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GRADE 11 BILINGUAL PURE MATHEMATICS Name:_______________ SEMESTER 1 Textbook: Advanced Maths as core for Edexcel (C1C2) Rosemary Emanuel & John Wood HW=homework (5 min) SQ=1 or 2 short questions (5 min) ST=short test(2 per semester 10marks) Max 20 minutes Oral UNIT 1 Algebra Quadratic Equations & Functions (Pg. 41 to 63) Chapter 3 (C1) Introduction – Pre-knowledge needed for grade 11 basics Solving linear equations Solve quadratic equations by : 1. Factoring and applications 2. Formula 3. Completing the square 4. Applications Sketching Quadratic Graphs 1. Max/min 2. Shape 3. Turning Point(vertex) and axis of symmetry 4. Intercepts with x and y axes Nature of roots (working with the discriminant) Review Booklet 3A Page 46 3B Pg. 55 3B Pg. 56 3B Pg. 55 3B Pg. 56 Solving Inequalities (Pg. 65 to 70) Chapter 4 (C1) Solving inequalities: 1. Linear 2. Quadratic Revise 4B Pg. 67 4C Pg. 70 4D Pg. 71 Solving Simultaneous equations (Pg. 72 – 87) Chapter 5 (C1) 1. 2 Linear 2. 1 linear & 1 quadratic Intersecting linear & Quadratic graphs, finding the point and dealing with. 3 cases of the discriminate related to this Review SS = selfstudy Lessons 1 2 1 2 SS Total 3 24/9/17 6 5A,B Pg. 75, 78 5C Pg. 83 5D Pg. 87 1 2 2 1/10/17 7 5E Pg. 87 SQ, Oral SS Total 5 8/10/17 8 3C Pg. 59 3D Pg. 62 3E Pg. 63 2 1 2 1 1 2 SS Total 10 Chapter 18 (C2) Revise exponents Definition – log exp Rules: ; WS 18A Pg. 313 18B Pg. 316 1 WS 1 ) Special cases: 3 ( ) Exponential function: Graphs Relationship between (log & exp). 1 Week 28/8/17 1 UNIT 2 Exponents & Logs (Pg. 312 – 323) ( Predicted Date Date Complete Change of base Solving equations Review Assessment 18C Pg. 319 18D Pg. 322 18E Pg. 323 HW, ST UNIT 3 – Geometry Coordinate Geometry (Pg. 92 – 110) Chapter 6 (C1) Revision: Coordinates, midpoint, gradient(+/-), (drawing and writing an equation), length of line segment joining 2 points. 6A Pg. 96 3 6B Pg. 105 2 6B Pg. 105 6B Pg. 106 1 2 Gradient of a line including: m=tanθ Special gradients (i.e. 0, 1, -1 and ∞) Parallel & perpendicular lines( Determine if points lie on a line or curve Equation of a straight line using 3 different formulae ( ) 1 2 SS 1 Total 9 15/10/17 9 12/11/17 13 26/11/17 15 ) ( ) Forms of equations of straight lines: 1 6B Pg. 106 Sketching straight lines Applications on straight lines Applications in co-ordinate geometry, use all previous knowledge Review Self-study Circle graph equation ( ) ( ) Tangents and normal to circle. Assessment & tests 6B Pg. 107 6C Pg. 109 6D Pg. 111 13A,B Pg. 204 2 1 3 SS 1 HW, Oral Total 15 UNIT 4 – Trigonometry Solving triangles, radians and applications Page 252 – 253 and 282 – 302 Radians definition radian degree Angles and quadrants ( Length of an Arc ( ) Area of a sector ( Area of a triangle ( Area of segment ( Solving triangles: Sin and cosine rules Assessment Chapter 16 (C2) Chapter 17 (C2) 16A Pg. 255 [leave 1 out angles greater than 360°/2π or less than 0° ] ) ) ) ) 17A Pg. 284, 285 17A Pg. 284, 285 1 2 17B Pg. 289, 290 17B Pg. 289, 290 17C, 17D, 17E, 17F Pg. 291 – 302 SQ, Oral 2 1 5 Total 12 Trig functions for any angle Page 254 - 268 Chapter 16 (C2) radian degree Trig ratio’s in all 4 quadrants (including reduction formulae: ) Special angles – special triangles (angles of any size) 16A Pg. 255 angles < 0°, >360° 16B Pg. 266 - 268; 2 1 16B Pg. 266 - 268 2 Graphs of trig functions Period, asymptotes, symmetry Transformation of trig graphs ( ) ( ) ( ) ( i.e.: ( ) 16B Pg. 266 – 268 2 16B Pg. 266 – 268 3 HW, SQ, ST 2 Total 10 ) Self-study, Assessment & tests SEMESTER 1 EXAM 3 LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: Aug/September MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Quadratics Equations PERIOD Week 1 CLASS WORK WORKBOOK AND TEXTBOOK Pre-knowledge: expressions, multiplication, factorisation Linear equations Solve and Booklet pages 5-9 Worksheet pages 9-11 Page 10 & Ex 3A no 1i,k,q, no 2a,b,f,h,k,l,n 1-2 Look at basic linear equations and then at fractions Quadratic equations Do a worksheet on basic factorization Explain common factors, diff of two squares, trinomials Worksheet pages 12-16 3 Factorization in equations Do in class 3B no 1a,d,g,t, no 2b, no 8b, Also show two ways to solve equations like Ex. 3B no 1k, l, o, r, 2 e, g, 7 c, d, 8 c, e 4-5 Consolidate quadratic factorization Ex. 3B no 8 a, b, d, f Completing the square for trinomials that do not factorize Ex 3B no 3 a, b, c, d, e, i, j Revise the quadratic formula Any quadratic equation can be solved by using the quadratic formula!! Ex 3B no 6 