UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

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1 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * * ADDITIONAL MATHEMATICS 0606/11 Paper 1 October/November hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator. READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is Total This document consists of 16 printed pages. DC (NH/JG) 50034/5 [Turn over

2 2 Mathematical mulae 1. ALGEBRA Quadratic Equation the equation ax 2 + bx + c = 0, b b ac x = 4 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A mulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /11/O/N/12

3 3 1 (i) Sketch the graph of y = 3 + 5x, showing the coordinates of the points where your graph meets the coordinate axes. [2] (ii) Solve the equation 3 + 5x = 2. [2] 2 Find the values of k for which the line y = k 6x is a tangent to the curve y = x(2x + k). [4] 0606/11/O/N/12 [Turn over

4 4 3 Given that p = log q 32, express, in terms of p, (i) log q 4, [2] (ii) log q 16q. [2] 4 Using the substitution u = 5 x, or otherwise, solve 5 2x+1 = 7(5 x ) 2. [5] 0606/11/O/N/12

5 5 5 Given that y = (i) x 2 cos 4x, find dy dx, [3] (ii) the approximate change in y when x increases from π 4 to π + p, where p is small. [2] /11/O/N/12 [Turn over

6 6 6 (i) Find the first 3 terms, in descending powers of x, in the expansion of x + 2 x 2 6. [3] (ii) Hence find the term independent of x in the expansion of 2 4 x 3 x + 2 x 2 6. [2] 0606/11/O/N/12

7 7 7 Do not use a calculator in any part of this question. (a) (i) Show that is a square root of [1] (ii) State the other square root of [1] (b) Express in the form a + b 6, where a and b are integers to be found. [4] /11/O/N/12 [Turn over

8 8 8 The points A( 3, 6), B(5, 2) and C lie on a straight line such that B is the mid-point of AC. (i) Find the coordinates of C. [2] The point D lies on the y-axis and the line CD is perpendicular to AC. (ii) Find the area of the triangle ACD. [5] 0606/11/O/N/12

9 9 1 9 A function g is such that g(x) = 2x 1 for 1 x 3. (i) Find the range of g. [1] (ii) Find g 1 (x). [2] (iii) Write down the domain of g 1 (x). [1] (iv) Solve g 2 (x) = 3. [3] 0606/11/O/N/12 [Turn over

10 10 10 The table shows values of the variables x and y. x y (i) Using the graph paper below, plot a suitable straight line graph to show that, for 10 x 80, y = A sin x + B, where A and B are positive constants. [4] 0606/11/O/N/12

11 11 (ii) your graph to find the value of A and of B. [3] (iii) Estimate the value of y when x = 50. [2] (iv) Estimate the value of x when y = 12. [2] 0606/11/O/N/12 [Turn over

12 12 11 (a) Solve cosec 2x π 3 = 2 for 0 < x < π radians. [4] (b) (i) Given that 5(cos y + sin y)(2 cos y sin y) = 7, show that 12 tan 2 y 5 tan y 3 = 0. [4] 0606/11/O/N/12

13 13 (ii) Hence solve 5(cos y + sin y)(2 cos y sin y) = 7 for 0 < x < 180. [3] 0606/11/O/N/12 [Turn over

14 14 12 Answer only one of the following two alternatives. EITHER y C y = (12 6x)(1 + x) 2 A O B x The diagram shows part of the graph of y = (12 6x)(1 + x) 2, which meets the x-axis at the points A and B. The point C is the maximum point of the curve. (i) Find the coordinates of each of A, B and C. [6] (ii) Find the area of the shaded region. [5] OR A y O dy The diagram shows part of a curve such that dx = 3x2 6x 9. Points A and B are stationary points of the curve and lines from A and B are drawn perpendicular to the x-axis. Given that the curve passes through the point (0, 30), find (i) the equation of the curve, [4] (ii) the x-coordinate of A and of B, [3] (iii) the area of the shaded region. [4] B x 0606/11/O/N/12

15 Start your answer to Question 12 here. Indicate which question you are answering. 15 EITHER OR 0606/11/O/N/12 [Turn over

16 16 Continue your answer here if necessary. 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. University of 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. 0606/11/O/N/12

17 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * * ADDITIONAL MATHEMATICS 0606/12 Paper 1 October/November hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator. READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is Total This document consists of 16 printed pages. DC (CW/SW) 50035/5 [Turn over

18 2 Mathematical mulae 1. ALGEBRA Quadratic Equation the equation ax 2 + bx + c = 0, b b ac x = 4 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A mulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /12/O/N/12

19 3 1 It is given that a = 4 3, b = 1 2 and c = (i) Find a + b + c. [2] (ii) Find λ and μ such that λ a + μ b = c. [3] 0606/12/O/N/12 [Turn over

20 4 2 (i) Find the inverse of the matrix [2] (ii) Hence find the matrix A such that A = [3] 3 (i) Show that cotθ + sinθ = cosecθ. [5] 1 + cosθ (ii) Explain why the equation cotθ + sinθ 1 + cosθ = 1 2 has no solution. [1] 0606/12/O/N/12

