = −4a sin θ cosθ = −4a. . ⇒ r 3 = −4axy ⇒ ( x 2 + y 2 2 = −4axy ) r r 1 1   r = 2a  cos 2θ cos π − sinn 2θ sin π  = 2a ( 0 − sin 2θ .1) = −2a sin 2θ 2  2  Answeer To otal: To otal: To otal: Guidancee 5 M1A1 M1A1 M1 OE 2 M1A1 AG 4 B1 Symmetry and shape correct B1 Loops in 2nd and 4th quadrant M1A1 AG Marks 331 Q18 11OR(ii) 11OR Question 11E(iv) x 2 + y 2 = a ( x + (x 2 + y 2 ))  r = a (1 + cosθ ) ⇒ x 2 + y 2 = a  1 +   Answer   x 2 + y 2  x (⇒ 2 tan θ = − tan 2θ ) 2 B1 Cardioid wiith indication of correct scale. 2 A1 Substitutes for r and cos(θ) M1 Craftear Guidance 3 ding at pole, in ap pproximately correect B1 Closed curve starting and end location. Marks Total: A1 AG M1 dy = −2a ( 2cos 2θ sin θ + sin 2θ cosθ ) = 0 dθ 2cos 2θ sin θ = − sin 2θ cos θ B1 y = r sin θ = −2a sin 2θ sin θ 332 11OR(iv) 11OR(iii) 1 π ) π 1 ( ) 2 π 2 2 2 0 1 π ∫ 2 + 2 cosθ dθ 1 π 3 θ 3  = a  4sin  = 2a 2 0  =a 0 ∫ a (1 + 2cosθ + cos θ ) + a ( −sin θ ) dθ 1 π 3 a2 4π + 9 3 16 Arc length = = sin 2θ  3 a 2  3θ =  + 2sin θ + 2 2 4  0 ∫ a cos 2θ  3 =  dθ  + 2cosθ + 2 0 2 2  2 3 1 a 1 + 2 cosθ + cos 2θ dθ Sector area = 2 0 ∫( 2 3 Craftear 5 M1A1 A1 M1A1 4 A1 M1 M1A1 333 334 Using x = r cos θ and y = r sin θ 4 r 2 = 8 cosec 2θ ⇒ r 2 = sinθ cos θ ⇒ r cos θ .r sin θ = 4 ⇒ xy = 4 (in simple form) Sketch: Curve in 1st quadrant with correct concavity, asymptotic to both axes. 1 π 3 ∫ π  1  = 2 ln 3 − ln  =2 ln3orln9 3  Q20 7 M1 A1 [3] B1B1 [2] 1 π 1 8 cosec 2θ dθ =  2 ln tan θ  31 π 21 6 6 B1 x2 + y 2 = 1 1− x ⇒ x2 + y 2 − x = 1 M1A1 A1 [3] M1A1 x2 + y2 ⇒ y 2 = 1 + 2 x or equivalent RHS  1  Sketch: Parabola symmetrical about x-axis. Intercepts at  − , 0  and ( 0 , ±1) .  2  Shading correct area A1 [3] Craftear Q19 4 B1 B1 B1 [3] 3π 2 Recognises 1 1 dθ as the area of the region between parabola and y-axis. 2 π (1 − cos θ )2 ∫ 2 0 = 2× 1 2 ∫ (1 + 2 x ) dx − 1 2 0 3 2 1  = 2  (1 + 2 x ) 2  = 3 −1 3 2 M1 M1A1 [4] 335 Q21 5  1 1  1 1 1 1 , π  (Accept 1.05 for π .) sin θ = ⇒ θ = π ; Intersection at  2 2 3 3  2 3  C1: Circle centre at pole and radius 1/√2 C2: Curve, approx. correct orientation, from (0,0) to (1,π). Completely correct correct shape. 1 1 1 13π 1 × π × − ∫ sin θ dθ 0 6 2 2 2 1 1 1  1π 1 3 π × − 2cos θ  3 = π + = −1 12 2 2 0 12 2 M1A1 (4) Total: 9 α e8θ + 16e8θ dθ = ∫ 0 α 0 17e 4θ dθ = (AEF) (LNR) M1A1 α  17 4θ  17 4α 17 =  e  ⇒ e − = 2015 4 4  4 0 ⇒ e4α = 1955.837 K ⇒ α = 1.89 CAO dM1A1 dM1A1 (6) Total 6 Craftear s=∫ B1 B1 B1 (3) M1A1 Q22 2 M1A1 (2) Q23 11O 1 1 ] Closed curve through pole with correct orientation. Completely correct. π π 3 1 2 × a 2 1 (1− 2cosθ + cos2 θ )dθ = a 2 1  − 2 π π 2 2 2 2  ∫ ∫ sθ + 1 2  s 2θ dθ  1 π 1 3  = a2  θ − 2 nθ + n 2θ  2 4   1π 1 2 3  = a2  π + 2 4   ds     dθ  2 = 2× ∫ = a 2 (1 − 2 sθ + = 2a 2 (1− 1 1 sθ ) = 2a 2 .2 n 2 θ = 4a 2 n 2 θ (AG) 2 2 π 1 2a π 2  = 4a  − 2  = 4 2a 1 n θ dθ 2 π 1  s θ 2  1π s2 θ + n 2 θ ) 1 ] 1 1 1 1 2 1 ] 4 l 336 Q24 1    3  Closed curve through (a, 0),  2a, π  , (a, π),  0, π  . 2    2  Completely correct shape for cardioid. 5 a2 Area = 2 a2 = 2 ∫ 2 π a2 2 3 1 (1+ sinθ ) dθ = π 2 3 ∫ 2 π 3  3 + 2sin θ − 1 cos 2θ  dθ  1  π 2 2  3  B1 [2] M1 (LNR) M1 (LR) M1 2 π 3 2 = ∫ 2 π 2 3 1 (1+ 2sin θ + sin θ )dθ π 3 B1 a 3 1 θ − 2 cos θ − sin 2θ   2 2 4  1π A1 [4] 3 1  1 = a 2  π +1 + 3  (CAO) 8  4 Q25 Use of r2 = x2 + y2 Use of x = r cos θ and y = r sin θ (both). Obtains r2 = a2 sin 2θ (AG) B1 B1 B1 [3] 1 3 Sketch with two loops, approximately symmetrical about θ = π and θ = − π . 4 4 1 2 ∫ 1 π 1 2 2 π   2 a 2 sin 2θ dθ = − a cos 2θ   0  4 0 = B1B1 [2] M1 (LNR) 1 2 a 2 A1 (2) Q26 8 Circle sketched Cardioid – correct location and orientation – correct indentation near pole. (a, π2 ) and (a, 32π ) (B1 for reverse, or a = a(1− cosθ ) ⇒ θ = π2 , 32π seen.) π 2 a 2 1 − cos θ 2 dθ 0 ) ∫ ( = πa + a ∫ (1 − 2 cos θ + cos θ ) dθ Area = 12 πa 2 + 2 × 12 π 2 2 0 π = 12 πa 2 + a 2 2 32 − 2 cos θ + 12 cos 2θ 0 1 2 2 (Half circle + Area of sector) 2 ∫ ( = 12 πa 2 + a 2 [32θ − 2 sin θ + 14 sin 2θ ] 02 = 12 πa 2 + a 2 ( 34 π − 2 ) = ( 54 π − 2 ) a 2 π ) dθ B1 B1B1 (3) B1B1 (2) B1M1 A1 (Use of double angle formula.) M1 (Integration) M1 (AG) A1 (6) [11] Craftear 4 Q27 1 337 1 2 r dθ 2 A= Use of double angle formula and attempt to integrate. = Use of ∫ 1 2 ∫ 2π 0 M1 4(1 + 2 cos θ + cos 2 θ ) dθ 2π M1 ∫ (3 + 4 cos θ + cos 2θ ) dθ 0 Integrates correctly. sin 2θ   = 3θ + 4sin θ + 2  0  2π A1 Finds value. = 6π (CWO) Accept 18.8 A1 [4] Q28 Uses identity. 1 1 1 4 4 1 4 1 1 2 4 √2 M1 2sin θ cos θ – π = sin 2θ – π + sin π Uses sine-cosine link. Obtains result. = cos π – 2θ+ π + Sketches graph. Closed loop through origin, in correct position. Obtains line of symmetry. For line of symmetry 2θ – π = 0 ⇒ θ = π. Uses area of sector formula. Rearranges. = cos 2θ – π + 4 3 1 π 3 3 4 8 3 1 (AG) √2 3 π 4 1 3 3 1 2 2 4 = 0 cos 4θ – π +√2 cos 2θ – π +1 dθ 2 1 = 16 = – Substitutes limits. = 3 3 8 sin 4θ – π + 1 16 1 3 4 8 2 1 + + π – π+1 16 1 2√2 1 – 3 3 θ 4π 4 2 0 sin 2θ – π + 3 B1B1 3 1 M1 A1 B1 A = 04 cos2 2θ – π +√2 cos 2θ – π + dθ 2 4 4 2 Integrates correctly. Obtains given answer. 3 3 M1 A1 Craftear 10 dM1A1 dM1 4 (AG) A1 [12] 6 N.B Method marks are dependent in final par N. If factor missing throughout – award M’s (Max 3) 1 3 3 1 1 If 2 × 08 r2 dθ , penultimate line is = π – – 8 8 2 2 Q29 1 (i) Uses area formula. π π [ ] M1 = eπ − e 3 ( = 20.3) A1 Area = 1 2 2θ 4e dθ = e 2θ π2 2 π6 6 ∫ π Obtains result. (ii) Uses arc length formula. Arc length = π 2 π 6 ∫ 4e 2θ + 4e 2θ dθ = 2 =2 π 2 π2 eθ dθ M1A1 6 π Obtains result. ∫ 2 [ ] 2 eθ π2 = 2 6 π  π  2 e 2 − e 6    (= 8.83) A1 3 [5] Q30 10 338 Differentiates wrt θ dr = 2 cos θ − 2 cos 2θ dθ Equates to zero 2c − 2 2c 2 − 1 = 0 ⇒ 2c 2 − c − 1 = 0 A1 And solves equation ⇒ (2c +1) (c − 1) = 0 M1 1 ⇒ c = − or 1 2 A1 2  3 3, π   3  2 A1 (5) Approximate shape and location Accurate scaling. B1 B1 (2) States required points on C Sketches C. ( 1 2 Uses r dθ 2 1 Area = 2 Obtains an integrable form = ∫ ∫ M1 ) ∫ π 4 4sin 2 θ 1 − 2cosθ + cos 2 θ 0 ( )dθ π 1 4 1 − cos 2θ − 4cosθ sin 2 θ + 1 − cos 4θ  0  4 [ M1 M1 ] dθ  A1A1 π Obtains result = 5 1 2 or 0.0103 π− − 16 2 3 M1 A1 (6) Craftear Integrates  5θ sin 2θ sin 3 θ sin 4θ  4 = − −4 −  2 3 16  0 4 [13] Q31 4 Draws sketch of C. Uses 1 β 2 r dθ 2 α ∫ Uses double angle formula. Shows (4,0) and (0,π) lie on C. Correct shape. (Full cardioid is B1 unless clear evidence of plotting up to 2π or –π to π.) 1 π (4 + 8cos θ + 4 cos 2 θ )dθ 2 0 ∫ π ∫ (2 + 4 cosθ + 2 cos θ )dθ = ∫ (3 + 4 cos θ + cos 2θ )dθ = B1 B1 2 M1 2 0 π 0 M1 π Integrates and obtains area. Finds areas. sin 2θ   = 3θ + 4sinθ + = 3π 2  0  (A1 for correct integral) 3π π π π + 4sin + sin cos = 4.712 5 5 5 5 3π − 4.712 = 4.713 A1A1 CWO 4 M1A1 A1 3 [9] 339 Q32 1 and uses compound angle formula. π  r cosθ −  = 2 4   cos θ sin θ   = 2 r  + 2   2 Changes to cartesian. ⇒ r cosθ + r sinθ = 2 ⇒ x + y = 2 or y = 2 − x . A1 Sketches graph. π to the initial line. 4 Point (2,0) clearly indicated. B1 Re-writes equation Straight line at – M1 A1 3 B1 2 B1 B1 2 [5] Q33 Sketches graph. Uses area of sector formula. Uses double angle formula Integrates. Correct shape and orientation. Passing through (0,0) and (3,0) 1 Area = 2 × 2 = = ∫ π 3 (1 + 2 cosθ ) 2 dθ 0 ∫ π 3 (1 + 4 cos θ + 4 cos 2 θ )dθ 0 ∫ π 3 (3 + 4 cos θ + 2 cos 2θ ) 0 = [3θ + 4 sin θ + sin 2θ ] Obtains result. M1 5   = π + 3 2   π 3 0 dθ M1 A1 A1 4 [6] Craftear 5 340 Q34 EITHE Uses x2 + y2 = r2, (x2 + y2)2 = a2(x2 – y2) x = r cos θ and y = r sin θ.  x2 y2  M1 ⇒ r 2 = a 2  2 − 2  r  r = a2 (cos2 θ – sin2 θ) = a2 cos 2θ (AG) One mark for each loop, or half of whole curve. Uses sector area formula. Sketches C. Area = π  1 4 2  π a cos 2θ dθ  = 2 −4  ∫ ∫ π  4 a 2 cos 2θ dθ   0  A1 2 B2,1,0 2 M1 π S. Omission of S.C. factor, but correct integration gets B1. Differentiates. Puts y′ = 0 . Obtains coordinates.  sin 2θ  4 a2 = a2  =  2  0 2 2( x 2 + y 2 )(2 x + 2 yy ′) = a 2 (2 x − 2 yy ′) y ′ = 0 ⇒ 2 x( x 2 + y 2 ) = a 2 x a ⇒ 2r 2 = a 2 ⇒ r = (r ≥ 0) 2 1 ⇒ cos 2θ = 2 B1B1 M1 ⇒θ = ± A1A1 i. Alternatively for last 7 marks: Obtains condition for tangent parallel to initial line. Differentiates equation of C. Forms equation. Solves for tanθ, and θ. Writes coordinates of points. A1A1 π 5π , ± 6 6 A1 M1 7 π 5π   a  a ,±  and  ,±   6 6   2   2 [14] dy dy dr =0⇒ = 0 ⇒ r cos θ + sin θ = 0 dx dθ dθ (M1A1) a2 dr dr = −2a 2 sin 2θ ⇒ = − sin 2θ r dθ dθ (A1) 2r 3 a2 sin 2θ sin θ = 0 r ∴ a 2 cos 2θ cosθ = a 2 sin 2θ sinθ 1 ∴ = tan θ tan 2θ 1− t2 ∴ = t ⇒ 2t 2 = 1 − t 2 ⇒ 3t 2 = 1 2t 1 ∴t = ± 3 π 5π ∴θ = ± , ± 6 6 5π  π  a  a ,±   ,±   6   2 6 ,  2 ∴ r cos θ − (Award final A1 if r found but result not written out.) Craftear 11 (M1) (M1) (A1) (A1) (7) [14] 341 Craftear VECTORS 342 Q1 6 Let t be a positive constant. The line l1 passes through the point with position vector ti + j and is parallel to the vector - 2i - j. The line l2 passes through the point with position vector j + tk and is parallel to the vector - 2j + k . It is given that the shortest distance between the lines l1 and l2 is 21. (a) Find the value of t. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The plane P 1 contains l1 and is parallel to l2 . (b) Write down an equation of P 1 , giving your answer in the form r = a + mb + nc . [1] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 Craftear ............................................................................................................................................................ 343 13 The plane P 2 has Cartesian equation 5x - 6y + 7z = 0 . (c) Find the acute angle between l2 and P 2 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Find the acute angle between P 1 and P 2 . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/M/J/21 [Turn over Craftear ............................................................................................................................................................ 344 Q2 6 The lines l1 and l2 have equations r =- i - 2j + k + s (2i - 3j) and r = 3i - 2k + t (3i - j + 3k) respectively. The plane P 1 contains l1 and the point P with position vector - 2i - 2j + 4k . (a) Find an equation of P 1 , giving your answer in the form r = a + mb + nc . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The plane P 2 contains l2 and is parallel to l1 . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 Craftear (b) Find an equation of P 2 , giving your answer in the form ax + by + cz = d . 345 (c) Find the acute angle between P 1 and P 2 . [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 [Turn over Craftear ............................................................................................................................................................ 346 (d) The point Q is such that OQ =- 5 OP . Find the position vector of the foot of the perpendicular from the point Q to P 2 . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 Craftear ............................................................................................................................................................ 347 Q3 5 8 The plane P has equation r = - 2i + 3j + 3k + m (i + k) + n (2i + 3j) . (a) Find a Cartesian equation of P , giving your answer in the form ax + by + cz = d . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The line l passes through the point P with position vector 2i - 3j + 5k and is parallel to the vector k. (b) Find the position vector of the point where l meets P . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 Craftear ............................................................................................................................................................ 348 (c) Find the acute angle between l and P . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Find the perpendicular distance from P to P . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 [Turn over Craftear ............................................................................................................................................................ 