a, d, j, Applications of quadratic equations Ex 3B no 9 and 10 5 6-7 8 9 4 PRE-KNOWLEDGE FROM IGCSE NEEDED FOR THIS SECTION: Expressions 5 6 Quadratic expressions: Perfect squares: ( ) ( ) ( ) ( Difference of perfect squares factorised: ) ( )( ) Trinomials: Three different types: Examples: (Factorise) ( )( ) add the factors ( )( ) add the factors ( )( ) look for difference Take factors of the last term and add them until the sum gives the coefficient of the center term. ( )( ). Because the last term is +, both signs in the brackets will be the same as the sign of the center term Take factors of the last term and add them until the sum gives the coefficient of the center term. ( )( ). Because the last term is +, both signs in the brackets will be the same as the sign of the center term Take factors of the last term - and look for the difference between the two factors to be the value of the coefficient of the center term. ( ). Because the last term is -, )( one signs in the brackets will be + and one sign will be ( )( ) 7 In quadratics where the is not 1, the factors of the first term and the last term must give the coefficient of the center term. An easy method to factorise quadratics of this type looks as follows: : multiply the coefficient of ( and the constant with each other, . ) : then find two factors of that will add up to the coefficient of , that is or : then breakdown the center term as and rewrite the quadratic as: (hint: it is easier to solve if you write the –coefficient first if there is a negative last term) : put brackets around the first two terms and brackets around the last two terms ( ) ) ( : take out common factors from each bracket if there are any ( ) ) ( : then take out the common brackets and put the other factors in the second bracket as follows: ( : multiply the coefficient of ( )( ) and the constant with each other, . ) : then find two factors of 18 that will add up to the coefficient of , that is : then breakdown the center term as and rewrite the quadratic as: : put brackets around the first two terms and brackets around the last two terms ( ) ) NOTE: if you put your own brackets after a – then ( the sign in the bracket must change. : take out common factors from each bracket if there are any ( ) ( ) : then take out the common brackets and put the other factors in the second bracket as follows: ( 8 )( ) When factoring quadratics, first always look if there is not a common factor to take out before you look at difference of two squares or trinomials Linear Equations: Linear equations can be written in the form: 9 3 4 10 11 Quadratic equations: Quadratic equations can be written in the form: 12 The Quadratic formula This formula can be used to solve x for any quadratic equation These are called EXACT solutions, as exact solutions have no decimal places 13 Completing the square: The quadratic equation can be rewritten as Complete the square in an expression: Rewrite Method: multiply the coefficient of ( ( ) ( ) by and square the answer ( ) ) add In bracket ( ) ______________________________________________________ First take out a common factor of 2, as completing the square can only happen when you have ( ) Method: multiply the coefficient of ( ) ( by and square the answer ) add In bracket 14 ( ) ( ) Now multiply the common factor with the two terms in the bracket ( ) _______________________________________________________ Complete the square in an equation: Rewrite ( ) and then solve for Method: multiply the coefficient of ( by and square the answer ( ) ) add In bracket ( ) ( ) √ √ or √ √ √ ________________________________________________________ As the equation = 0, first divide by 2, as completing the square can only happen when you have Method: multiply the coefficient of 15 by and square the answer ( ) ( ) add In bracket ( ) ( ) √ √ or √ √ √ √ Complete the square of the following expressions: a b c Solve for x, by completing the square method: a √ b c 16 √ LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: September MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Parabola graphs PERIOD 1 2 CLASS WORK HOME WORK Sketch parabola graphs Show that the stationary point or vertex can be determined by differentiation as well as the completing the square Worksheet pages 19 Sketch more graphs, and look at graphs that do not cut the x-axis Ex 3C page 59 no 2 b, c, e Quadratic graphs: Parabola graphs from IGCSE were sketched using points In grade 11 you will sketch parabola graphs using x-intercepts, y-intercept, turning point 17 x-intercepts: Axis of symmetry and vertex / turning point: Turning