21 5 4 Given that log a pq = 9 and log a p 2 q = 15, find the value of (i) log a p and of log a q, [4] (ii) log p a + log q a. [2] 0606/12/O/N/12 [Turn over

22 6 5 The line x 2y = 6 intersects the curve x 2 + xy + 10y + 4y 2 = 156 at the points A and B. Find the length of AB. [7] 0606/12/O/N/12

23 Using sin15 = 2 4 ( 3 1) and without using a calculator, find the value of sinθ in the form a + b 2, where a and b are integers. [5] 0606/12/O/N/12 [Turn over

24 8 7 Solutions to this question by accurate drawing will not be accepted. y E D C (6, 8) A ( 5, 4) B (8, 4) O x The vertices of the trapezium ABCD are the points A( 5, 4), B(8, 4), C(6, 8) and D. The line AB is parallel to the line DC. The lines AD and BC are extended to meet at E and angle AEB = 90. (i) Find the coordinates of D and of E. [6] 0606/12/O/N/12

25 9 (ii) Find the area of the trapezium ABCD. [2] 0606/12/O/N/12 [Turn over

26 10 8 O 10 cm 1.5 rad A 18 cm C B D The diagram shows an isosceles triangle OBD in which OB = OD = 18 cm and angle BOD = 1.5 radians. An arc of the circle, centre O and radius 10 cm, meets OB at A and OD at C. (i) Find the area of the shaded region. [3] (ii) Find the perimeter of the shaded region. [4] 0606/12/O/N/12

27 11 9 (a) (i) Using the axes below, sketch for 0 x π, the graphs of y = sin 2x and y = 1 + cos 2x. [4] y O x (ii) Write down the solutions of the equation sin 2x cos 2x = 1, for 0 x π. [2] (b) (i) Write down the amplitude and period of 5 cos 4x 3. [2] (ii) Write down the period of 4 tan 3x. [1] 0606/12/O/N/12 [Turn over

28 12 10 A function f is such that f(x) = 4x 3 + 4x 2 + ax + b. It is given that 2x 1 is a factor of both f(x) and f ʹ(x). (i) Show that b = 2 and find the value of a. [5] Using the values of a and b from part (i), (ii) find the remainder when f(x) is divided by x + 3, [2] 0606/12/O/N/12

29 (iii) 13 express f(x) in the form f(x) = (2x 1)(px 2 + qx + r), where p, q and r are integers to be found, [2] (iv) find the values of x for which f(x) = 0. [2] 0606/12/O/N/12 [Turn over

30 14 11 Answer only one of the following two alternatives. EITHER A curve is such that y = 5x x 2. (i) (ii) (iii) dy Show that dx = kx (1 + x 2 ) 2, where k is an integer to be found. [4] Find the coordinates of the stationary point on the curve and determine the nature of this stationary point. [3] By using your result from part (i), find x (1 + x 2 ) 2 dx and hence evaluate 2 x (1 + x 2 ) 2 dx. 1 [4] OR A curve is such that y = Ax2 + B x 2 2, where A and B are constants. (i) Show that dy 2x(2A = + B) dx (x 2 2) 2. [4] It is given that y = 3 and dy = 10 when x = 1. dx (ii) Find the value of A and of B. [3] (iii) Using your values of A and B, find the coordinates of the stationary point on the curve, and determine the nature of this stationary point. [4] Start your answer to Question 11 here. Indicate which question you are answering. EITHER OR /12/O/N/12

31 15 Continue your answer here /12/O/N/12 [Turn over

32 16 Continue your answer here if necessary 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. University of 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. 0606/12/O/N/12

33 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * * ADDITIONAL MATHEMATICS 0606/13 Paper 1 October/November hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is Total This document consists of 16 printed pages. DC (SJF/CGW) 50036/5 [Turn over

34 2 Mathematical mulae 1. ALGEBRA Quadratic Equation the equation ax 2 + bx + c = 0, b b ac x = 4 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A mulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /13/O/N/12

35 3 1 (a) On the Venn diagrams below, shade the region corresponding to the set given below each Venn diagram. P Q P Q R R P (Q R) (b) It is given that sets, B, S and F are such that = {students in a school}, B = {students who are boys}, S = {students in the swimming team}, F = {students in the football team}. Express each of the following statements in set notation. P (Q R) [2] (i) All students in the football team are boys. [1] (ii) There are no students who are in both the swimming team and the football team. [1] 0606/13/O/N/12 [Turn over

36 4 2 The rate of change of a variable x with respect to time t is 4cos 2 t. (i) Find the rate of change of x with respect to t when t = π 6. [1] The rate of change of a variable y with respect to time t is 3sint. (ii) Using your result from part (i), find the rate of change of y with respect to x when t = π 6. [3] 3 A committee of 7 members is to be selected from 6 women and 9 men. Find the number of different committees that may be selected if (i) there are no restrictions, [1] (ii) the committee must consist of 2 women and 5 men, [2] (iii) the committee must contain at least 1 woman. [3] 0606/13/O/N/12

