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12 05.04.09 sundayherald MATHEMATICS STANDARDGRADE : GENERAL LEVEL (2008 PAPER 1) 1. To enter the castle she needs the correct four digit code. (b) 6.39 × 9 (c) 8.74 ÷ 200 The computer gives her some clues: (d) 5 of 420 6 2. Marks Marks 5. Samantha is playing the computer game “Castle Challenge”. Carry out the following calculations. (a) 12.76 – 3.18 + 4.59 Marks In the “Fame Show”, the percentage of telephone votes cast for each act is shown below. Plastik Money Brian Martins Starshine Carrie Gordon • • • only digits 1 to 9 can be used each digit is greater than the one before the sum of all four digits is 14. (a) The first code Samantha found was 1, 3, 4, 6. 23% 35% 30% 12% Use the clues to list all the possible codes in the table below. 1 3 4 6 Altogether 15 000 000 votes were cast. How many votes did Starshine receive? 3. Marks AB and BC are two sides of a kite ABCD. y 6 4 B 3 C 2 (b) The computer gives Samantha another clue. • –6 –4 –2 O 2 4 x 6 three of the digits in the code are prime numbers What is the four digit code Samantha needs to enter the castle? –2 A Marks 6. –4 -3 –6 12 -9 -8 (a) Plot point D to complete kite ABCD. 1 5 7 -11 (b) Reflect kite ABCD in the y-axis. 4. Marks Europe is the world’s second smallest continent. Find the three numbers from the circle which add up to –10. Write this number in scientific notation. You must show your working. 3. ANSWERS 1. y 6 3. (c)0·0437 (d)350 4500000 4. 5. 5 4 C 3 B (a)14·17 (b)57·51 2. The circle above contains seven numbers. Its area is approximately 10 400 000 square kilometres. 2 1 0 –6 –5 –4 -3 –2 –1 –1 –2 A D 1 2 3 4 5 6 x 6. 7. 8. –3 –4 –5 –6 9. 1·04 × 107 (a) 1238, 1247, 12 56, 2345 (b) 2345 –9, –8, 7 £1·22 (a) 23 (b) 220° sundayherald 05.04.09 7. 13 Marks The cost of sending a letter depends on the size of the letter and the weight of the letter. HIGHER (2008 PAPER 1, SECTION A) SECTION A Weight Format Letter Large Letter ALL questions should be attempted. Cost 1st Class Mail 2nd Class Mail 0–100 g 34p 24p 0–100 g 48p 40p 101–250 g 70p 60p 251–500 g 98p 83p 501–750 g 142p 120p 1. A sequence is defined by the recurrence relation un+1 = 0.3un + 6 with u10 = 10. What is the value of u12? A C 2. 6.6 8.7 7. 8 8 7 9. 6 B C D The x-axis is a tangent to a circle with centre (–7, 6) as shown in the diagram. y Claire sends a letter weighing 50 g by 2nd class mail. She also sends a large letter weighing 375 g by 1st class mail. C(–7, 6) Use the table above to calculate the total cost. 8. Marks Four girls and two boys decide to organise a tennis tournament for themselves. O Each name is written on a plastic token and put in a bag. What is the equation of the A circle? (x + 7) + (y – 6) = 1 (a) What is the probability that the first token drawn from the bag has a girl’s name on it? A B (x + 7)2 + (y – 6)2 = 1 B (x + 7) + (y – 6) 49 C 2 (x –+ 7) 7)2 + + (y (y + – 6) 49 (x 6) = 36 C (x – 7)2 + (y + 6)2 = 36 D (x + 7)2 + (y – 6)2 = 36 (b) The first token drawn from the bag has a girl’s name on it. This token is not returned to the bag. What is the probability that the next token drawn from the bag has a boy’s name on it? "k" 3. Marks 9. C 4. O A 70 ° T In the diagram above: x "−# The vectors u = −1 "1$ "0" and v = "4# are perpendicular. "k$ What is the value of k? A 0 B 3 C 4 D 5 A sequence is generated by the recurrence relation un+1 = 0.4un – 240. What is the limit of this sequence as n → ∞ ? A – 800 B – 400 C 200 D 400 B 5. The diagram shows a circle, centre (2, 5) and a tangent drawn at the point (7, 9). What is the equation of this tangent? • O is the centre of the circle • AB is a tangent to the circle at T • angle BTC = 70 °. y (7, 9) Calculate the size of the shaded angle TOC. O 6. (2, 5) x A y – 9 = − 5 (x – 7) B 4 C y–7= 4 (x – 9) 5 D 4 y + 9 = − (x + 7) 5 y+9= 5 (x + 7) 4 What is the solution of the equation 2 sin x − 3 = 0 where A π 6 C 3π 4 2π 3 5π D 6 B π ≤ x ≤ π? 2 14 05.04.09 sundayherald The adiagram shows a line L; theLangle between L and the 12. positive direction ofRSTU, the 7. The diagram shows line L; the angle between and the positive direction of the In the diagram VWXY represents a cuboid. e direction of the x-axis is 135°, as shown. → → →" SR represents vector f, ST represents vector g and SW represents vector h. y → Express VT in terms of f, g and h. Y L V 135° O X W x h U R What is the gradient of line L? 8. f 3 2 A 11 −− 22 B − C −− 11 D 1 1 2 2 A C g S → VT = f + g + h → → VT = − f + g − h B D T → VT = f − g + h → VT = − f − g + h The diagram shows part of the graph of a function with equation y = f(x). 