349 Q4 7 The points A, B, C have position vectors 2i + 2j, - j + k and 2i + j - 7k respectively, relative to the origin O. (a) Find an equation of the plane OAB, giving your answer in the form r.n = p . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ The plane P has equation x - 3y - 2z = 1. (b) Find the perpendicular distance of P from the origin. [1] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 Craftear ............................................................................................................................................................ 350 (c) Find the acute angle between the planes OAB and P . [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (d) Find an equation for the common perpendicular to the lines OC and AB. [10] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 [Turn over Craftear ............................................................................................................................................................ 351 ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/12/O/N/21 Craftear ............................................................................................................................................................ 352 Q5 4 The points A, B, C have position vectors - i + j + 2k , - 2i - j, 2i + 2k , respectively, relative to the origin O. (a) Find the equation of the plane ABC, giving your answer in the form ax + by + cz = d. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 Craftear ............................................................................................................................................................ 353 (b) Find the perpendicular distance from O to the plane ABC. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find the acute angle between the planes OAB and ABC. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 [Turn over Craftear ............................................................................................................................................................ 354 Q6 7 The points A, B, C have position vectors - 2i + 2j - k , - 2i + j + 2k , - 2j + k , respectively, relative to the origin O. (a) Find the equation of the plane ABC, giving your answer in the form ax + by + cz = d. [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the acute angle between the planes OBC and ABC. [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 Craftear ............................................................................................................................................................ 355 The point D has position vector ti - j. (c) Given that the shortest distance between the lines AB and CD is 10, find the value of t. [6] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/12/O/N/20 Craftear ............................................................................................................................................................ 356 Q7 5 The lines l1 and l2 have equations r = 3i + 3k + m (i + 4 j + 4k) and r = 3i - 5j - 6k + n (5j + 6k) respectively. (a) Find the shortest distance between l1 and l2 . [5] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 Craftear ............................................................................................................................................................ 357 11 The plane P contains l1 and is parallel to the vector i + k . (b) Find the equation of P, giving your answer in the form ax + by + cz = d . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (c) Find the acute angle between l2 and P. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/11/M/J/20 [Turn over Craftear ............................................................................................................................................................ 358 Q8 7 The lines l1 and l2 have equations r = - 5j + m (5i + 2k) and r = 4i + 2j - 2k + n (j + k) respectively. The plane P contains l1 and is parallel to l2 . (a) Find the equation of P, giving your answer in the form ax + by + cz = d . [4] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Find the distance between l2 and P. [3] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 Craftear ............................................................................................................................................................ 359 15 The point P on l1 and the point Q on l2 are such that PQ is perpendicular to both l1 and l2 . 22 (c) Show that P has position vector 55 27 i - 5j + 27 k and state a vector equation for PQ. [8] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 Craftear ............................................................................................................................................................ 360 Q9 6 With O as the origin, the points A, B, C have position vectors i − j, 2i + j + 7k, i−j+k respectively. (i) Find the shortest distance between the lines OC and AB. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 Craftear ........................................................................................................................................................ 361 (ii) Find the cartesian equation of the plane containing the line OC and the common perpendicular of the lines OC and AB. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 [Turn over Craftear ........................................................................................................................................................ 362 Q10 3 The lines l1 and l2 have equations r = 6i + 2j + 7k + , i + j and r = 4i + 4j + - −6j + k respectively. The point P on l1 and the point Q on l2 are such that PQ is perpendicular to both l1 and l2 . Find the position vectors of P and Q. [8] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ 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................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 Craftear ................................................................................................................................................................ 363 ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ 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................................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 [Turn over Craftear ................................................................................................................................................................ 364 Q11 7 The line l1 passes through the points A −3, 1, 4 and B −1, 5, 9. The line l2 passes through the points C −2, 6, 5 and D −1, 7, 5. (i) Find the shortest distance between the lines l1 and l2 . [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 Craftear ........................................................................................................................................................ 365 (ii) Find the acute angle between the line l2 and the plane containing A, B and D. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 [Turn over Craftear ........................................................................................................................................................ 366 Q12 10 The line l1 is parallel to the vector ai − j + k, where a is a constant, and passes through the point whose position vector is 9j + 2k. The line l2 is parallel to the vector −ai + 2j + 4k and passes through the point whose position vector is −6i − 5j + 10k. (i) It is given that l1 and l2 intersect. 6. (a) Show that a = − 13 [3] ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 Craftear ................................................................................................................................................ 367 (b) Find a cartesian equation of the plane containing l1 and l2 . [4] ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 [Turn over Craftear ................................................................................................................................................ 368 (ii) Given instead that the perpendicular distance between l1 and l2 is 3 30, find the value of a. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 Craftear ........................................................................................................................................................ 369 Q13 7 The lines l1 and l2 have vector equations r = ai + 9j + 13k + , i + 2j + 3k and r = −3i + 7j − 2k + - −i + 2j − 3k respectively. It is given that l1 and l2 intersect. (i) Find the value of the constant a. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The point P has position vector 3i + j + 6k. (ii) Find the perpendicular distance from P to the plane containing l1 and l2 . [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 Craftear ........................................................................................................................................................ 370 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 [Turn over Craftear (iii) Find the perpendicular distance from P to l2 . 371 Q14 8 The plane 1 has equation ` a ` a ` a 5 −4 0 r= 1 +s 1 +t 1 . 0 3 2 (i) Find a cartesian equation of 1 . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The plane 2 has equation 3x + y − z = 3. (ii) Find the acute angle between 1 and 2 , giving your answer in degrees. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 Craftear ........................................................................................................................................................ 372 (iii) Find an equation of the line of intersection of 1 and 2 , giving your answer in the form r = a + ,b. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 [Turn over Craftear ........................................................................................................................................................ 373 Q15 10 18 The position vectors of the points A, B, C, D are i + j + 3k, 3i + 4j + 5k, −i + 3k, mj + 4k, respectively, where m is a constant. (i) Show that the lines AB and CD are parallel when m = 32 . [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Given that m ≠ 32 , find the shortest distance between the lines AB and CD. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 Craftear ........................................................................................................................................................ 374 (iii) When m = 2, find the acute angle between the planes ABC and ABD, giving your answer in degrees. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 [Turn over Craftear ........................................................................................................................................................ 375 Q16 OR The position vectors of the points A, B, C, D are i + j + 3k, 3i − j + 5k, 3i − j + k, 5i − 5j + !k, respectively, where ! is a positive integer. It is given that the shortest distance between the line AB and the line CD is equal to 2ï2. (i) Show that the possible values of ! are 3 and 5. [7] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 Craftear ........................................................................................................................................................ 376 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Using ! = 3, find the shortest distance of the point D from the line AC, giving your answer correct to 3 significant figures. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 [Turn over Craftear ........................................................................................................................................................ 