point = ( ) ( ) 18 Determine whether a point is on the graph of a parabola (quadratic equation) 19 LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: September MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Nature of roots PERIOD 1 2 CLASS WORK HOME WORK Link quadratic formula and the nature of the roots of an equation On page 22 do the 4 questions. Look at sketching parabola and nature of roots given Work out the unknown value if roots are given On pages 24 answer the questions. When you solve a quadratic equation in the form , the solutions of are called the roots or zeroes. The graphs of a quadratic equation look as follows: OR Any quadratic equation in the form formula: can be solved by using the quadratic √ The NATURE OF THE ROOTS are all about the values underneath the √ equation. The answer of in the will be called the discriminant or delta and we show it as . o If the answer under the square root is 4, e.g. √ then the answers from the square root will be ±2. The 4 is a perfect square, so the answers will be real and rational. o If the answer under the square root is 5, e.g. √ then the answers from the square root will be ±2.236067977. The 5 is not a perfect square, so the answers will be real and irrational. o If the answer under the square root is -6, e.g. √ your calculator. The answer will thus be non-real. 20 then the answers will be an error on The three questions you must answer when talking about the nature of the roots are: o Real OR non-real o Rational OR irrational o Equal OR unequal SUMMARY: : Real roots <0: non-real roots > 0 : real, unequal roots =0 : real, equal roots = perfect square : real, rational roots perfect square : real, irrational roots 21 √ Roots : 22 Nature of roots Sketches of parabola graphs (quadratic equations) and the nature of the roots (x-intercepts): - delta is: Roots: nature of and how many xintercepts Two real roots >0 The answer under the √ is a positive value. x-intercepts Parabola cuts the xaxis at two separate places y 0 x = 0 One real root, as the Parabola only touches two answers (roots) of the x-axis yonce. the equation will be The answer under the the same. √ is equal to zero. 0 <0 Non-real roots The answer under the √ is a negative value. x Parabola will not touch or go through the xaxis y 0 23 x Examples of the types of questions that can be asked: o A.) Determine the nature of the roots of the following equations. o B.) Determine the value of an unknown (e.g. k, m, t) if the nature of the roots is known. o C.) Show OR prove that the nature of the roots are:………….. (e.g. real, rational) To answer any of the above types of questions the quadratic equation must always be written in standard form: A. Determine the nature of the roots of the following equations. Without solving the equations, determine the nature of the roots of: (equal, unequal, rational, irrational or non-real) Example: Exercise: Determine the nature of the roots without solving the equations (1) (2) B. Determine the value of an unknown (e.g. k, m, t) if the nature of the roots as known. Example: For which value(s) of will the equation of Exercise: For which value(s) of have equal roots? will the equation of have equal roots? C. Show OR prove that the nature of the roots are…………..: Remember: ( ) ( Example: Show that the roots of ) will be real for all real values of Hint: use the “completing the square method” on the expression of the discriminant Exercise: Prove that for all real values of the roots of ( will be non-real. 24 ) ( ) LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: September MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Quadratic inequalities PERIOD 1 2 CLASS WORK HOME WORK Linear inequalities Quadratic inequalities in the book-explain Do all the questions in the book on also the special cases and things we do pages 28-30 not do in inequalities Stress the concept of positive shaped parabola graphs Ex 4B page 67 no no 1 k,l,m and n 3c and d Linear inequalities: Solving algebraic inequalities is just like solving equations, add, subtract, multiply or divide to, or from for example both sides. The exception is if you multiply or divide both sides by a negative value, then you must turn the inequality sign around. Example: Solve for 25 Quadratic Inequalities: Terminology: When a value is bigger than 0, then we have a positive interval When a value is smaller than 0, then we have a negative interval Steps: If there is a in an inequality, place all the values on the one side so that the other side is . Arrange the variables in descending order, and then factorise