13. y The diagram shows part of the graph of a quadratic function y = f(x). The graph has an equation of the form y = k(x – a)(x – b). (0, 4) y O x y = f(x) 12 (3, –3) Which of the following diagrams shows the graph with equation y = –f(x – 2)? y A B (2, 4) O y (–2, –4) D 10. (5, 3) (2, –4) sin a = 15. 3 , find an expression for sin(x + a). 5 4 3 sin x + cos x 5 5 2 3 sin x − cos x 5 5 Here are two statements about the roots of the equation x + x + 1 = 0: 17. Which of the following is true? A Neither statement is correct. A Neither statement is correct. ly statement (1) B C Only statement (1) is correct. Only statement (2) is correct. Only statement (2) is correct. D Both statements are correct. E(–2, –1, 4), P(1, 5, 7) and F(7, 17, 13) are three collinear points. P lies between E and F. What is the ratio in which P divides EF? B y = 3(x + 1)(x + 4) y = 12(x – 1)(x – 4) D y = 12(x + 1)(x + 4) Find ∫ 4 sin (2x + 3) dx. cos ( 3) A – 4cos 3) + c cos ((2x + 3) B C –2cos 3) + + cc 4cos (2x (2x + + 3) C 4cos (2x + 3) + c D 8cos (2x + 3) + c What is the derivative of (x3 + 4)2? 1 3 A (3x2 + 4)2 B (x + 4)3 3 ( + 4) 6x2(x3 + 4) D 2(3x2 + 4)–1 2x2 + 4x + 7 is expressed in the form 2(x + p)2 + q. What is the value of q? 2 C y = 3(x – 1)(x – 4) C C 16. (1) the roots are equal; (2) the roots are real. 11. 14. x O Given that 0 ≤ a ≤ π and 2 3 A sin x + B 5 3 4 C sin x − cos x D 5 5 A y x (0, –2) x 4 What is the equation of the A graph? y 3(x – 1)(x – 4) x O (3, 5) O 9. O 1 (1, 3) x (5, –3) C y A 5 B 7 C 9 D 11 2 A function f is given by f(x) = 9 − x . What is a suitable domain A x ≥ of 3 f? 18. A x≥3 B C x ≤≤3x ≤ 3 –3 C –3 ≤ x ≤ 3 D –9 ≤ x ≤ 9 Vectors p and q are such that |p| = 3, |q| = 4 and p.q = 10. Find the valueAof q.(p 0 + q). A 1:1 B 1:2 A B 0 14 B C 14 26 C 1:4 D 1:6 C 26 D 28 sundayherald 05.04.09 19. 15 x The diagram shows part of the graph whose equation is of the form y = 2m . ANSWERS What is the value of m? y SECTIONA (3, 54) x O A 3 C 8 2.D 3.C 4.B 5.A 6.B 7.C 8.D 9.B 10.A 11.B 12.C 13.A 14.B 15.C 16.A 2 B 1.C 17.C 18.C 19.B 20.D SECTIONB D 18 21.(a) (–1,4)maximum 20. (1,0)minimum The diagram shows part of the graph of y = log3(x – 4). The point (q, 2) lies on the graph. y (b) so(x–1)isafactor (q, 2) (5, 0) (i)x=1,f(x)=0 y = log3(x – 4) (c) x O (ii)(x–1)(x–1)(x+2) ( ) ( )( )( ) y (c) (–1, 4) What is the value of q? A (0, 2) 6 B 7 C 8 (–2, 0) D 13 [END OF SECTION A] SECTION B ALL questions should be attempted. 22. 21. A function f is defined on the set of real numbers by f(x) = x3 – 3x + 2. (b)(1,3) ) Find the coordinates of stationary the stationary points (a) Find the coordinates of the points on the curve y = f(x) and ) and determine their nature. (b) (i) Show that (x – 1) is a factor of x3 – 3x + 2. 23. (ii) Hence or otherwise factorise x3 – 3x + 2 fully. meets thethe axes and hence sketch the curve. )both State coordinates of thewhere points where curve with equation (c) State the coordinates of the points the curvethe with equation y = f(x) y = f(x) y = f(x) meets both the axes and hence sketch the curve. 22. The diagram shows a sketch of the curve with equation y = x3 – 6x2 + 8x. (a) Find the coordinates of the points on the curve where the gradient of the tangent is –1. y y = x3 – 6x2 + 8x O (b) The line y = 4 – x is a tangent to this curve at a point A. Find the coordinates of A. 23. Functions f, g and h are defined on suitable domains by f(x) = x2 – x + 10, g(x) = 5 – x and h(x) = log2 x. (a) Find expressions for h(f(x)) and h(g(x)). (b) Hence solve h(f(x)) – h(g(x)) = 3. [END OF SECTION B] (a)(1,3),(3,–3) x (a)h(f(x))=log2(x2–x+10) h(g(x))=log2(5–x) (b)x=3,–10 (1, 0) x