377 (iii) Using ! = 3, find the acute angle between the planes ABC and ABD, giving your answer in degrees. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © UCLES 2017 9231/11/M/J/17 Craftear ........................................................................................................................................................ 378 Q17 9 The plane 1 passes through the points 1, 2, 1 and 5, −2, 9 and is parallel to the vector i + 2j + 3k. (i) Find the cartesian equation of 1 . [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The plane 2 contains the lines r = 2i − 3j + k + , i − 2j − k and r = 2i − 3j + k + - 2i + 3j − k. (ii) Find the cartesian equation of 2 . [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 Craftear ........................................................................................................................................................ 379 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find the acute angle between 1 and 2 . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 [Turn over Craftear ........................................................................................................................................................ 380 Q18 6 The points A, B and C have position vectors 2i − j + k, 3i + 4j − k and −i + 2j + 4k respectively. (i) Find the area of the triangle ABC. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 Craftear ........................................................................................................................................................ 381 (ii) Find the perpendicular distance of the point A from the line BC. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find the cartesian equation of the plane through A, B and C. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 [Turn over Craftear ........................................................................................................................................................ Q19 Find a cartesian equation of the plane 1 passing through the points with coordinates 2, −1, 3, 4, 2, −5 and −1, 3, −2. [4] The plane 2 has cartesian equation 3x − y + 2z = 5. Find the acute angle between 1 and 2 . [3] Find a vector equation of the line of intersection of the planes 1 and 2 . [4] Craftear 8 382 © UCLES 2016 9231/11/M/J/16 [Turn over 383 Write down matrices P and D such that P−1AP = D, where D is a diagonal matrix, and find P−1 . [5] Write down a matrix C such that C2 = D, and deduce a matrix B such that B2 = A. [4] OR Q20 11: The position vectors of the points A, B, C, D are a = 2i + ,j − 3k, b = 6i + 3j − 2k, c = i + 2j − k, d = i + 7j + 4k respectively. It is given that the shortest distance between the lines AB and CD is 3. (i) Show that ,2 + , − 20 = 0. [7] Craftear (ii) The planes p1 and p2 are the planes through A, B and D corresponding to the two values of , satisfying the equation in part (i). Find the acute angle between p1 and p2 . [7] © UCLES 2016 9231/13/M/J/16 [Turn over Q21 384 11: EITHER The lines l1 and l2 have equations r = 6i − 3j + s 3i − 4j − 2k and r = 2i − j − 4k + t i − 3j − k respectively. The point P on l1 and the point Q on l2 are such that PQ is perpendicular to both l1 and [7] l2 . Show that the position vector of P is 3i + j + 2k and find the position vector of Q. Find, in the form r = a + ,b + -c, an equation of the plane which passes through P and is perpendicular to l1 . [3] The plane meets the plane r = p i + q j in the line l3 . Find a vector equation of l3 . [4] OR Craftear A curve C has parametric equations © UCLES 2016 9231/11/O/N/16 [Turn over 385 Q22 11: The lines l1 and l2 have equations r = 8i + 2j + 3k + , i − 2j and r = 5i + 3j − 14k + - 2j − 3k respectively. The point P on l1 and the point Q on l2 are such that PQ is perpendicular to both l1 and l2 . Find the position vector of the point P and the position vector of the point Q. [8] The points with position vectors 8i + 2j + 3k and 5i + 3j − 14k are denoted by A and B respectively. Find −−→ −−→ (i) AP × AQ and hence the area of the triangle APQ, Craftear (ii) the volume of the tetrahedron APQB. (You are given that the volume of a tetrahedron is 1 × area of base × perpendicular height.) 3 [6] © UCLES 2015 9231/11/M/J/15 [Turn over Q23 8 386 A line, passing through the point A 3, 0, 2, has vector equation r = 3i + 2k + , 2i + j − 2k. It meets the plane , which has equation r. i + 2j + k = 3, at the point P. Find the coordinates of P. [3] Write down a vector n which is perpendicular to , and calculate the vector w, where w = n × 2i + j − 2k. 3 Craftear The point Q lies in and is the foot of the perpendicular from A to . Use the vector w to determine [4] an equation of the line PQ in the form r = u + -v. © UCLES 2015 9231/13/M/J/15 [Turn over Q24 387 11 EITHER The points A, B and C have position vectors i, 2j and 4k respectively, relative to an origin O. The point N is the foot of the perpendicular from O to the plane ABC. The point P on the line-segment ON is such that OP = 34 ON . The line AP meets the plane OBC at Q. Find a vector perpendicular to 4 . [4] the plane ABC and show that the length of ON is 21 [5] Show that the acute angle between the planes ABC and ABQ is cos−1 23 . [5] Craftear Find the position vector of the point Q. © UCLES 2015 9231/11/O/N/15 [Turn over Q25 The line l1 passes through the points A 2, 3, −5 and B 8, 7, −13. The line l2 passes through the points C −2, 1, 8 and D 3, −1, 4. Find the shortest distance between the lines l1 and l2 . [5] The plane 1 passes through the points A, B and D. The plane 2 passes though the points A, C and D. Find the acute angle between 1 and 2 , giving your answer in degrees. [6] Craftear 11 388 © UCLES 2014 9231/11/M/J/14 [Turn over Q26 389 11: OR With respect to an origin O, the point A has position vector 4i − 2j + 2k and the plane 1 has equation r = 4 + , + 3-i + −2 + 7, + -j + 2 + , − -k, −−→ −−→ where , and - are real. The point L is such that OL = 3OA and 2 is the plane through L which is −−→ −−→ parallel to 1 . The point M is such that AM = 3ML. (i) Show that A is in 1 . [1] (ii) Find a vector perpendicular to 2 . [2] (iii) Find the position vector of the point N in 2 such that ON is perpendicular to 2 . [5] Craftear (iv) Show that the position vector of M is 10i − 5j + 5k and find the perpendicular distance of M from the line through O and N , giving your answer correct to 3 significant figures. [6] © UCLES 2014 9231/13/M/J/14 [Turn over Q27 10 390 The line l1 is parallel to the vector i − 2j − 3k and passes through the point A, whose position vector is 3i + 3j − 4k. The line l2 is parallel to the vector −2i + j + 3k and passes through the point B, whose position vector is −3i − j + 2k. The point P on l1 and the point Q on l2 are such that PQ is perpendicular to both l1 and l2 . Find (i) the length PQ, (ii) the cartesian equation of the plane containing PQ and l2 , . [4] [3] Craftear (iii) the perpendicular distance of A from [5] © UCLES 2014 9231/11/O/N/14 [Turn over Hence find the exact value of S. Q28 391 [2] 11: OR The points A, B, C and D have coordinates as follows: A 2, 1, −2, B 4, 1, −1, C 3, −2, −1 and D 3, 6, 2. The plane 1 passes through the points A, B and C. Find a cartesian equation of 1 . [4] Find the area of triangle ABC and hence, or otherwise, find the volume of the tetrahedron ABCD. [6] [The volume of a tetrahedron is 31 × area of base × perpendicular height.] [4] Craftear The plane 2 passes through the points A, B and D. Find the acute angle between 1 and 2 . © UCLES 2013 9231/11/M/J/13 [Turn over Q29 392 11 EITHER The line l1 passes through the point A whose position vector is 4i + 7j − k and is parallel to the vector 3i + 2j − k. The line l2 passes through the point B whose position vector is i + 7j + 11k and is parallel to the vector i − 6j − 2k. The points P on l1 and Q on l2 are such that PQ is perpendicular to both l1 and l2 . Find the position vectors of P and Q. [8] Craftear Find the shortest distance between the line through A and B and the line through P and Q, giving your answer correct to 3 significant figures. [6] © UCLES 2013 9231/13/M/J/13 [Turn over Q30 8 393 a ` a ` a 2 1 1 The plane 1 has equation r = 3 + s 0 + t −1 . Find a cartesian equation of 1 . −1 1 −2 ` The plane 2 has equation 2x − y + Ï = 10. Find the acute angle between 1 and 2 . [3] [2] Find an equation of the line of intersection of 1 and 2 , giving your answer in the form r = a + ,b. [5] The curve C has parametric equations Craftear 9 © UCLES 2013 9231/11/O/N/13 [Turn over Sketch C. [3] Q31 8 394 The points A, B, C have position vectors 4i + 5j + 6k, 5i + 7j + 8k, 2i + 6j + 4k, respectively, relative to the origin O. Find a cartesian equation of the plane ABC. [4] The point D has position vector 6i + 3j + 6k. Find the coordinates of E, the point of intersection of the line OD with the plane ABC. [4] [3] Craftear Find the acute angle between the line ED and the plane ABC. © UCLES 2013 9231/13/O/N/13 [Turn over Q32 395 11 OR The position vectors of the points A, B, C, D are 2i + 4j − 3k, −2i + 5j − 4k, i + 4j + k, i + 5j + mk, respectively, where m is an integer. It is given that the shortest distance between the line through A and B and the line through C and D is 3. Show that the only possible value of m is 2. [7] [3] 1 Show that the acute angle between the planes ACD and BCD is cos−1 √ . 3 [4] Craftear Find the shortest distance of D from the line through A and C. © UCLES 2012 9231/11/M/J/12 [Turn over Q33 9 396 The plane Π has equation r = 2i + 3j − k + λ (i − 2j + 2k) + µ (3i + j − 2k). The line l, which does not lie in Π , has equation r = 3i + 6j + 12k + t(8i + 5j − 8k). Show that l is parallel to Π . [4] Find the position vector of the point at which the line with equation r = 5i − 4j + 7k + s(2i − j + k) meets Π . [4] [4] Craftear Find the perpendicular distance from the point with position vector 9i + 11j + 2k to l. © UCLES 2012 9231/12/O/N/12 [Turn over Q34 4 397 The points A, B and C have position vectors i + 2j + 2k, 2i + 4j + 5k and 2i + 3j + 4k respectively. −−→ −−→ Find AB × AC. [3] Deduce, in either order, the exact value of (i) the area of the triangle ABC, (ii) the perpendicular distance from C to AB. Craftear [3] © UCLES 2012 9231/13/O/N/12 [Turn over 398 Q35 9 The plane Π1 has parametric equation r = 2i − 3j + k + λ (i − 2j − k) + µ (i + 2j − 2k). [4] The plane Π2 has cartesian equation 3x − 2y − 3ß = 4. Find the acute angle between Π1 and Π2 . [3] Find a vector equation of the line of intersection of Π1 and Π2 . [4] Craftear Find a cartesian equation of Π1 . © UCLES 2012 9231/13/M/J/12 [Turn over VECTORS MS Craftear 399 6(d)  0 5 19     −2  −6  = 5 110 cos α leading to cos α = 5 110  1 7    6(c) 76.8°  −1  5  11     2  −6  = 21 110 cos α leading to cos α = 21 110 47    Acute angle between l2 and Π 2 is 90 − α = 54.1° r = 21 i + j + λ ( 2i + j) + μ ( −2 j + k ) 5  −1  −1 t    0  2 = 21  t = 21 = 4.2 5 21     14 i j k  −1   −2 −1 0 =  2  0 −2 1  4   0   t   −1       1 − 1 = t  0   t  0  1        Answer 6(b) 6(a) Question Q1 Craftear Guidance 3 A1 M1 Dot product using their normal to Π1 . A1 FT 3 A1 M1 A1 Uses dot product of −2j + k and normal. 1 B1 FT Using their value of t. 5 M1 A1 Uses formula for shortest distance. M1 A1 Finds common perpendicular. B1 Finds vector from any point on l1 to any point on l2. Marks 400 6(c) 6(b) 6(a) Question Q2 5 A1 0.845 rad DM1 Dot product using their normal vectors. A1 FT 3  9  32     2  6  = 14 166 cos α leading to cos α = 14 166  1   −7     48.4° M1 A1 Finds vector perpendicular to Π1 . i j k 9  3     2 −3 0 =  6  ~  2  1 0 −3  3   1  4 A1 9 x + 6 y − 7 z = 41 M1 A1 OE. Finds vector perpendicular to Π 2 . 