to find the zeros ( roots or x-intercepts) Sketch a x-axis and indicate the roots on the x-axis: Now choose a value in between the two x-intercepts, and substitute the value into the inequality, and see whether the answer is positive or negative. Then take values on the left and right of the x-intercepts to find the signs of the intervals. Examples: 1. Solve for Solution: ( )( ) So the -intercepts are: 1 5 Choose a value between 1 and 5, and substitute it into the equation the value of the sign, for example: 26 to find , now indicate it on the number line ( ) ( ) 1 5 Choose a value smaller than 1, and substitute it into the equation to find the value of the sign, for example: , now indicate it on the number line ( ) ( ) + 1 5 Choose a value bigger than 5, and substitute it into the equation to find the value of the sign, for example: , now indicate it on the number line ( ) ( ) - + From the question ( 1 )( ) + 5 we can see that we have to look for the positive intervals, so the final answer: 2. Solve for Solution: ( ) ( )( ) So the -intercepts are: -7 1 Choose a value between -7 and 1, and substitute it into the equation to find the value of the sign, for example: , now indicate it on the number line ( ) ( ) -7 1 Choose a value smaller than -7, and substitute it into the equation the value of the sign, for example: ( ) ( ) , now indicate it on the number line + -7 1 27 to find Choose a value bigger than 1, and substitute it into the equation to find the value of the sign, for example: ( ) , now indicate it on the number line ( ) - + -7 From the question ( )( + 1 we can see that we have to look for the negative ) intervals, so the final answer: Exercise: Solve for (1) (2) (3) (4) ( )( ) SPECIAL CASES: (1) Solve for ( ) As (any base)² is always positive, we can enter any value in the place of , and the answer after we squared will be greater than 0 (positive). In this case there is only one x-intercept of the quadratic equation 2 If you substitute a value less than 2 or a value bigger than 2 into the equation, bith answers will be positive + + 2 So, the final answer: (2) Solve for ( ) As (any base)² is always positive, we can enter any value in the place of , and the answer after we squared will be greater than 0 (positive). In this case there is only one x-intercept of the quadratic equation 2 28 If you substitute a value less than 2 or a value bigger than 2 into the equation, bith answers will be positive + + 2 There are no places where the interval is negative, and there is one solution where it is = to zero, so the final answer is: (3) Solve for If there is only in an inequality, then we use the signs of the values to determine what the sign of the unknown must be: + - ? Which sign must be divided into a ( + ) to give us a ( – ) answer? ______________ So the denominator must be Final solution: ______________________ Note: here we can’t have = 0 as the unknown is in the denominator! (4) Solve for If there is only in an inequality, then we use the signs of the values to determine what the sign of the unknown must be: ? + - Which sign must be divided by a ( -) to give us a ( + ) answer? ______________ So the numerator must be Final solution: (4) ______________________ can be solved in the same way as ( 29 )( ) All that you have to remember is that the denominator can’t = 0 in a fraction!!! DO NOT DO THE FOLLOWING IN INEQUALITIES!!!!! (1) If you are given: but if do NOT square root both sides, as: √ will be was a negative value, then the sign should change to the opposite sign!!!! So rather take all to one side so that the other side has a 0, and then factorise. (2) If you have do NOT multiply with whether the on both sides, as you do not know is positive or negative, and if it was negative the sign should have changed. Rather take all to one side, make other side 0, put all on the same denominator on the side where the fraction is, and then solve the inequality. Exercise: Solve for (1) (2) (3) 30 (4) , LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: October MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Simultaneous equations PERIOD CLASS WORK Revise linear simultaneous equations, using the elimination method as well as the substitution method Ex 5B no 1a Ex 5B no 1 b and c – use only the substitution method Do a problem on substitution with fractions – first with linear equations, and then with linear and quadratic equations Ex 5C no 1 a,f , 2d, 3c This is one of the most important skills in maths and will be in both exams, so practice more questions!! Three cases of the nature of roots of the intersections of linear and quadratic equations Page 33 1 2-3 4-5 HOME WORK Two linear equations Different sign we add Same sign we subtract _________________________________________________________ 31 One linear and one quadratic equation 32 Nature of roots questions can also be asked when working with the intersections of graphs: Exercise: (Exam question) Determine how many intersections the curves will have with each other Simultaneous equations and then discriminant value 33 Quadratic Applications 34 LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: October MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Exponents and Logs PERIOD CLASS WORK 1-2 HOME WORK Revise exponent rules, do applications of exponents and surds and the relationship between the two. Concentrate on negative exponents and fraction exponents and roots Worksheets pages 36-39 3 Discuss log laws Ex 18B no 1e,f,g,h,k,l, no 2c,d,i,j,k,l 4 Exponential graphs Asymptotes transformations Ex 18 D no 10 Ex 18C no 1a,c,d,f, The use of the calculator and logs. What does , Log x, lg x, and ln x mean on the calculator. The laws / rules of logs Change of base rule Rules: and Ex 18C no 2b,c,d,e Equations of logs and inequalities Ex 18D no 1a,d,e,f,i, 2a,d,e,3,a,b,g,h, 4a,b 5&6 7 8&9 35 36 37 Example: exponents are used in algebraic solving of fractions 38 Exponential equations x as a base x as a power Quadratic equations in exponents NOTE: ( ) Let ( Then Factorise: ( )( ) 0 39 ) Inside Roots or surds Exponents can be written as roots: power √ Outside power Logarithms (logs) 40 Expanding – reversing the laws 41 Exponential equations where the bases can’t be written as the same base, can be solved using logs Solve for x: Finding the value of a log-expression or log-equation can also be solved by changing the base Example: 42 Example: Solving log equations 43 Exponential graph: All exponential graphs have a horizontal asymptote. In the equation the equation of the asymptote is . An asymptote is a straight line towards which the graph goes but it will never touch or intersect the line 44 LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: October/Nov MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Coordinate Geometry PERIOD CLASS WORK HOME WORK 1 Distance formula Ex 6A no 1a,d,e,h 2 Midpoint of line segment Ex 6A no 2a,d,e,h,4,7 Gradients of line, parallel and perpendicular lines m=0,m=-1,m=1,m=, sketching straight lines m=tanθ angle of inclination when does a point lie on a curve or straight line? Ex 6B no 1a,b,c,i,2 all, 3a,b, Ex 6B no 9b,c,8b,c,11 all, 4 (any 3) Equations of lines – 3 different equations Do the examples as well as when lines are parallel and perpendicular to other lines Ex 6B no 14a,d,15b,d,16a,c Applications of straight lines and coordinate geometry Circle, tangents and normals 13A no 1a,b,c,2a,b,3a,b,4a,b 13B no 2a,3a 3-4 5-6 7-8 45 Distance formula 46 Midpoint of a line segment 47 1 2 48 Gradient of a line 49 50 51 Equations of straight lines 52 53 2 3 4 Sketching straight lines 54 Circles: We use the distance formula to work out the length of the radius from the centre of the circle to the circumference of the circle. Circle centre at (0,0) Circle centre at (a,b) - somewhere in the four quadrants ( 55 ) ( ) Finding equation of a circle Finding centre coordinates and radius A tangent is a straight line that is always perpendicular to the radius of the circle. A normal is perpendicular to a tangent and the normal is the equation of the radius of the circle. Tangent 56 LESSON PLAN GRADE: 11 bilingual YEAR: 2017 DATE: November MATHEMATICS TEXTBOOK: Advanced Maths C1C2 as core for Edexcel SECTION: Trigonometry PERIOD 1 2 3,4,5 CLASS WORK HOME WORK Convert radians to degrees and vice versa Ex 16A all Length of an arc in a circle Ex 17A no 3, 4, 5 Area of segment, sector and triangle Ex 17A no 2 all, Ex 17B no 1 a,b,c 4a,b Ex 16B no 1 all the negative angles, 2 all, 3, 4. Worksheet page 63-64 Ex 16B no 5-13 Worksheet page 75 6-9 10-11 CAST Diagram – draw and explain all 4 quadrants Reduction formulae, angles bigger than 360°, negative angles Special angles Solving triangles: sine, cosine, area rules 12-13 Trig graphs Transformation of graphs 57 the sine-rule: ex 17D no 1a,b,c,d 2a The cosine-rule ex 17E no 1a,b, 2a,b The area-rule ex 17F no 17G no 2a,b Consolidation with word sums: Ex 17F selected examples Radians: 58 59 Sector: Segment: 60 C C C 61 TRIGONOMETRY: CAST DIAGRAM S A T C 62 Th The reciprocal functions are: cosec θ, secθ and cotθ REDUCTION FORMULAE: Indicate the reduced function with correct signs (+ or -): 3 Example: sin(180°+θ)= - sinθ . NOTE: all numbers in a indicate the quadrant number. cos(180°-θ)= tanθ= cot(180°+θ)= cosec(180°-θ)= sec(180°-θ)= sec(360°-θ)= tan(180°-θ)= cos(180°+θ)= secθ= tan(180°+θ)= sinθ= cot(360°-θ)= sec(180°+θ)= cosec(360°-θ)= cosecθ= cos(360°-θ)= cotθ= sin(180°+θ)= cosθ= cos(180°-θ)= sin(360°+θ)= tan(360°-θ)= cos(180°+θ)= cos(θ+360°)= sin(180°-θ)= sin(360°-θ)= tan(360°+θ)= cosec(180°+θ)= cot(180°-θ)= cot(360°+θ)= 63 Reduce the following angles to be acute angles: 3 3 Example: cos 240° = cos(180°+60°)= -cos 60° cos 120°= tan 225°= cos 300°= sin 20°= cos 60°= sin 160°= tan 315°= sin 200°= tan 135°= sin 340°= tan 45°= cos 40°= EVERY FUNCTION HAS TWO QUADRANTS WHERE IT IS POSITIVE AND TWO QUADRANTS WHERE IT IS NEGATIVE. Complete: In which quadrant does θ lie if: a) sinθ > 0 and cosθ < 0 __________________________ b) cosθ > 0 and sinθ < 0 __________________________ c) tanθ > 0 and sinθ < 0 __________________________ d) tanθ > 0 and cosθ > 0 __________________________ e) sinθ = f) 13 cosθ – 12 = 0 and 180˚< θ < 360˚ __________________________ √ and θ ε [180˚ ; 270˚ ] __________________________ Rules: Angles bigger than 360 θ is an acute angle which is in the first quadrant, so all angles 90˚ bigger than 360° are firstly in the first quadrant. sin(360+) = sin cos(360+) = cos θ tan(360+) = tan 180˚ 0˚ 270˚ Examples: a) sin 780 = sin(2360 + 60) = sin 60 b) tan 405 = tan(1360 + 45) = tan 45 c) cos 960 = cos 240 = cos (180+60) = -cos60 64 NOTE: Angles bigger than 360° are sometimes reduces twice: always first the first quadrant (because the angle is bigger than 360°), and then the angle must be checked again, and if it is bigger than 90°, it must be reduced again. Examples: 1 1 b) tan 600° a) tan 480° 2 3 = tan 120° = tan 240° = - tan 60° = tan 60° Negative angles -270˚ Quadrant 1 Quadrant 2 -360˚ -180˚ 0˚ Quadrant 3 Quadrant 4 Negative angles are measured clockwise starting from the -90˚ x-axis In the first negative quadrant (which is the fourth quadrant for positive angles), only the cos θ function has a positive ratio. Rules: 4 sin ( -θ ) = - sin θ 4 cos ( -θ ) = cos θ Examples: 4 cos(-20˚) = cos 20˚ 4 3 3 cos(-210˚) = cos 210˚ = cos(180˚+30˚)= - cos30˚ sin(-210˚) = - sin 210˚ = - sin(180˚+30˚)= - (-sin30˚)= sin30˚ 4 4 3 3 3 3 tan(-210˚) = - tan 210˚ = - tan(180˚+30˚)= - (+tan30˚)= -tan30˚ 65 4 tan ( -θ ) = - tan θ The Unit Circle: 66 67 68 In the sine-rule there is an ambiguous case: When two sides and one angle is given, and the side opposite the given angle is the shortest of the two sides, then there are two possibilities for the angle opposite the longer side. 69 70 71 72 Trigonometric graphs: 73 Transformation of graphs 74 2 75 PORTFOLIO WORK: Name: Oral Grade 1 Domain Communication Taxonomy Date Description Mark Using language of mathematics Presenting his or her mathematical thinking coherently and clearly to peers Analysing and evaluating the mathematical thinking strategies of others Giving accurate answers to questions of knowledge Giving accurate answers to questions of application Giving accurate answers to questions of reasoning Total Students grade 1 2 OVER ALL ORAL 1 2 3 3 1 2 1 10 TOTAL score Oral Grade 2 Domain Communication Taxonomy Date Description Mark Using language of mathematics Presenting his or her mathematical thinking coherently and clearly to peers Analysing and evaluating the mathematical thinking strategies of others Giving accurate answers to questions of knowledge Giving accurate answers to questions of application Giving accurate answers to questions of reasoning Total Oral Grade 3 Domain Communication Taxonomy Students grade 1 2 3 1 2 1 10 Date Description Mark Using language of mathematics Presenting his or her mathematical thinking coherently and clearly to peers Analysing and evaluating the mathematical thinking strategies of others Giving accurate answers to questions of knowledge Giving accurate answers to questions of application Giving accurate answers to questions of reasoning Total 76 1 2 3 1 2 1 10 Students grade MARK NAME:___________________________________________ Grade 11 Homework 1 (5 min) Ministry Portfolio Requirement Exponents and Logs Solve for (a) (b) (c) (d) 77 3 Marks NAME:___________________________________________ Grade 11 Homework 2 (5 min) Ministry Portfolio Requirement Coordinate Geometry (a) The points A and B have coordinates (3,-1) and (11,-7) respectively. a. Find the coordinates of the midpoint of AB Given that AB is the diameter of a circle: b. Find the length of the diameter of the circle. c. What is the equation of the line AB? (b) The points A and