2 B1 FT OE. FT their i – 3k M1 Uses point on Π 2 . Craftear Guidance B1 OE. Finds direction vector from P to a point of l1. Marks −9(3) − 6(0) + 7( −2) = −41 i j k  −9    2 −3 0 =  −6  3 −1 3  7  r = −i − 2 j + k + λ ( 2i − 3 j ) + μ ( i − 3k ) r = −2i − 2 j + 4k + λ ( 2i − 3j) + μ ( i − 3k ) − ( −2i − 2 j + 4k ) + ( −i − 2 j + k ) = i − 3k Answer 401 5(a) Question Q3 6(d) 4 A1 −3 x + 2 y + 3z = 21 Craftear Guidance M1 A1 Finds vector perpendicular to Π. Marks M1 Substitutes point on Π. Answer 4 A1 DM1 Substitutes into the equation of Π 2 . M1 A1 FT Scales OP by a factor of ±5 and uses multiple of their normal to Π 2 . −3(−2) + 2(3) + 3(3) = 21 i j k  −3    1 0 1 = 2  2 3 0  3   −7    2  t = − 32 leading to OF =  1   − 19   2 9 (10 + 9t ) + 6 (10 + 6t ) − 7 ( −20 − 7t ) = 41 leading to 290 + 166t = 41  −2   9   10 + 9t            OF = OQ + QF = −5  −2  + t  6  =  10 + 6t   4   −7   −20 − 7t        402 5(d) 5(c) 5(b) Craftear A1  2    −3   11     −2   −3  1     18 3  2 = = 3.84 22     22 2   3   0   2   −2         0  −  −3  =  3  7  5   2        Acute angle between l and Π is 90 − α = 39.8°  0   −3  3     0  2  = 1 22 cos α leading to cos α = 22 1   3     M1 Substitutes into the equation for Π. −3(2) + 2(−3) + 3(5 + t ) = 21 leading to t = 6 3 M1 A1 Uses dot product of their direction and normal vectors. B1 Finds direction vector from P to plane. 3 A1 M1 Uses dot product of k and their normal. A1 FT 3 B1 Forms general point on line (given as a single vector).  2     −3  5 + t    403 7(c) 7(b) 7(a) Question Q4 = 14 1 22.2° 1 1  6     −1 −3  = 3 14 cos α leading to cos α = 3 14  −1   − 2     12 + 32 + 22 1 1   r.  −1 = 0  −1   i j k  2 1     n = 2 2 0 =  −2  ~  −1  0 −1 1  −2   −1 Answer Craftear Guidance 3 A1 Accept 0.388 radians. Mark final answer. M1 Takes dot product of normal vectors. A1 FT 1 B1 Divides by magnitude of the normal to Π 0.267 3 A1 M1 A1 Finds common perpendicular. Marks 404 7(d)  −0.2   5     r =  −0.1  + k  −3  1  0.7       2  1 1  λ = − leading to OP = −  1  10 10    −7   2 − 2 μ − 2λ   2      2 − 3μ − λ  1  = 0  14 μ + 54λ = 6     μ + 7 λ   −7  Craftear 10 B1 FT FT using their common perpendicular.   M1 A1 Solves for λ and substitutes into OP . A1 Deduces second equation. A1 Deduces one equation.   M1 Uses that dot product of PQ with line direction is zero,  or, alternatively, PQ is a multiple of the common perpendicular (parameter k not 1).  M1 A1 Finds PQ .  2  2 − 2μ   2 − 2 μ − 2λ        leading to    OP = λ  1  , OQ =  2 − 3μ  PQ =  2 − 3μ − λ   −7   μ   μ + 7λ         2 − 2 μ − 2 λ   −2   2 − 2 μ − 2λ   5         2 − 3μ − λ  −3  = 0 or  2 − 3μ − λ  = k  −3  1  μ + 7λ   1   μ + 7λ         M1 A1 Finds direction of common perpendicular. i j k  −20   5      2 1 −7 =  12  ~  −3  −2 −3 1  −4   1  405 4(c) 4(b) 4(a) Question Q5 2 = 89 10 Craftear 4 A1 36.2 M1 A1 Finds normal to the plane OAB. 2 M1 A1 Divides by magnitude of normal vector. 1.06… M1 Uses dot product correctly. OE 5 A1 A1 A1 A1 OE M1  −2   2      −6  −4  = 89 29 cos θ 7  3     i j k 2    −1 1 2 =  −4  −2 −1 0  3  22 + 62 + 10 −2 x − 6 y + 7 z = 10 Setting up 3 equations using points given. Alternative method for question 4(a) M1 A1 Substitutes point. −2(−1) − 6(1) + 7(2) = 10  −2 x − 6 y + 7z = 10 B1 Finds direction vectors of two lines in the plane.  BC = 4i + j + 2k Guidance M1 A1 Finds normal to the plane ABC.  AC = 3i − j Marks i j k  −2    1 2 2 =  −6  3 −1 0  7   AB = −i − 2 j − 2k Answer 406 Q6 7(c) 7(b) 7(a) Question t = 103 4 + 10t 2 −4 −10t ( = 10  (4 + 10t) 2 = 10 4 + 10t 2  2   −2  −4 −10t    −4  3t  = 2  4 +10t  2   t  4 +10t 2    1 ) Craftear 4 Finds common perpendicular. 6 A1 M1 Sets equal to 10 and solves for t. M1 A1 Uses formula for perpendicular distance. M1 A1 A1 0.49(0) rad 28.1° i j k  −2    0 −1 3 =  3t  t 1 −1  t  M1 Uses dot product of normal vectors. M1 A1 Finds normal to the plane OBC. 5 5      3  2  = 35 45 cosθ 1   4     i j k 5   −2 1 2 =  2  0 −2 1  4  5 M1 A1 Substitutes point. OE 5(0) + 3(−2) +1(1) = −5  5 x + 3 y + z = −5 B1 Finds F direction vectors of two lines in the plane.  BC = 2i − 3j − k Guidance M1 A1 Finds normal to the plane ABC. OE  AC = 2i − 4 j + 2k Marks i j k 5   0 −1 3 =  3  1 −2 1 1   AB = −j + 3k Answer 407 Q7 5(c) 5(b) 5(a) Question A1 Acute angle between l2 and Π is α − 90 = 10.4° 3 M1 A1FT  0  4  −9     5  3  = 61 41cos α  cos α = 61 41  6   −4     (A1 FT their normal) 4 M1 A1 4(3) + 3(0) − 4(3) = 0  4 x + 3 y − 4z = 0 5 M1 A1 M1 A1 Craftear B1 M1 A1 Marks i j k 4    1 4 4 = 3  1 0 1  −4  0  4  1    15 −5  −6  = = 1.71  77     77  −9   5  i j k 4    1 4 4 =  −6  0 5 6  5  3  3 0         −5  −  0  =  −5   −6   3   −9        Answer 408 Q8 7(b) 7(a) Question  −4   −2  1    53 −7  −5  = = 7.21  √ 54     √ 54  2 5  0   4   −4         −5  −  2  =  −7   0   −2   2        Craftear A1 −2 x − 5 y + 5z = 25 3 M1 A1 M1 4 M1 M1 A1 Marks −5(−5) = 25 i j k  −2    5 0 2 =  −5  0 1 1  5  Answer 409 7(c) Craftear B1 FT  55   −2  1     r =  −135  + k  −5  27      22   5 (B1 FT their common perpendicular) 8 M1 A1 A1  55   1  55 22 11 i − 5j + k λ =  OP =  −135  = 27 27 27 27    22   4 − 5λ   0      7 + μ 1  = 0  −2λ + 2μ = −5  −2 + μ − 2λ  1     M1  4 − 5λ   5   4 − 5λ   −2          7 + μ  0  = 0 or  7 + μ  = k  −5     −2 + μ − 2λ   2   −2 + μ − 2λ       5 A1 M1 A1  5λ   4   4 − 5λ             OP =  −5  , OQ =  2 + μ   PQ =  7 + μ   2λ   −2 + μ         −2 + μ − 2λ  410 Q9 6(ii) 6(i) Question Craftear 4 A1 AEF −x + 4 y + 5z = 0 M1 A1 Finds normal to plane. M1 Uses point on plane. 2 k  −1   1 =t 4  −1  5  − (0) + 4 ( 0) + 5 ( 0) = 0 3 i j n = 1 −1 5 M1 A1 Uses formula for shortest distance. 1 3      −1 ⋅  2   0   −1      = 1 = 0.267 14 32 + 22 +12 Guidance M1 A1 Finds direction of common perpendicular. B1 Marks i j k  −9   3  JJJG JJJG     OC × AB = 1 −1 1 =  −6  = t  2  1 2 7  3   −1 1 JJJG   AB =  2  7   Answer 411 Q10 3 Question 3  4 JJJG   JJJG   OP =  −1 , OQ =  −2  7     1 λ = −3, µ = 1 8 JJJG JJJG A1 States OP and OQ . M1 A1 Solves simultaneous equations. A1 Deduces second equation. CWO.  −2 − λ  0      2 − λ − 6µ   −6  = 0 ⇒ 6λ + 37 µ = 19     −7 + µ  1  ˙ A1 Deduces one equation. CWO. Craftear Guidance JJJG M1 Uses that dot product of PQ with line directions is 0 . JJJG Or, alternatively, PQ is multiple of common perpendicular. JJJG M1 A1 Finds PQ . Marks −2λ − 6µ = 0  −2 − λ  1      2 − λ − 6 µ  1  = 0     −7 + µ  0  i j k  −2 − λ  1     Or  2 − λ − 6 µ  = k 1 1 0 = k  −1   −7 + µ   −6  0 −6 1     ˙ 6 + λ   4   −2 − λ  JJJG  JJJG   JJJG    OP =  2 + λ  , OQ =  4 − 6 µ  ⇒ PQ =  2 − λ − 6 µ   µ   7   −7 + µ        Answer 412 Q11 7(ii) 7(i) Question = 54 18 2 2 2 13 + 4 + 2 2 378 9 =− ( or 0.481rad ) =− = 6 = 2.45  42  cos −1  −  − 90 = 27.6°  14  1 +1 + 0 2  1   −13      1 ⋅ 4  0  2      j k 2  −26   −13      2 4 5 = 8 ~ 4  2 6 1  4   2  i 2 5 +5 +2 2  1   −5       5 ⋅  5   1   −2      i j k  −5    2 4 5 = 5  1 1 0  −2   2 1 JJJG   JJJG   AB =  4  , CD =  1  5  0     Answer 42 14 Craftear Guidance 5 A1 M1 A1 FT Uses dot product of their normal with direction of line. M1 A1 Finds normal to the plane. 5 M1 A1 Uses formula for shortest distance. M1 A1 Finds direction of common perpendicular. Allow any nonzero scalar multiple. B1 Find the directions of the lines. Marks 413 Q12 10(i)(b) 10(i)(a) Question 6 13 Craftear A1 AEF Equation of plane: −13 x + 5 y − z = 43 4 M1 Substitutes a point M1 A1 Uses cross product to find normal to the plane AEF 3 A1 AG Using point on plane: e.g. 5 ( 9 ) − 2 = 43 i j k 6 30 6 −1 1 = −6i + j − k ~ −13i + 5 j − k − 13 13 13 6 2 4 13 Normal to the plane: ⇒ µ = 1, λ =12, a = − M1 Equates coordinates of points  λ a   −6 − µ a      Point of intersection:  9 − λ  =  −5 + 2 µ       2 + λ   10 + 4 µ  Guidance B1 Marks  λa   −6 − µ a      Point on l1 is  9 − λ  and on l2 is  −5 + 2µ  2+ λ  10 + 4µ      Answer 414 Q13 7(i) Question 10(ii) Answer Craftear A1 Use third equation to obtain a = 2 . M1 Marks 5 A1 9 + 2λ = 7 + 2 µ 13 + 3λ = −2 − 3µ ⇒a =3 to obtain λ = −3 and µ = −2 . Solve two equations 2 ⇒ 26 ( a − 3) = 0 2 Guidance A1 ) ⇒ 15 18 + 13a 2 = ( 6 + 13a ) ( M1 Puts distance equal to 3 30 M1 A1 Takes dot product of the correct vectors B1 Finds the magnitude of the cross product of the direction vectors of the lines 3 30 36 + 26a 2 = 36 + 78a  −6   −6       −5 − 9  ⋅  −5a  = 36 + 78a  10 − 2   a       −6    2  −5a  = 36 + 26a  a    i j k a −1 1 = −6i − 5aj + ak −a 2 4 415 Q14 8(i) Question 7(iii) 7(ii) M1 A1  −6   −12  1     Perpendicular distance = 6  ⋅  0  = 10 = 3.16  160      −8   4  −x + 8 y − 4z = 3 i j k −4 1 3 = −i + 8 j − 4k 0 1 2 −i + 2 j − 3k 2i +10 j + 6k Answer = 10 = 3.16  −6   −1   −2         6  ×  2  =  −10   −8   −3   −6        Craftear Guidance 3 M1 A1 Uses point on plane to find cartesian equation, AEF. M1 Finds normal to Π1 . Marks 11 M1 A1 Find length PN M1, A1 M1, A1 Perpendicular distance from P to l2 is = 10 M1 A1 Alt method: Find N (foot of perpendicular) in terms of parameter and uses scalar product with n to find 3   parameter M1, A1 so PN =  0   −1   9 +1 3 × 3 + 0 ×1− 6 + 7 Cross product of direction of P to l2 with direction of l2 Use formula Alt method: Finds equation of plane M1, A1: Finds foot of perpendicular from P to plane M1 Hence length A1 Alt method: Find equation of plane 3x – z + 7 = 0 M1, A1 M1 A1 Normal to the plane is n = −12i + 4k 416 Q15 10(i) Qu uestion 8(iii) 8(ii)  2  1  JJJG   JJJG   1 JJJG AB =  3  , CD =  1.5  = AB  2  1  2     Craftear 5 Guiidance 1 B1 Or shows if parallel,, then m=3/2 Marks A1 States vector equation of line. M1 A1 Finds point common to both planes. Point on both planes is, e.g. (1,1,1) . r = i + j + k + λ ( 4i + 13j + 25k ) M1 A1 Finds direction of line of intersection. i j k 3 1 −1 = 4i +13j + 25k −1 8 −4 Answer A1 Accept 1.26 rad. ⇒ θ = 72.5° 2 M1 Uses scalar product.  −1   3       8  ⋅  1  = 9 11cos θ  −4   −1     417 10(iii) 10(ii) 2 2 = 2 = ( 3 − 2m ) 2 + ( 2m − 3 ) 2 −2 ( 3 − 2m ) + 0 + 0 o o. 19   =  2 2 2  378  1 +4 +5  1 + 8 +10 12 + 22 + 22 ⇒ θ = 12.2° cosθ = i j k  1   2 3 2 =  −4  o. o   −1 1 1  5  i j k  2 1     2 3 2 =  −4  ~  −2  o. o     −2 −1 0  4   2  = n AC.n i j k  3 − 2m  1     n = 2 3 2 =  0  ( so parallel to  0  )  −1  1 m 1  2m − 3    Craftear  −1   −4   2 1  −2  JJJG   JJJG   JJJG       AB =  3  , CD =  m  and AC =  −1  or AD =  m −1 or BC=  −4     −2   2 1  0  1         6 A1 CAO. M1A1FT Uses formula for angle between two lines. A1 Finds normal to ABD (AEF). M1A1 Finds normal to plane ABC (AEF). 5 A1 M1 Uses formula for shortest distance. M1A1 Finds common perpendicular using cross product. B1 418 Q16 12O(ii) 12O(i) Question  2 uuur   AB =  −2     2 2  uuur   CD =  − 4     α −1 2   2  uuur     or CD =  −4    2     i j k 1 Distance of D from AC = 4 −6 0 = 1 +1 +1 1 −1 −1  4 uuur   AD =  −6   0   (AG) Craftear 56 = 4.32 (or with CD) 3 = 2 2 ⇒ 2 2 2α 2 − 16α + 38 = 8 ⇒ (α − 3)(α − 5 ) = 0 ⇒ α = 3 or 5 2 (α − 5 ) + ( α − 3 ) + 4  2  α − 5      −2  α − 3   −2   2     i j k  5 − α  α − 5 uuur uuur     AB × CD = 1 −1 1 =  3 − α  ~ α − 3 2 −4 α −1  −2   2   2 uuur   AC =  −2     −2  Answer Total: Total: , instead of AC Guidance 3 M1A1 Use DP.AC = 0 to find Find length A1 (= -5/3) B1 Alt method: Let P be point on AC with parameter 7 A1 A1 M1 M1 M1A1 Substitutes their vectors into correct formula M1A1 B1 May find Marks 419 Q17 9(ii) 9(i) Question 12O(iii) k Craftear Total: 4 M1A1 5 × 2 − ( −3) + 7 × 1 = 20 ⇒ 5 x − y + 7 z = 20 Total: M1A1 i j k 5   1 −2 −1 =  −1 2 3 −1  7  4 A1 M1A1 Marks 7 × 1 + 2 − 3 × 1 = 6 ⇒ 7 x + y − 3z = 6 4 M1 Total: M1A1 B1B1 Cartesian equation of Π1 is 7 x + y − 3z = const i j k  −7    1 −1 2 =  −1  1 2 3  3  Answer i j k 3   ABD: n 2 = 1 −1 1 =  2  2 −3 0  −1 cos θ = −32− 2+0 ⇒ θ = 19.1° or 0.333 rads 14 j  −1   ABC: n1 = 1 −1 − =  −1 1 −1 1  0  i Guidance 420 Q18 6(i) Question 9(iii) Craftear 4 M1A1) Find area using Area = ½ ab sin C or equivalent A1 M1A1 OE B1 2 correct required (M1A1 1  3 212 + 32 + 182 = 13.9  86  2 2   JJJG AC = −3i + 3 j + 3k Marks Alt method: Use scalar product to find angle Area of triangle ABC = JJJG JJJG AB × BC = 21i + 3j + 18k (*) JJJG BC = −4i − 2j + 5k Answer 13 ⇒ θ = 78.7° or 1.37 rad. 59 75 JJJG AB = i + 5 j − 2k ⇒ cos θ = cos θ = 75     1  . −1     −3   7  49 + 1 + 9 25 +1 + 49 Guidance Total: 3 A1 M1M1 421 6(iii) 6(ii) Through ( 2, –1 , 1) Hence 7 + + 6 = const. + 6 = 19 From (*) Cartesian equation is 7 + 2 A1 M1 3 (M1A1) A1 Area triangle = sin C × |AC| M1A1 Alt method: Find angle at C Craftear Alt method: Use equation of BC to find D (foot of perpendicular) in terms of parameter and scalar product to find parameter , λ= 8/15. Find length 1  = 4.15  430  5  JJJG JJJG AB × BC 212 + 32 + 182 d= = JJJG BC 4 2 + 2 2 + 52 422 423 i j k  17   1      AB × AC = 2 3 −8 =  34  ~  2  −3 4 −5  17   1  So x + 2 y + z = const ⇒ const = 2 − 2 + 3 = 3 ( using a point ) ⇒ x + 2y + z = 3 1  3  3 3     9 + 1 + 4 1 + 4 +1 cosθ =  2  .  −1 ⇒ cos θ = = 14 6 84     1  2  ⇒θ = 70.9° or 1.24 radians i j k  5   Direction of line of intersection is 1 2 1 =  1  3 −1 2  −7  1 13  13 4  Finds point common to both planes is ( −1, 0 ,4 ) or  , ,0  or ( 0, , ) 5 5 7 7   −1  5      Equation of line of intersection is eg r =  0  + t  1  .  4   −7      M1 A1 M1 A1 [4] M1 M1 A1 [3] M1A1 M1 A1 [4] Craftear Q19 8 Q20 424 Qu u Soluti 11 (o) (i) Direction perpendicular to AB and CD: i j k λ − 2   n= 0 1 1 = 4  4 3 − λ 1  −4   5   DB =  −4  . Hence  −6    2 M1A1  5  λ − 2      4  − 4  .   −6   −4      or equivalent = 3 ( λ − 2 )2 + 1 6 + 1 6 ( ⇒ ( 5λ − 2 ) = 9 λ 2 − 4λ + 36 ) M1A1 M1M1 A1 [7] 2 ⇒ … ⇒ λ + λ − 20 = 0 (AG) ( λ + 5)( λ − 4 ) = 0 ⇒ λ = −5, 4 B1 i j k  −10  2     λ = 4 ⇒ a =  4  ⇒ Normal to ABD = 4 −1 1 =  −29   −3  −1 3 7  11    M1A1 i j k  44   2     λ = −5 ⇒ a =  −5  ⇒ Normal to ABD = 4 8 1 =  −29    −1 12 7  56   −3  A1 cos θ = −440 + 841 + 616 2 2 2 10 + 29 + 11 ⇒ θ = 66.1° 2 2 44 + 29 + 56 2 = 1017 1062 5913 M1A1 A1 [7] Craftear (ii) Marks Q21 11 E JJJG JJJG JJJG M1A1 A1 Solves: s = –1 , t = –1 , k = 1 p = 3i + j + 2k and q = i + 2 j − 3k (Both required) B1B1  −22  2  3       r =  1  + λ  −19  + µ  −1  2  5        5 Craftear B1 i j k  −22    3 −4 − =  −19  2 −1 5  5  ALT METHOD: Find PQ in terms of s,t M1 Calculate scalar products direction vectors for l1 and l2 M1,A1,A1 Solve simultaneous equations M1 A1 p and q A1 M1A1 M1A1 t − 3s + 2k = 4 −3t + 4s − k = −2 −t + 2s + 5k = 4 Obtains three equations. E.g. from OP + PQ = OQ i j k 2   1 −3 −1 =  −1 3 −4 −2  5  [3] [7] 425 M1 A1 B1 i j k  90   3      Normal to Π is 22 19 −5 =  −120  =  −4  2 −1 5  −60   −2  Cartesian equation of Π is 3x – 4y – 2z = 1 (Since P lies in Π )  −1  4     Vector equation of l3 is r =  −1 + λ  3  (OE) (Since z = 0 .) 0 0     Craftear B1 i j k  −4   4      Direction of l3 is 3 −4 − =  −3  ~  3  0 0 1  0   0  [4] 426 427 110 (i) (ii) (a) 5 3+λ   8+λ           p =  2 − 2λ  , q =  3 + 2µ  ⇒ QP =  − 1 − 2λ − 2µ   3   17 + 3µ   − 14 − 3µ        M1A1 3+λ   1        − 2  .  − 1 − 2λ − 2µ  = 0 ⇒ 5λ + 4µ = − 5  0   17 + 3µ      M1A1 3+λ   0       2  .  − 1 − 2λ − 2µ  = 0 ⇒ − 4λ − 13µ = 53  − 3   17 + 3µ      A1 Solving : λ = 3, µ = – 5 M1A1 Whence:  11   5     p =  − 4 , q =  − 7 .  3  1      A1 (8) − 3  3      AP =  − 6  , AQ =  − 9   − 2  0      i j k  12    3 − 6 0 =  6  (CAO) ;  −45  −3 − 9 − 2   (b)  −3  uuur  1 1  AB =  1  ⇒ Volume = × 3 2  −17    B1 Area = 1 2205 2 (=23.5)  − 3   12      1  . 6   −17   −45    = 735 (= 122.5) 2205  6 2205 M1A1 A1 M1A1 (6) Total: 14 Alternative: for marks 3,4,5,6 and 7 in part (i) i j 1 −2 0 k 6   0 = 3  2 2 −3   6k − λ = 3 3k + 2λ + 2 = −1 2k − 3µ = 17 Solving; k = 1 , λ = 3 and µ = – 5 (If k missing, or assumed to be 1, deduct 1 mark.) (M1A1) (A1) (M1A1) Craftear Q22 Q23  3 + 2λ   1       λ  .  2 = 3  2 − 2λ   1      3 + 2λ + 2λ + 2 − 2λ = 3 ⇒ λ = −1 P is (1, –1, 4) (Accept position vector.) 1   n =  2 1   i M1 A1 A1 (3) B1 j k  −5    w= 1 2 1 =  4   −3  2 1 −2   i j k  10   5      Direction of PQ is − 5 4 − 3 =  2  ~  1  (OE) 1 2 1  − 14   − 7  1  5     Equation of PQ is r =  −1 + µ  1  4      −7  [N.B. For methods not using w at most three B1 marks.] M1A1 (3) M1A1 dM1A1 (4) Total 10 Craftear 8 428 Q24 429 11 E  8  4     −1 2 0 =  4 ~  2 − 1 0 4  2   1  E. j k 1  4     0 . 2   0  1    4 2 + 2 2 + 12 = 4 (AG) 21 3  4i + 2 j + k  1  = (4i + 2 j + k )  21  21  7  1   − 3     Line AP: r =  0  + t  2   0  1      p= M1A1 M1A1 [4] B1 M1A1 0 1 1  For Q 1 − 3t = 0 ⇒ t = ⇒ q =  2  3 3  1 M1A1 [5] E.  − 1  0    1  AB =  2  , BQ =  − 4  3  0    1  B1 i j  2   −1 2 0 =  1  0 − 4 1  4  cos k  4  2     2 . 1     −1  1   4  21. 21 = cos −1 M1A1 8+ 2+ 4 14 2 = cos −1 = cos −1 (AG) 21 21 3 M1A1 [5] Total 14 Craftear i Q25 430  6   5      AB =  4  CD =  − 2   − 8  − 4     B1 i j  − 16   2      Common perpendicular is 3 2 − 4 =  − 8  ~  1  5 − 2 − 4  − 16   2   1    AD =  − 4  ⇒ shortest distance =  9    i  1   2      − 4 ⋅  1  9   2     4 +1 + 4 = 16 or 5.33 3  − 2   Normal to Π1 is 1 − 4 9 =  31  3 2 − 4  14  i j k M1A1 [5] k  34    Normal to Π2 is 1 − 4 9 =  49  5 − 2 − 4  18  cos θ = j M1A1 M1A1 k − 2 × 34 + 31× 49 +14 × 18 2 2 + 312 + 14 2 34 2 + 49 2 + 182 ⇒ θ = 36.7° (CAO) A1 M1A1 A1 [6] Q26 OR (i) r = 4i – 2j + 2k + λ(i + 7j + k) + µ(3i + j – k) ⇒ A is in Π1. (ii) i j k  −8   2      1 7 1 =  4  ~  − 1 3 1 − 1  − 20   5  (iii) L is (12, –6, 6) 2x – y + 5z = 24 + 6 + 30 = 60 n = t (2i – j + 5k) ⇒ 4t + t + 25t = 60 ⇒ t = 2 n = 4i – 2j + 10k (iv) 3 (8i – 4j + 4k) = 10i – 5j + 5k (AG) 4 M is (10, –5, 5) ⇒ NM = 6i – 3j – 5k (6i – 3j – 5k) × (2i – j + 5k) = – i + 2j 20(−i + 2j 20 Perpendicular distance is = = 8.16 30 6 (Mark various alternative methods in a similar manner.) m = 4i – 2j + 2k + B1 [1] M1A1 [2] B1 B1 M1A1 A1 [5] B1 B1 M1A1 M1A1 [6] Craftear 11 Q27 431 i (i) k  − 3  1      1 − 2 − 3 =  3  ~  − 1  − 3  1  −2 1 3     M1A1 BA = 6i + 4j − 6k Shortest distance B1  6  1      4 . −1  −6   1     12 +12 + = 43 ( = 2.31) M1A1 (5) Alternative  − 6 − 2λ − µ   1       − 4 + λ + 2µ  .  − 2  = 0 ⇒ −13λ − 14µ = 16  6 + 3λ + 3µ   − 3      (M1)  − 6 − 2λ − µ   − 2       − 4 + λ + 2µ  .  1  = 0 ⇒ 14λ +113µ = −26  6 + 3λ + 3µ   3      (A1) ⇒ λ = − 529 , µ = 389 Shortest distance = i (ii) j (M1A1) 4 3 . (A1) (= 2.31 k  − 4  4     1 −1 1 =  − 5 ~  5 − 2 1 3  − 1   1  M1A1 Cartesian equation of Π: 4x + 5y + z = −12 − 5 + 2 = −15 (iii) Distance of A from Π:  6   4      4  . 5  − 6  1     M1A1 4 2 + 52 + 12 = 38 42 M1A1 (4) ( = 5.86) A1 (3) [12] Craftear 10 j Q28 k  3    1 =  − 1 1  − 6  M1A1 Obtains cartesian equation 3× 2 + 1× (−1) + (−2) × (−6) = 17 ⇒ 3x − y − 6z = 17 M1 A1 Obtains area of triangle ABC. 1 2 1 3 + 12 + 6 2 = 46 (=3.39) 2 2 M1 A1 Obtains length of perpendicular from D to triangle ABC. 9 − 6 − 12 − 17 M1A1 1 × Base 3 area × Height. i 2 1 j 0 −3 3 2 + 12 + 6 2 = 26 46 1 1 26 13 × 46 × = 3 2 3 46 Uses Either or triple scalar product method. 1  3  1     26 13 Or e.g.  5  ⋅  − 1  = = 6    6 3  4  − 6 Obtains normal to ABD. i 1 2 j 5 0 k  5    4 = 7  1  − 10  Uses scalar product 2 2 3 +1 + 6 to find angle between normals and hence angle between Π1 and Π2. ⇒ cos θ = 2 4 M1A1 6 M1A1  3   5      5 + 7 + 10 cos θ =  − 1  ⋅  7   − 6   − 10      M1 68 A1 2 46 174 2 2 ⇒ θ = 40.5º Craftear 11 O Vector perpendicular to Π1 432 [14] 433 Q29 Finds PQ. Finds direction of common perpendicular.  − 3 + µ − 3λ    PQ =  − 6 µ − 2λ   12 − 2 µ + λ    i j M1A1 k  − 10  2     3 2 − 1 =  5  ~  − 1 4 1 − 6 − 2  − 20    M1 (Or uses scalar product of PQ with the direction vector of each line.) Obtains two correct equations. e.g. Solves. Finds p and q. Finds common perpendicular. − 3 + µ − 3λ = 12µ + 4λ − 24µ − 8λ = −12 + 2µ − λ A1 A1 µ = 1 , λ = −2 M1A1  4 + 3λ   − 2   1+ µ   2         p =  7 + 2λ  =  3  q =  7 − 6 µ  =  1   −1− λ   1  11 − 2 µ   9          i AB × PQ = − 1 2 j 0 k 4   4 = 12  − 1 4  1   6  2 3 –1   PA  4  QA 6 PB 4 QB 6  − 2 –10 10 2   Uses triple scalar product to find shortest distance.  6  4      4  ⋅ 12   − 2  1      16 +144 + 1 Alternative for last 4 marks: 24 + 48 − 2 161 = 70 161 A1 M1A1 Finds PA (or QA or PB or QB) = 8 = 5.52 Plane through e.g. P in direction PA : 4 12 29 0 . Award M1A1. Then use of distance of point from line 70 . Award M1A1. formula to get 161 Craftear 11 E B1 M1A1 A1 6 [14] Q30 434 8 Finds normal to Π1 . i j k 1 0 1 = i + 3j − k M1A1 1 −1 − 2 Finds Cartesian equation. Equation of Π1 : Finds angle between normals, using scalar product. cosθ = Finds direction of line of intersection, using vector product. i Finds point common to both planes. States vector equation. 3 12 2 − 3 −1 11 6 2 = ⇒ θ = 75.7° or 1.32 rad. 66 j A1 3 M1 A1 2 k 1 3 − 1 = 2i − 3 j − 7 k 2 −1 1 M1A1 Point on both planes is e.g. (6,2,0) M1A1 r = 6i + 2j + t (2i − 3j − 7k ) (OE) A1 5 [10] 8 Finds normal AB = i + 2j + 2k, AC = −2i + j − 2k and cartesian equation AB × AC = −6i − 2j + 5k M1A1 6x + 2 y − 5z = constant = 24 + 10− 30 = 4 M1A1 Finds where OD meets plane Obtains angles between planes Equation of OD: r = t (6i + 3j + 6k ) 1 ⇒ 36t + 6t − 30t = 12t = 4 ⇒ t = 3 E is the point (2,1,2). Craftear Q31 (4) B1 M1A1 A1 (4) Using (6i + 2j − 5k ) . 2i + j + 2k ⇒12 + 2 −10 = 36 + 4 + 25 4 + 1 + 4 sinθ M1A1 ⇒ θ = 9.5o (0.166 rad). A1 (3) [11] 435 11 OR Obtains direction of common perpendicular. Uses result for shortest distance between lines. Solves equation. i j k n = 4 −1 1 0 m −1 1 = –mi + 4(1– m)j + 4k  1  − m      0  4 − 4m   − 4  4     m 2 + 16(1 − 2m + m 2 ) + 16 M1A1 =3 M1A1 A1 ⇒ … ⇒ 19m 2 − 40m + 4 = 0 ⇒ (19m − 2)(m − 2) = 0 M1 A1 ⇒ m = 2, since m is an integer. (AG) Finds relevant vectors.  1  0  − 1       CA =  0  and CD =  1  or AD =  1   − 4 1 5       B1 Use of cross-product. i j k  − 4 1 1   0 1 1 =  1  17 17   1 0 −4  −1  M1 Obtains shortest distance. 1 18 4 2 +12 + 12 = 17 17 A1 Finds 2nd vector in BCD (CD may already have been found.) Finds normal vector to BCD. (Normal to ACD already found.) Finds angle between planes = angle between normal vectors. (= 1.03) BC = 3i – j +5k i j cos θ = 1 3 B1 k n = 3 − 1 5 = –6i – 3j +3k ~ 2i + j – k 0 7 Craftear Q32 M1 1 (4i − j + k) ⋅ (2i + j − k) 16 + 1 + 1 4 + 1 + 1 = 6 18 6  1  ∴ Angle between planes = cos–1   (AG)  3 = 1 3 M1 A1 4 [14] 436 OR OR Alternatives for middle part: Or (a) Vector from D to any point on AC  1+ t     −1   − 5 − 4t    (B1) Uses orthogonality to obtain t.  1+ t   1     21  − 1 . 0  = 0 ⇒ t = − 17  − 5 − 4t   − 4     (M1) 1 18 4 2 +12 + 12 = 17 17 (A1) Finds magnitude of perpendicular. Or (b) Finds length of AD (or CD) Finds projection of AD (or CD) onto AC. Finds perpendicular by Pythagoras. (= 1.03) AD = 27  − 1  1      1 . 0   5   − 4    4 2 +12 27 − (3) (B1) = 21 17 441 18 = 17 17 (M1) (= 1.03) (A1) (3) Craftear 11 437 Finds vector normal to Π. i Cartesian equation of Π. Substitutes general point of l2. Finds value of parameter. Finds p.v. of intersection. Finds distance along l from known point to foot of perpendicular from given point to l. F.t. on nonhypotenuse side (must be real). Writes a set of three equations in three unknowns for the intersection of l with Π. Solves the set of equations. Finds p.v. of intersection. 2 = 2i + 8 j + 7 k 1 M1A1 −2  8   2  3 + 8t   2           6 + 5t  ⋅  8  = 138 or  5  ⋅  8  = 0  − 8  7  12 − 8t   7          A1 Independent of t ⇒ parallel, or ⇒ parallel. A1 4 Π: 2x + 8y + 7z = 21 Sub. x = 5 + 2s , y = −4 − s , z = 7 + s ⇒ s = −2 and line meets Π at point with p.v. i – 2j + 5k B1 M1 A1 A1 4 Take (9,11,2) as A, (3,6,12) as B and let C be foot of perpendicular from A to l. B1 AB = 62 + 52 +102 = 161 BC = Finds distance from point to known point on l. k n= 1 −2 3 Dot product of this with general point on l1. Deduces result. j 1 2 2 2 (6 5 + 8 )(6i + 5j−10k)(8i + 5j−8k) AC = 161−153 = 8or 2 2 = (= 2.83) 153 = 153 153 M1 A1 A1 Alternatively: 5 + 2 s = 2 + λ + 3µ (B1) − 4 − s = 3 − 2λ + µ 7 + s = −1 + 2λ − 2 µ ⇒ s = −2 , ⇒ λ = 2 , µ = −1 and line meets Π at point with p.v. i – 2j + 5k (M1A1) (A1) 4 [12] Craftear 9 438 Q33 9 Finds vector BA . B1 BA = 6i + 5j – 10j i 1 j k 6 5 − 10 82 + 52 + 8 8 5 − 8 M1A1 1224 = 8 or 2 2 ( = 2.83) 153 = A1 (4) Alternatively: (A) Take (9,11,2) as A, (3,6,12) as B and let C be foot of perpendicular from A to l. Finds distance from point to known point on l. (B1) F. on non-hypotenuse si F.t. (must be real). AB = BC = 2 2 6 + 5 + 10 = 161 1 2 2 2 (6i + 5j − 10k)(8i + 5j − 8k ) 8 +5 +8 153 = = 153 153 AC = 161−153 = 8 or 2 2 (M1) (A1) (A1 ) (4) ( = 2.83) Craftear Finds distance along l from known point to foot of perpendicular from given point to l. 2 (B)  8t − 6    AC =  5t − 5  10 − 8t    Finds vector . AC Uses AC perpendicular to l to find t.  8t − 6   8       5t − 5  .  5  = 0 ⇒ t = 1 10 − 8t   − 8      Finds length AC. AC = 22 + 02 + 22 = 8 (B1) (M1A1) (A1) [12] (4) Q34 4 Finds vector product. Finds area of triangle. Finds length of perpendicular. 1   AB =  2   3   1   AC =  1   2   B1 i j k 1   AB × AC = 1 2 3 =  1  1 1 2  − 1 M1A1 1 1  1 Area of triangle ABC =  1  = 3 2  2  − 1 M1A1 1 2 1 3 1 + 2 2 + 32 d = 3⇒d = 2 2 14 A1 3 3 [6] 439 Q35 Finds normal vector to plane. x = 2+λ +µ 6   1 − 2 − 1 =  1  or y = −3 − 2λ + 2 µ z = 1− λ − 2 µ 1 2 − 2  4  Uses known point to find constant term. Plane equation is 6x + y + 4z = constant. Substitute (2, –3,1) ⇒ 12 – 3 + 4 = 13. Or eliminate λ and µ. ⇒ Π1 : 6x + y + 4z = 13 Angle between normals is equal to angle between planes. Solve plane equation simultaneously. (Note: may find direction from vector product and use with one point.) i j cosθ = k (6i + j + 4k).(3i − 2 j − 3k) 2 2 6 +1 + 4 2 2 2 3 +2 +3 2 = M1A1 M1 4 53 22 A1 4 M1A1 A1 3 ⇒ θ = 83.3º or 1.45 rad. 6x + y + 4z = 13 and 3x – 2y – 3z = 4 Obtains e.g. y + 2z = 1 and 3x + z = 6 Or two of (0,–11,6) , (11/6,0,1/2) , (2,1,0)  x   2  −1        y  =  1  + t  − 6  (OE)  z   0  3        Alternatively: Direction of line from vector product. Finds a point on line. States equation of line. M1 A1A1 A1 (M1A1) (A1) (A1) 4 [11] Craftear 9 440 Craftear PROOF BY INDUCTION 441 Q1 3 1 Prove by mathematical induction that 2 4n + 31 n - 2 is divisible by 15 for all positive integers n. 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.................................................................................................................................................................... .................................................................................................................................................................... © UCLES 2021 9231/11/M/J/21 [Turn over Craftear .................................................................................................................................................................... 442 Q2 3 6 (a) Prove by mathematical induction that, for all positive integers n, n /`5r + r j = n (n + 1) (2n + 1) . r =1 4 2 1 2 2 2 [6] ............................................................................................................................................................ ............................................................................................................................................................ 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............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 Craftear ............................................................................................................................................................ 443 7 (b) Use the result given in part (a) together with the List of formulae (MF19) to find n / r in terms r =1 of n, fully factorising your answer. 4 [3] 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............................................................................................................................................................ © UCLES 2021 9231/13/M/J/21 [Turn over Craftear ............................................................................................................................................................ 444 Q3 3 5 The sequence of real numbers a1 , a2 , a3 , … is such that a1 = 1 and a n +1 = ea n + 3 1 . an o (a) Prove by mathematical induction that ln an H 3 n -1 ln 2 for all integers n H 2 . [6] 1 [You may use the fact that ln bx + l 2 ln x for x 2 0 .] x ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ (b) Show that ln an +1 - ln an 2 3 n -1 ln 4 for n H 2 . [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2021 9231/11/O/N/21 [Turn over Craftear ............................................................................................................................................................ 445 Q4 2 It is given that y = xe ax , where a is a constant. Prove by mathematical induction that, for all positive integers n, dny = à n x + na n-1j e ax . dx n [6] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... 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.................................................................................................................................................................... © UCLES 2021 9231/12/O/N/21 [Turn over Craftear .................................................................................................................................................................... 446 Q5 5 Prove by mathematical induction that, for every positive integer n, d 2n-1 (x sin x) = (-1) n - 1 `x cos x + (2n - 1) sin xj . dx 2n-1 [7] .................................................................................................................................................................... .................................................................................................................................................................... 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............................................................................................................................................................................ ............................................................................................................................................................................ ............................................................................................................................................................................ © UCLES 2020 9231/11/O/N/20 [Turn over Craftear ............................................................................................................................................................................ 448 Q6 2 4 Prove by mathematical induction that 7 2n - 1 is divisible by 12 for every positive integer n. [5] .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... .................................................................................................................................................................... © UCLES 2020 9231/12/O/N/20 Craftear .................................................................................................................................................................... 449 Q7 2 The sequence u1, u 2, u3, f is such that u1 = 1 and un + 1 = 2un +1 for n H 1. (a) Prove by induction that un = 2 n - 1 for all positive integers n. 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(b) Deduce that u2 n is divisible by un for n H 1. [2] ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ © UCLES 2020 9231/13/M/J/20 [Turn over Craftear ............................................................................................................................................................ 450 Q8 2 It is given that y = ln ax + 1, where a is a positive constant. Prove by mathematical induction that, for every positive integer n, n dn y n−1 n − 1!a = −1 . [6] dxn ax + 1n ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ 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................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2019 9231/11/O/N/19 [Turn over Craftear ................................................................................................................................................................ 451 Q9 8 (i) Prove by mathematical induction that, for z ≠ 1 and all positive integers n, 1 + z + z2 + à + zn−1 = zn − 1 . z−1 [5] 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........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2019 9231/11/M/J/19 Craftear ........................................................................................................................................................ 452 Q10 1 3 Prove by mathematical induction that 33n − 1 is divisible by 13 for every positive integer n. [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2019 9231/13/M/J/19 [Turn over Craftear ................................................................................................................................................................ 453 Q11 2 It is given that f n = 23n + 8n−1 . By simplifying f k + f k + 1, or otherwise, prove by mathematical induction that f n is divisible by 9 for every positive integer n. [6] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2018 9231/11/M/J/18 [Turn over Craftear ................................................................................................................................................................ 454 Q12 9 18 For the sequence u1 , u2 , u3 , à, it is given that u1 = 8 and ur+1 = 5ur − 3 4 for all r . (i) Prove by mathematical induction that n un = 4 54 + 3, for all positive integers n. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 Craftear ........................................................................................................................................................ 455 (ii) Deduce the set of values of x for which the infinite series u1 − 3x + u2 − 3x2 + à + ur − 3xr + à is convergent. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Ð ln un − 3 = N 2 ln a + N ln b. N n=1 [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/13/M/J/18 [Turn over Craftear (iii) Use the result given in part (i) to find surds a and b such that 456 Q13 3 4 The sequence of positive numbers u1 , u2 , u3 , à is such that u1 < 3 and, for n ≥ 1, un+1 = 4un + 9 . un + 4 (i) By considering 3 − un+1 , or otherwise, prove by mathematical induction that un < 3 for all positive integers n. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 Craftear ........................................................................................................................................................ 457 (ii) Show that un+1 > un for n ≥ 1. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2018 9231/11/O/N/18 [Turn over Craftear ........................................................................................................................................................ 458 Q14 6 dr u dr y and are denoted by dxr dxr y r and u r respectively. Prove by mathematical induction that, for all positive integers n, Q P@ A @ A @ A @ A @ A n n n n n n 1 2 r n x y =e u+ u + u +à+ u +à+ u . [8] 0 1 2 r n @ A @ A @ A k k k+1 [You may use without proof the result + = .] r r−1 r It is given that y = ex u, where u is a function of x. The rth derivatives ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ 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................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2018 9231/12/O/N/18 Craftear ................................................................................................................................................................ 459 ................................................................................................................................................................ 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................................................................................................................................................................ 