B have coordinates (3,-1) and (6,-4) respectively. d. Find the coordinates of the C if B is the midpoint of AC Given that AB is the radius of a circle: e. Find the length of the radius of the circle. f. What is the equation of the line AC? 78 3 Marks NAME:___________________________________________ Grade 11 Homework 3 (5 min) Ministry Portfolio Requirement Trig functions and angles 3 Marks (a) A- Given that is an acute angle measured in degrees, express in terms of cos: cos(180°+) B- Find, as an exact value, the value of tan 120° (b) A- Given that is an acute angle measured in degrees, express in terms of cos: sin(360°+) B- Find, as an exact value, the value of cos 150° (c) A- Given that is an acute angle measured in degrees, express in terms of cos: tan(180°-) B- Find, as an exact value, the value of sin 120° (d) A- Given that is an acute angle measured in degrees, express in terms of cos: cos(360°-) B- Find, as an exact value, the value of tan 240° 79 NAME:___________________________________________ Grade 11 Short Questions 1 (5 min) Ministry Portfolio Requirement Equations Solve the simultaneous equations: (a) and (b) and (c) and (d) and 80 2 Marks NAME:___________________________________________ Grade 11 Short Questions 2 (5 min) Ministry Portfolio Requirement Trigonometry 2 Marks (a) Find the area of triangle ABC, rounding your answer to 1 decimal place. A 102° 12 cm C 9 cm B (b) Find the length of BC in triangle ABC, rounding your answer to 1 decimal place. A 102° 12 cm C 9 cm B (c) Find the size of angle C in the triangle ABC, rounding your answer to 1 decimal place. A 102° C 9 cm 22 cm B 81 NAME:___________________________________________ Grade 11 Short Questions 3 (5 min) Ministry Portfolio Requirement Trigonometry 2 Marks (a) -1440° -1080° -720° -360° 0° 360° 720° 1080° 1440° Write down the equation of the trigonometric graph shown above (b) --2π -π π 2π Write down the equation of the trigonometric graph on the left (c) --2π -π π 82 2π Write down the equation of the trigonometric graph on the left NAME: Grade 11 (Bilingual: Pure Maths) MATHEMATICS TEST Equations Exponents and logs Date: October Time: 20 minutes Total: 10 Ministry Portfolio Requirement INSTRUCTIONS: 1. Work neatly and legibly. 2. Answer all questions. 3. Show all your workings within the designated spaces. 4. Write your name in the given space at the top of this page. The following skills are being tested: TOPIC STUDENT COMMENT Linear equations Quadratic equations Simultaneous equations Exponents and logs OVERAL IMPRESSION CORRECTION MARK Comment:________________________________________________________________ ________________________________________________________________________ _____________________________________________________________________________________ Teacher signature:______________________ Parent signature:_________________________________ 83 Question 1 Multiple choice questions Shade the circle next to the correct answer: 1. Factorise ( )( ( ) ( )( ) ) ( )( ) )( 2. If ( )( ) then 3. What are the points of intersection of the curves of (-3,-4) and (3,4) (-3,4) and (3,-4) 4. Express (-4,-3) and (4,3) (-4,3) and (4,-3) as a single logarithm ( ) (4 marks) 84 Question 2 Extended response a) Solve by completing the square. (Answer to 3 significant digits). (3 marks) b) Solve the inequality: (1 mark) 85 c) The rectangle has a perimeter of 30 cm, and an area of 44 cm² Write down the set of simultaneous equations which you should use to find the length and width of the rectangle. NOTE: YOU DON’T HAVE TO SOLVE THE TWO EQUATIONS! x y (1 mark) d) Solve for : . (Show steps and leave answer correct to 2 decimal places) ( 1 mark) 86 NAME: Grade 11 (Bilingual: Pure Maths) MATHEMATICS TEST Coordinate Geometry Trigonometry Date: December Time: 20 minutes Total: 10 Ministry Portfolio Requirement INSTRUCTIONS: 5. Work neatly and legibly. 6. Answer all questions. 7. Show all your workings within the designated spaces. 8. Write your name in the given space at the top of this page. The following skills are being tested: TOPIC STUDENT COMMENT Coordinate Geometry Radians and degrees Trig functions OVERAL IMPRESSION CORRECTION MARK Comment:________________________________________________________________ ________________________________________________________________________ _____________________________________________________________________________________ Teacher signature:______________________ Parent signature:_________________________________ 87 Question 1 Multiple choice questions Shade the circle next to the correct answer: 5. Find the length of the line joining the points (-1,-1) and (-3,2) √ √ √ √ 6. The gradient of a straight line joining K( -1 , 2 ) and L( 3 , y ) is 2. What is the value of y? 7. Convert 30° 150° to degrees 60° 45° 8. The arc AB of a circle with centre O, which subtends an angle of 1.87C at the centre O is 10cm. Find the radius of the circle. 