460 Q15 2 4 Prove, by mathematical induction, that 5n + 3 is divisible by 4 for all non-negative integers n. [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2017 9231/11/M/J/17 Craftear ................................................................................................................................................................ 461 Q16 Ð @ n 3 Prove, by mathematical induction, that r=1 r ln A @ A r+1 n + 1n = ln for all positive integers n. [6] r n! ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ © UCLES 2017 9231/13/M/J/17 Craftear ................................................................................................................................................................ 462 Q17 3 4 (i) Show that dn n dn+1 n+1 n x ln x = n x + n + 1x ln x . n+1 dx dx [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Prove by mathematical induction that, for all positive integers n, @ A 1 1 dn n x ln x = n! ln x + 1 + + à + . dxn 2 n [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ © UCLES 2017 9231/11/O/N/17 Craftear ........................................................................................................................................................ Q18 Prove by mathematical induction that, for all positive integers n, 10n + 3 × 4n+2 + 5 is divisible by 9. [6] Craftear 3 463 © UCLES 2016 9231/11/M/J/16 [Turn over Q19 It is given that a diagonal of a polygon is a line joining two non-adjacent vertices. Prove, by mathematical induction, that an n-sided polygon has 12 n n − 3 diagonals, where n ≥ 3. [6] Craftear 2 464 © UCLES 2016 9231/13/M/J/16 [Turn over 465 Q20 4 @ Using factorials, show that A @ A @ A n n n+1 + = . r−1 r r [2] Hence prove by mathematical induction that @ A @ A @ A @ A n n n n−1 n n−r r n n a + xn = a + a x+à+ a x +à+ x 0 1 r n for every positive integer n. Craftear [4] © UCLES 2016 9231/11/O/N/16 [Turn over Q21 3 466 4an 5 + for every positive integer n. 5 an Prove by mathematical induction that an > 5 for every positive integer n. [5] The sequence a1 , a2 , a3 , à is such that a1 > 5 and an+1 = [2] Craftear Prove also that an > an+1 for every positive integer n. © UCLES 2015 9231/11/M/J/15 [Turn over 467 Q22 3 r=1 Ð 2r − 1 . ∞ State the value of Ð 2r − 1 = 2n + 1 . n Prove by mathematical induction that, for all positive integers n, 2 n [6] [1] 2 Craftear r=1 1 1 © UCLES 2015 9231/13/M/J/15 [Turn over Q23 3 468 Given that a is a constant, prove by mathematical induction that, for every positive integer n, 6 Craftear dn xeax = nan−1 eax + an xeax . dxn © UCLES 2015 9231/11/O/N/15 [Turn over Q24 3 469 Prove by mathematical induction that, for all non-negative integers n, 112n + 25n + 22 [6] Craftear is divisible by 24. © UCLES 2014 9231/11/M/J/14 [Turn over Q25 It is given that φ n = 5n 4n + 1 − 1, for n = 1, 2, 3, à . Prove, by mathematical induction, that φ n is divisible by 8, for every positive integer n. [7] Craftear 3 470 © UCLES 2014 9231/13/M/J/14 [Turn over Q26 3 471 It is given that ur = r × r! for r = 1, 2, 3, . Let Sn = u1 + u2 + u3 + + un . Write down the values of 2! − S1 , 3! − S2 , 4! − S3 , 5! − S4 . 2 [1] Prove, by mathematical induction, a formula for Sn , for all positive integers n. [4] Craftear Conjecture a formula for Sn . © UCLES 2014 9231/11/O/N/14 [Turn over 472 Q27 Prove by mathematical induction that 52n − 1 is divisible by 8 for every positive integer n. [5] Craftear 2 © UCLES 2013 9231/11/M/J/13 [Turn over Q28 3 473 Prove by mathematical induction that, for every positive integer n, 7 Craftear dn x e sin x = ï2n ex sin x + 14 n0. dxn © UCLES 2013 9231/13/M/J/13 [Turn over 474 Q29 5 It is given that y = 1 + x2 ln 1 + x. Find d3 y . dx3 [3] Prove by mathematical induction that, for every integer n ≥ 3, 5 Craftear dn y n−1 2 n − 3! . n = −1 dx 1 + xn−2 © UCLES 2013 9231/11/O/N/13 [Turn over Q30 9 475 Prove by mathematical induction that, for every positive integer n, cos 1 + i sin 1n = cos n1 + i sin n1. 5 Craftear Express sin5 1 in the form p sin 51 + q sin 31 + r sin 1, where p, q and r are rational numbers to be determined. [6] © UCLES 2013 9231/13/O/N/13 [Turn over Q31 2 476 Prove, by mathematical induction, that, for integers n ≥ 2, [5] Craftear 4n > 2n + 3n . © UCLES 2012 9231/11/M/J/12 [Turn over 477 Q32 5 Let In denote ã 0 xn e−2x dx. Show that In = 12 nIn−1 , for n ≥ 1. n! . 2n+1 [6] Craftear Prove by mathematical induction that, for all positive integers n, In = [2] © UCLES 2012 9231/12/O/N/12 [Turn over Q33 1 2 3 N + + + ... + . Prove by mathematical induction that, for all positive (N + 1)! 2! 3! 4! integers N , 1 SN = 1 − . [5] (N + 1)! Let SN = Craftear 3 478 © UCLES 2012 9231/13/O/N/12 [Turn over Q34 3ur − 2 for all r. Prove by 4 n 3 mathematical induction that un = 4 4 − 2, for all positive integers n. [5] For the sequence u1 , u2 , u3 , . . . , it is given that u1 = 1 and ur+1 = Craftear 2 479 © UCLES 2012 9231/13/M/J/12 [Turn over PROOF BY INDUCTION MS Craftear 480 1 Question Q1 Craftear A1 Hence, by induction, true for every positive integer n. 6 A1 M1 A1 Separates 2 4 k + 31k − 2 or considers difference. is divisible by 15 because 15 × 24 k + 30 × 31k is divisible by 15. Then 24k + 4 + 31k +1 − 2 = (15 +1)24 k + (30 +1)31k − 2 B1 States inductive hypothesis. Assume that 2 4 k + 31k − 2 is divisible by 15 for some positive integer k. Guidance B1 Checks base case. Marks 24 + 31 − 2 = 45 is divisible by 15 Answer 481 Q2 3(b) 3(a) Question 4 r =1 2 ) 2 4  5r 4 + r 2  = 1 k 2 (k +1)2 ( 2k +1)   2 ( k 2 M1 Take out the factor of 𝑘 + 1 OR expands the summation expression and the target expression for 𝑘 + 1 and collects like terms for both. n r 4 + 16 n(n +1)(2n +1) = 12 n 2 (n +1) 2 (2n +1) 4 1 30 2 r =1  r = n(n +1)(2n + 1)(3n + 3n − 1) n r =1 [5]  r 4 = 16 n(n +1)(2n +1)(3n(n + 1) − 1) r =1 n  5 Craftear 3 A1 4  r the subject and takes out all linear 2 r . CAO factors and the remaining term is of correct form. M1 Makes M1 Uses correct formula for 6 A1 States conclusion. Implication must be clearly expressed. 2 So H k +1 is true. By induction, H n is true for all positive integers n. 2 M1 Considers sum to k +1. A1 Factorises or having expanded, checks explicitly. At least one intermediate step seen following the award of M1 before reaching the answer. 3 ( 2k + k + 10(k + 1) + 2 ) Guidance B1 States inductive hypothesis [for some k] including the algebraic form. If says for ALL k, then B0. B1 Checks base case. Marks 1 (k + 1)2 (2k 3 + 11k 2 + 20k + 12) = 12 (k + 1)2 (k + 2) 2 (2k + 3) 2 1 ( k + 1) 2 2 r =1 1  ( 5r + ) = 2 k (k +1) ( 2k +1) + 5(k + 1) + (k + 1) k +1 r2 Assume that 1 5 × 14 + 12 = (2)2 ( 2 + 1) [ = 6] so H1 is true. 2 Answer 482 Q4 Q3 2 Question 3(b) 3(a) Question ln 2. 3k ln 2 Marks ( ) dk y = a k x + ka k −1 e ax . dx k ) ( ) ( Craftear So true when n = k + 1. By induction, true for all positive integers n. ( d k +1 y = a a k x + ka k −1 e ax + e ax a k = a k +1 + ( k + 1) a k e ax k +1 dx Assume that dy = axeax + eax = (ax +1)eax so true when n = 1. dx Guidance 6 A1 States conclusion. M1 A1 Differentiates kth derivative. B1 States inductive hypothesis. M1 A1 Differentiates once using the product rule. 2 A1 AG > 2 × 3n −1 ln 2 = 3n −1 ln 4 6 A1 States conclusion, all previous marks must be gained. M1 A1 Applies inductive hypothesis and writes in required form. M1 Applies result given in part (a). B1 States inductive hypothesis. M1 Applies results given in parts (a) and (b). ) Guidance B1 Checks base case. Must be n =2 and exact. Marks  1  3ln  an +  − ln an > 2ln an an   Answer So true when n = k + 1. By induction, true for all integers n ≥ 2. ≥  1  ln ak +1 = 3ln  ak +  > 3ln ak ak   Assume that ln ak ≥ 3 k −1 a2 = 23 leading to ln a2 = 3ln 2 so true when n = 2. Answer 483 Q6 Q5 2 Question 5 Question d 2k x sin x ) = (−1) k −1 (− x sin x + 2k cos x) 2k ( dx A1 A1 Depends on previous M1 and A1. 7 2 k + 2 − 1 is divisible by 12. So if is true for n = k (could be written earlier) it is also true for n = k +1 . Hence, by induction, true for every positive integer n. Craftear M1 Separates 7 2 k − 1. Then 7 2 k +2 − 1 = 49 ⋅ 7 2 k −1 = 48 ⋅ 7 2 k + 7 2 k −1 Or (7 2 k + 2 − 1) − (7 2 k −1) = 48 ⋅ 7 2 k 5 B1 States inductive hypothesis. Assume that 7 2k − 1 is divisible by 12 for some positive integer k. Guidance B1 Checks base case. Marks 7 A1 States conclusion. M1 A1 Differentiates again. M1 A1 Differentiates once. Must have correct LHS for A1. 7 2 − 1 = 48 is divisible by 12. Answer So, it is also true for n = k + 1 . Hence, by induction, true for all positive integers. d 2 k +1 x sin x ) = ( −1) k −1 (− x cos x − sin x − 2k sin x) 2 k +1 ( dx = (−1)k (x cos x + (2k +1)sin x) Then B1 States inductive hypothesis. Assume true for n = k , so d 2 k −1 k −1 x sin x ) = ( −1) ( x cos x + (2k −1)sin x ) 2 k −1 ( dx Guidance B1 Checks base case using product rule. Marks d ( x sin x ) = x cos x + sin x = (−1)0 ( x cos x + (2(1) − 1)sin x) dx Answer 484 Q8 Q7 2 Question 2(b) 2(a) Question ( )( ) ) By induction, true for every positive integer n . so true for n = k +1 . k d k +1 y k!a k +1 k −1 (k −1)! a k = − ka (−1) = ( − 1) dx k +1 (ax + 1) k +1 (ax + 1) k +1 Then Craftear k dk y k −1 (k −1)! a = ( − 1) k (ax + 1) k for some positive integer k. Assume that dx so true for n = 1 . dy a 0!a1 0 = = ( −1) dx ax + 1 ( ax + 1)1 Answer n n u2 n 2 2 n − 1 2 − 1 2 + 1 = n = = 2n + 1 n un 2 −1 2 −1 ( Guidance 6 A1 States conclusion. M1 A1 Differentiates k th derivative. B1 States inductive hypothesis. M1 A1 Proves base case. Marks So, it is also true for n = k + 1 . Hence, by induction, true for all positive integers. 2 M1 A1 5 A1 M1 A1 B1 Assume that it is true for n = k , so uk = 2k − 1 . Then uk +1 = 2(2k − 1) + 1 = 2k +1 − 1 B1 Marks u1 = 1 = 21 −1 Answer 485 Q10 Q9 1 Question 8(i) Question k k k +1 z − 1 k z − 1 + z ( z − 1) z − 1 +z = = , z −1 z −1 z −1 k zk −1 z −1 Guidance A1 A1 States conclusion. is divisible by 13 H k ⇒ H k +1 By induction, 33n − 1 is divisible by 13 for every positive integer n . Craftear M1 Separates 33k − 1 Then 33k + 3 − 1 = 3333k − 1 = 26 ⋅ 33k + 33k − 1 5 B1 States inductive hypothesis. Assume that 33k − 1 is divisible by 13 for some positive integer k Guidance B1 Checks base case. Marks 5 A1 States conclusion. M1 A1 Combines fractions. B1 States inductive hypothesis. B1 Shows base case. Marks 33 − 1 = 26 is divisible by 13 Answer H k → H k +1 Hence, by induction, true for all positive integers. so true when n = k +1 1 + z + … + z k −1 + z k = Then Assume that 1 + z + …+ z k −1 = z1 − 1 z −1 So true when n =1 . 1= Answer 486 Q12 Q11 9(i) Question 2 Question n + 3, Craftear So Pk ⇒ Pk+1. Therefore, by induction, Pn is true for all positive integers. k +1 A1 States conclusion. A1 M1 Proves inductive step. k   1  5 uk +1 =  5  4   + 3  − 3  = correct step   4    4     5 = 4  4 B1 States inductive hypothesis. Proves base case. B1 States proposition. Assume Pk is true for some k . Then 5 Pn : un = 4   + 3 4 5 Let n = 1 then 4   + 3 = 8 ⇒ P1 true. 4 Marks Guidance A1 = 23.23k + 8.8k–1 M1 = 8 f(k) A1 So if f (k) is divisible by 9, so is f (k +1), (and f (1) is divisible by 9), f ( n ) is divisible by 9 for every integer n ⩾1 Answer A1 Alt method: f (k + 1) = 23k + 3 + 8k = 9 23k + 8k −1 OE so f ( k +1) is divisible by 9. 6 A1 Correct split of powers 23.23k + 8.8k–1 + 23k + 8k–1 OE ) M1 Uses expansion of f ( k + 1) . f ( k + 1) + f ( k ) = 23k +3 + 8k + 23k + 8k −1 = ( B1 Makes general statement. Assume that f ( k ) is divisible by 9. Guidance B1 Checks base case. Marks We have that f (1) = 9 is divisible by 9. Answer 487 9(iii) 9(ii) n N ln(un − 3) = N ∑   5 n  ln  4      n =1  4  N ( ) 1 5 N ( N + 1) ln + N ln 4 2 4 n =1 N 1 2 Use ∑n = N ( N +1) . 4 5x 4 <1⇒− < x < 4 5 5 n = N 2 ln Craftear 5 5 + N ln 2 5 ⇒ a = , b = 2 5 oe 2 2 Alt method: Writes series as an AP M1, uses summation formula M1 Correct answer A1 = ∑ ∑  5 =  ln  n + ln 4  4  n =1 n =1 n =1 ∑ N So series is convergent for −1 < n5 5x 5x ( un − 3) x = 4 x   = 4   so r =    4  4  4  n 10 A1 M1 = N 2 ln = Nln4 + ∑ n ( ln( ) 5 ) M1 4 ∑n 5 5 + N ln 2 5 ⇒ a = , b=2 5 2 2 2 N ( N + 1) 4 5 = Nln4+ ln   = ln 4N ∏   4 n M1 5 1   N 5 ln(un − 3) = ln ∏ 4   4 n=1 N M1 Alt method: A1 M1 A1 488 Q13 3(ii) 3(i) Question 4uk + 9 −uk + 3 = > 0 ⇒ uk +1 < 3 uk + 4 uk + 4 4un + 9 −u 2 + 9 − un = n un + 4 un + 4 So un < 3 ⇒ un+1 − un > 0 . un +1 − un = Hence, by induction, un < 3 for all n.1 . 3 − uk +1 = 3 − Then Craftear 3 B1 Uses un < 3 M1 A1 Considers un+1 − un . 5 B1 States conclusion. M1 A1 B1 States inductive hypothesis. Assume that uk < 3 Guidance B1 States base case. Marks u1 < 3 ( given ) Answer 489 Q14 6 Question k  k r k k  k  1 = ex    u +   u( ) + " +   u( ) + " +   u( )  + 1 r k  0  k  1  k  2  k  r +1  k  k +1  e x    u( ) +   u( ) + " +   u( ) + " +   u( )  1 r k  0  k +1) Guidance Craftear A1 States conclusion. So H k implies H k +1 so, by induction, H n is true for all n .1 . 8 B1 Shows reasoning for first and last term correctly M1A1 Shows application of  k   k   k + 1  + = .  r   r −1  r  M1 Differentiates using product rule B1 States inductive hypothesis. M1A1 Shows base case using product rule Marks   k +1  k +1 r  k + 1 ( k +1)  = ex     u + …  u + … u  r   k + 1  0    k  k   k  r k = e x    u + …   +  u + …  u ( k +1)      k   r   r − 1   0  y( Then  k  k 1 k r k k  k H k : y( ) = ex    u +   u( ) + " +   u( ) + " +   u( )  1 r k  0  Assume that 1  1 1  1 1 y ( ) = e x u ( ) + ue x = e x    u +   u ( )  ⇒ H1 is true  1  0  Answer 490 Q15 2 Question Craftear P0 is true and Pk ⇒Pk+1, hence Pn is true for all non-negative integers n. (or shows that 5k +1 + 3 = 5.5k + 5.3 − 4.3 = 5 5k + 3 − 4.3 ) ) Total: 5 A1 A1 = 20α −12 = 4 ( 5α − 3) ( M1 Alt method: Use f(k+1) – f(k) 5k +1 + 3 = 5 ( 4α − 3) + 3 M1 A1 B1 or e.g. 5k +3 = 4α for 2nd B1 Assume that Pk is true for some non-negative integer k. Guidance B1 Some explanation of what Pk being true means Marks Let Pn be the proposition that 5n+3 is divisible by 4 50 +3 =4 ⇒ P0 is true (allow P1) Answer 491 Q17 Q16 3(i) Question 3 Question  21  1 × ln2 = ln2 = ln   ⇒ (H1 is true )  1!   ( k +1)k   r +1  k +2  + ( k + 1) ln   rln  = ln     k!  r   k +1  r =1   ( ) ( dn n x + ( n +1) x n lnx n dx ) d n +1 n+1 d n  n +1 1  x . + ( n + 1) x n lnx  = x ln x = n +1 n  x dx dx   Answer Craftear 2 M1A1 Marks AG Guidance 6 A1 Thus Hk ⇒ Hk+1 and hence by PMI Hn is true for all positive integers. Total: A1 = ln k k +1 k + 1) ( k + 2 ) ( = ln ( k + 1)!( k + 1)k k +1 k + 2) ( = ln ( k + 1)! B1 M1 ∑ k +1 B1 B1 Marks ( k + 1)k ( k + 2 )k +1 k +1 k !( k +1) Hence ∑ k  [ k +1]k   r +1   rln  Assume, for some positive integer k, that  = ln  k!   r  r =1   When n =1 Answer Guidance 492 3(ii) ( ) ) Craftear 5 A1 A1 1 1   = ( k +1)!lnx + 1+ + … +  ⇒ Hk+1 is true k +1  2  is true; hence, by PMI, A1 M1 B1 Statement of Hk seen 1 1  = k !+ [ k +1] k !lnx + 1 + + … +  2 k  ( Check H1 is true and Hk is true ⇒ H Hn is true for all positive integers n. ( ) dk 1 1  x k lnx = k !lnx + 1 + + … +  k 2 k dx  d k +1 k +1 dk x x ln = x k + [ k + 1] x k lnx k +1 k dx dx Assume Hk is true ⇒ 493 494 Q18 3 For n =1 10 +192 + 5 = 207= 9 × 23 ⇒ H1 is true. B1 Assume Hk is true for some positive integer k ⇒ 10n + 3.4n +2 + 5 = 9α Let f(n) = 10 n + 3.4n + 2 + 5 Hence f ( n +1) − f ( n ) =10n (10 −1) + 3.4n+2 ( 4 − 1) B1 ( = 9 10 n + 4 n + 2 = 9β M1 ) A1 Hence f ( n + 1) ( = 9 ( β + α ) ) ⇒ Hk+1 is true A1 H1 is true and Hk ⇒ Hk+1 , hence by PMI Hn is true for all positive integers n. ( ) N.B. Or can show f ( n +1) = 9 10α − 2.4 n +2 − 5 for M1A1A1. (3rd ,4th&5th A1 [6] marks) Q19 1 n ( n − 3) = 0 2 A triangle has no diagonals ⇒ H3 is true. 1 Assume Hk is true: A k-gon has k ( k − 3) diagonals for some integer ≥ 3 2 Adding an extra vertex, a further (k – 1) diagonals can be drawn. 1 k 2 − 3k + 2k − 2 ( k + 1)( k − 2 ) k ( k − 3) + k − 1 = = 2 2 2 1 = ( k + 1)( k +1− 3) (So Hk ⇒ Hk+1) 2 ⇒Hn is true for all integers n 3 . With n = 3 , M1 A1 B1 M1 A1 A1 [6] Craftear 2 Q20 4  1  1 k 1 k r k k  k   k   k + 1 k − r +1 r x is:  Multiplying by ( a + x ) , the coefficient of a +  =   r −1  r   r  Hence Hn is true for all positive integers. ⇒ Hk+1 is true. k Craftear ( a + x ) =   a k +   a k −1 x +…+   a k −r x r +…+   x k k 0 Assume Hk is true,i.e. 1 ( a + x ) =   a +   x = a + x ⇒ H1 is true. 1 0  n  n n! n! n! 1  1 + = +   +  =   r −1  r  ( r − 1)!( n − r + 1) ! r !( n − r ) ! ( r − 1) !( n − r ) !  n − r + 1 r   r + n − r +1  ( n + 1)! =  n +1 n! =   = ( r − 1)!( n − r )!  r ( n − r +1)  r !( n − r + 1)!  r  A1 M1 B1 B1 A1 M1 [4] [2] 495 Q21 496 3 a1 > 5 (given) ⇒ H1 is true. Assume Hk is true for some positive integer k, i.e. ak = 5 + δ, where δ > 0. 2 2 4a + 25 − 25ak (4ak − 5)(ak − 5) 4a + 25 ak +1 − 5 = k −5= k = > 0 , ⇒ ak +1 > 5 5ak 5ak 5ak Or 5 4 4 δ δ2 ak +1 = (5 + δ ) + , = 4 + δ + (1 − + − ...) for 0 < δ < 5 5 5+δ 5 5 25 3 = 5 + δ + 0(δ 2 ) [ ak+1 > 5, ( δ [ 5 is trivial). 5 Hk ⇒ Hk +1 and H1 is true, hence by mathematical induction, the result is true for all=== n ∈ Z + =(N.B. The minimum requirement is ‘true for all positive integers’.) 5 1 a k +1 − a k = − ak ak 5 1 5 < 1 and ak > 1 ⇒ ak +1 − ak < 0 ⇒ ak +1 < ak 5 ak B1 B1 M1A1 (M1) (A1) A1 (5) M1 A1 (2) Total: 7 Q22 Hk : ∑ k 1 r = 1 (2r ) 2 − 1 = k is true for some integer k. 2k +1 1 1 1 = = ⇒ H1 is true. 2 − 1 3 2 ×1 + 1 1 k k 1 2k 2 + 3k + 1 + = + = 2k + 1 (2k + 2) 2 − 1 2k + 1 (2k + 1)(2k + 3) (2k + 1)(2k + 3) (2k + 1)(k + 1) k +1 = = (2k + 1)(2k + 3) 2[k + 1] + 1 ∴ H k ⇒ H k +1 ∴ (By Principle of Mathematical Induction) Hn is true for all positive integers n. (This mark requires all previous marks.) 2 1 1 = 2 r = 1 (2r) −1 2 ∑∞ B1 B1 M1A1 A1 A1 (6) B1 (1) Total 7 Q23 3 n =1 in formula gives a 0e ax + axe ax = e ax + axe ax d xe ax = e ax × 1 + x.aeax = eax + axe ax ⇒ H1 is true oe dx ( ) Assume Hk is true, i.e. d k +1 xe ax dx k +1 dk xe ax = ka k −1e ax + a k xe ax . k dx ( ) ( ) = ka e + a e + a xe k ax Craftear 3 k ax k +1 ax = (k +1)a k eax + a k +1xeax ⇒ H k +1 is true, hence by PMI Hn is true for all positive integers n. B1 B1 B1 M1 A1 A1 [6] Total 6 Q24 3 497 Hk: f(k) = 112k + 25k + 22 = 24λ B1 f(0) = 1 + 1 + 22 = 24 = 1 × 24 ⇒ H is true. B1 f(k + 1) – f(k) = 112k + 2 + 25k + 1 + 22 – (112k + 25k + 22) = 112k(121 – 1) + 25k(25 – 1) = 112k × 24 × 5 + 25k × 24 = 24µ M1 A1 A1 Alternatively: f(k + 1) = 112k + 2 + 25k + 1 + 22 = 121.112k + 25.25k + 22 = (120 + 1)112k + (24 + 1)25k + 22 (OE) = 120.112k + 24.25k + 24λ = 24µ (M1) (A1) (A1) ⇒ f(k + 1) = 24µ + 24λ = 24(µ + λ) ⇒ Hk + 1 is true. Hence by PMI Hn is true for all non-negative integers. (Must see non-negative integers.) CSO: Final mark requires all previous marks. A1 [6] 3 φ (1) = 5 × 5 – 1 = 24 which is divisible by 8 ⇒ H1 is true. B1 Assume Pk is true for some positive integer k ⇒ φ (k) = 8l φ (k + 1) – φ (k) = 5k + 1(4k + 5) – 1 – 5k(4k + 1) + 1 = 5k(20k + 25 – 4k – 1) = 5k(16k + 24) = 8m ∴φ (k + 1) = 8(l + m) Hence, by PMI, true for all positive integers n. (CWO – all previous marks required.) B1 M1 A1 A1 A1 A1 [7] Alternatively φ (k + 1) = 5k + 1(4k + 5) – 1 = 5. (4k.5k) + 25.5k – 1 = 5(8l – 5k + 1) + 25.5k – 1 = 40l + 20.5k + 4 = 40l + 24.5k – 4.5k+ 4 = 40l + 24.5k – 4(5k– 1) = 40l + 24.5k – 4(8l – 4k.5k) = 8l + 24.5k + 16k.5k = 8m (M1A1) (A1) (A1) Q26 3 2! – S1 = 1 , 3! – S2 = 1 , 4! – S3 = 1 , 5! – S4 = 1 (Two correct B1, all four correct B2) Sn = (n +1)! – 1 2! – 1 = 2 – 1 = 1 ⇒ H1 is true. Hk: Sk = (k +1)! – 1 (k +1)! – 1 + (k + 1) × (k + 1)! = (k + 1)!(1 + k +1) – 1 = ([k +1] + 1)! – 1 Hence Hk ⇒ Hk+1 So result holds for all positive integers (by PMI). Craftear Q25 B2,1,0 (2) B1 (1) B1 B1 M1 A1 (4) [7] Q27 2 498 Proves base case. Pn: 5 2 n − 1 is divisible by 8. 5 2 −1 = 24 = 3× 8 ⇒ P1 is true B1 B1 States inductive hypothesis. Assume Pk is true: 5 2 k − 1 = 8λ for some k. 5 2 k + 2 − 1 = 25.5 2 k − 1 = 24.5 2 k + 5 2 k − 1 = 3 × 8.5 2 k + 8λ M1 Proves inductive step. ∴ Pk ⇒ Pk+1 A1 States conclusion. (Since P1 is true and Pk ⇒ Pk+1)∴ Pn is for every positive integer n (by PMI). A1 5 [5] Q28 Differentiates once. Rearranges. Shows true for n = 1 . States inductive hypothesis. (May be seen by implication.) ( ) M1 A1 ( ) B1 ( 2 )  sin x + 14 kπ e + e cos x + 14 kπ   M1 = ( 2 ) e  12 sin x + 14 kπ  + 12 cos x + 14 kπ   A1 = ( 2 ) e sin x + 14 (k +1)π  ⇒ H A1 Differentiates. d k +1 = dx k +1 Rearranges. Shows H k ⇒ H k +1 and states conclusion. B1 d x e sin x = e x sin x + e x cos x dx  1  1 cos x  = 2e x  sin x + 2  2  1   = 2e x sin x + π  ⇒ H1 true . 4   k k 1  d  H k : k (e x sin x) = 2 e x sin  x + kπ  4  dx  k +1 k x x   x  k +1 x  k +1 true. Craftear 3 [7] 7 ∴ ,by PMI, true for all positive integers n. (CWO) Q29 5 (i) Differentiates once, twice and three times. (ii) y′ = 2(1+ x )ln (1+ x ) + (1+ x ) y′′ = 2ln (1+ x ) + 3 2 y′′′ = 1+ x (Allow B1 if constant term in previous line incorrect.) d 3 y (− 1) .2.0! 2 = = ⇒ H 3 is true. 3 1+ x 1+ x dx k −1. d k y (− 1) .2.(k − 3)! Hk : k = for some k . dx (1 + x )k − 2 B1 B1 B1 3 2 Proves base case. States inductive hypothesis. Differentiates Proves inductive step and states conclusion. d k +1 y k −1 − ( k −1) = (−1) .2(k − 3)!(−1)(k − 2 )(1 + x ) dx k +1 (− 1)k .2.(k − 2)! ⇒ H is true = k +1 (1 + x )k −1 Hence by PMI Hn is true for all integers [=3 B1 B1 M1 A1 A1 5 [8] 499 Q30 9 States inductive hypothesis Proves H k ⇒ H k +1 Hk: (cosθ + i sinθ ) = cos kθ + i sin kθ for some k k (cosθ + isinθ )k +1 = (cosθ + isin θ )(cos kθ + isin kθ ) = (cosθ coskθ − sinθ sin kθ ) + i(sinθ cos k θ + cosθ sin kθ ) = cos [k + 1]θ + 1sin[k +1]θ ∴Hk ⇒ H k+1 States conclusion H1 is trivially true, so true for all positive integers Uses de M’s Thm. to find 2isin nθ 1 = (cos nθ + i sin nθ ) − (cos nθ − i sin nθ ) zn = 2i sin nθ zn − B1 M1 A1 A1 A1 (5) B1 5 and groups = z 5 − 5 z 3 + 10 z − Applies above result and obtains sin5 B1 10 5 1 + − 5 3 z z z 1   1 1   =  z 5 − 5  − 5 z 3 − 3  + 10 z −  z z   z    M1 A1 32i sin5 θ = 2i sin5θ −10i sin3θ + 20i sinθ M1 sin 5 θ = A1 ( 1 sin 5 θ − 5 sin 3θ + 10 sin θ 16 ) (6) [11] Q31 2 Craftear Uses binominal expansion 1  5 5  z −  = (2i sin θ ) = 32i sin θ z  (States proposition.) (Pn: 4n > 2n + 3n) Proves base case. Let n = 2, 16 > 4 + 9 ⇒ P2 is true. B1 States inductive hypothesis. Proves inductive step. Assume Pk is true ⇒ 4k > 2k + 3k B1 4 k +1 = 4.4 k > 4(2 k + 3 k ) = 4.2 k + 4.3 k > 2.2 k + 3.3 k = 2 k +1 + 3 k +1 M1 A1 ∴ Pk ⇒ Pk+1 States conclusion. Hence result true, by PMI, for all integers n ≥ 2. A1 (CWO) 5 [5] 500 Q32 5 Integrates by parts to obtain reduction formula. ∞ ∞ ∫ xe n −2 x 0 -2x ∞  − e −2 x  n −1 e dx =  x n + nx dx  0 2 0 2  n = I n −1 (AG) 2 ∫ (States proposition.) Pn : I n = n! 2 n +1 Proves base case. n=1 I0 = Shows Pk ⇒ Pk +1 . States conclusion. ∞ 1  1  e -2x dx = − e -2x  = 0  2 0 2 1 1 1 1! I1 = × = = 2 ∴ P1 true. 2 2 4 2 ∫ ∞ k! for some integer k. 2 k +1 (k + 1)! k + 1 k! ∴ I k +1 = × k +1 = k +2 2 2 2 ∴ Pk ⇒ Pk +1 Pk : I k = Hence by PMI Pn is true for all positive integers n. M1 A1 2 B1 B1 B1 M1A1 A1 [8] 6 3 1 ( N + 1)! Proves base case. S = 1 = 1 = 1 − 1 ⇒ H1 is true. 1 2! 2 2! States inductive 1 Hk : Assume S k = 1 − is true. hypothesis. (k + 1)! Proves inductive ⇒ S = 1 − 1 + k + 1 = (k + 2)!−(k + 2) + (k + 1) k +1 step. (k + 1)! (k + 2)! (k + 2)! 1 ∴ Hk ⇒ Hk+1. ⇒ S k +1 = 1 − (k + 2)! (By PMI Hn is) true for all positive integers N. States conclusion. ∴ Proposition. Craftear Q33 HN : S N = 1 − B1 B1 M1 A1 A1 5 [5] Q34 2 n (States proposition.) Proves base case. States Inductive hypothesis. Proves inductive step. 3 (Pn : un = 4  − 2) 4 3 Let n = 1 4 × – 2 = 3 –2 = 1 ⇒ P1 true. 4 Assume Pk is true for some k.    3  k 34  − 2 − 2 k    4  3 3 6+2 = 4. .  − u k +1 = 4 4 4 4 B1 B1 M1 k +1 States conclusion. 3 = 4.  − 2 ∴ Pk ⇒ Pk +1 4 ∴ By PMI Pn is true ∀ positive integers. A1 A1 5 [5]

Further Math P1 Topical Workbook (2012-2021)

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