0.19 5.35 10.7 18.7 (4 marks) 88 Question 2 a) Write your answers for each of the questions in the space provided. Be sure to show all your work and correct units where applicable. Extended response Find the point of intersection of the equations: (2 marks) 89 b) Find the missing angle in the triangle ABC below (3 significant places) A 18cm 47° B C 11.2cm (2 marks) 90 c) Find the value of x 120° (x+2) cm (x-2) cm (2x-2) cm (2 marks) 91 NAME: Section Marks Multiple Choice ______ 24 Extended Response ______ 36 Date: January Total ______ 60 Time: 2 hours 30 minutes Grade Bilingual Diploma Mathematics Grade 11 Instructions to Candidates Answer ALL the questions. You must write your answer to each question in the space following the question. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates Full marks may be obtained for ALL questions. The marks for individual questions, and the parts of questions are shown in round brackets: e.g.(2). There are 16 questions in this question paper. The total for this paper is 60. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. 92 Section A Multiple choice: For each of the questions below shade the box with the correct answer 1. The points A and B have the coordinates ( -1 , -1 ) and ( -2 , 4) respectively. The line t passes through the point A and is perpendicular to the line AB. a) Find the midpoint of the line AB ( ( ) ) ( ) ( ) (2) b) Find the equation of line t in the form , where are integers. (2) 93 2. The equation Find the value of , where is a constant, has equal roots. 2 -2 3. Find the coordinates of the point of intersection of the line with equation (7 , 4) (-7 , 4) (-4 , 15) (-4 , 7) (2) and the line (2) 94 4. Find, in questions a) and b), giving your answer(s) correct to 3 significant figures where appropriate, the value of for which a) 1 -2 -3 -4 (2) b) ( ) ) (2) 95 5. Railway track A B Path=70m 44 m 44 m 1.84 C C The shape ABC shown is a design of a railway track with a straight path of 70m that connects two points A and B on the track. A and B are equidistant from point C. The size of angle ACB is 1.84 radians. Find: a) The length of the arc AB, to the nearest m. 81 1781 40 70 (2) b) The shortest distance from C to the path AB, to the nearest m. 56 27 54 28 (2) c) The area of sector ABC, to the nearest m² 81 1781 40 70 (2) 96 6. The diagram shows a sketch of the curve for Y 1 -270° -180° -90° 0° 90° 180° 270° x -1 The exact value of a) Write down the exact value of i. ii. ( ) i) ii) i)- ii) - i)- ii) i) ii) - (2) b) What is the correct equation of the graph below? -4π -2π O 2π 4π 6π ( ( 8π ) ) (2) 97 7. The circle A has centre ( -2 , 3 ) and passes through the point ( 2 , 0 ) Find the equation for A. ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) (2) Section B Extended response – please answer the questions in the space below. Be sure to show all your work and correct units where applicable. 1. a) Work out ( 00 ) (2) b) If and find the value of (1) 98 c) Evaluate without using your calculator. (2) 2. a) Convert to degrees. ( to 3 significant figures) b) Convert 266° to radian measure. ( to 3 significant figures) (2) 3. Determine the exact value of: (show all working) a) ( ) (2) b) ( ) (2) 99 c) If and √ , in which quadrant is (1) 4. a) Solve for by completing the square (4) b) Find the range of values of . (2) 100 5. Find the number of points of intersection of and (2) 6. a) Find the distance between K( 1 , 4 ) and L( -2, -1 ) , ( to 3 significant figures) (2) 101 b) Find the equation of the line KL in the form (3) 7. The equation of a circle is ( ) a) Find the centre coordinates and the value of the radius. b) Determine whether the point ( 1 , 5 ) lies on the circle. (3) 102 8. The points A( -3 , 1 ), B( 1 , 2 ), C( 0 , -1 ), D( -4 , -2 ) are given. Show that ABCD is a parallelogram. (2) S The diagram shows a plan for a patio. 9. √ m P Q 4m 4m The patio PRQS is in the shape of a sector of a circle with centre R and radius 4 m. It is also given that the length of the straight line PQ is √ m. R a) a) Show through calculations that the size of angle PRQ is 2.17 radians (2) 103 b) Find the area of patio PRQS ( to 3 significant figures) (2) c) Determine the area of the segment PQS ( to 3 significant figures) (2) (End of Paper) 104