37 5 4 (i) On the axes below sketch, for 0 x π, the graphs of y = tan x and y = 1 + 3sin 2x. [3] y O 2 x Write down (ii) the coordinates of the stationary points on the curve y = 1 + 3sin 2x for 0 x π, [2] (iii) the number of solutions of the equation tan x = 1 + 3sin 2x for 0 x π. [1] 0606/13/O/N/12 [Turn over

38 6 5 A pilot flies his plane directly from a point A to a point B, a distance of 450 km. The bearing of B from A is 030. A wind of 80 km h 1 is blowing from the east. Given that the plane can travel at 320 km h 1 in still air, find (i) the bearing on which the plane must be steered, [4] (ii) the time taken to fly from A to B. [4] 0606/13/O/N/12

39 7 6 In the expansion of (p + x) 6, where p is a positive integer, the coefficient of x 2 is equal to 1.5 times the coefficient of x 3. (i) Find the value of p. [4] (ii) your value of p to find the term independent of x in the expansion of (p + x) x 2. [3] 0606/13/O/N/12 [Turn over

40 8 7 A particle P moves along the x-axis such that its distance, x m, from the origin O at time t s is t given by x = t 2 for t (i) Find the greatest distance of P from O. [4] (ii) Find the acceleration of P at the instant when P is at its greatest distance from O. [3] 0606/13/O/N/12

41 9 8 (i) Given that 3x 3 + 5x 2 + px + 8 (x 2)(ax 2 + bx + c), find the value of each of the integers a, b, c and p. [5] (ii) Using the values found in part (i), factorise completely 3x 3 + 5x 2 + px + 8. [2] 0606/13/O/N/12 [Turn over

42 10 9 D 20 cm 10 cm A O 6 rad 20 cm 10 cm B C The diagram shows four straight lines, AD, BC, AC and BD. Lines AC and BD intersect at O such that angle AOB is π 6 radians. AB is an arc of the circle, centre O and radius 10 cm, and CD is an arc of the circle, centre O and radius 20 cm. (i) Find the perimeter of ABCD. [4] 0606/13/O/N/12

43 11 (ii) Find the area of ABCD. [4] 0606/13/O/N/12 [Turn over

44 12 10 (i) Solve tan 2 x 2sec x + 1 = 0 for 0 x 360. [4] (ii) Solve cos 2 3y = 5sin 2 3y for 0 y 2 radians. [4] 0606/13/O/N/12

45 13 (iii) Solve 2cosec z + π 4 = 5 for 0 z 6 radians. [4] 0606/13/O/N/12 [Turn over

46 14 11 Answer only one of the following two alternatives. EITHER The tangent to the curve y = 5e x + 3e x at the point where x = 1n 3, meets the x-axis at the 5 point P. (i) Find the coordinates of P. [5] The area of the region enclosed by the curve y = 5e x + 3e x, the y-axis, the positive x-axis and the line x = a is 12 square units. (ii) Show that 5e 2a 14e a 3 = 0. [3] (iii) Hence find the value of a. [3] OR (i) Given that y = 3e 2x 1 + e 2x, show that dy dx = Ae 2x (1 + e 2x ) 2, where A is a constant to be found. [4] (ii) Find the equation of the tangent to the curve y = at the point where the curve 1 + e2x crosses the y-axis. [3] (iii) Using your result from part (i), find e 2x 3e 2x (1 + e 2x ) 2 dx and hence evaluate 0 1n3 e 2x (1 + e 2x ) 2 dx. [4] Start your answer to Question 11 here. Indicate which question you are answering. EITHER OR 0606/13/O/N/12

47 15 Continue your answer here. 0606/13/O/N/12 [Turn over

48 16 Continue your answer here if necessary. 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. University of 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. 0606/13/O/N/12

49 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * * ADDITIONAL MATHEMATICS 0606/21 Paper 2 October/November hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is Total This document consists of 17 printed pages and 3 blank pages. DC (LEO/JG) 50026/6 [Turn over

50 2 Mathematical mulae 1. ALGEBRA Quadratic Equation the equation ax 2 + bx + c = 0, b b ac x = 4 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A mulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /21/O/N/12

51 3 1 Solve the inequality 4x 9 > 4x(5 x). [4] 0606/21/O/N/12 [Turn over

52 4 2 (a) It is given that is the set of integers, P is the set of prime numbers between 10 and 50, F is the set of multiples of 5, and T is the set of multiples of 10. Write the following statements using set notation. (i) There are 11 prime numbers between 10 and 50. [1] (ii) 18 is not a multiple of 5. [1] (iii) All multiples of 10 are multiples of 5. [1] (b) (i) In the Venn diagram below shade the region that represents (A B) (A B ). [1] A B (ii) In the Venn diagram below shade the region that represents Q (R S ). [1] Q R S 0606/21/O/N/12

53 5 3 (i) On the grid below draw, for 0 x 360, the graphs of y = 3 sin 2x and y = 2 + cos x. [4] (ii) State the number of values of x for which 3 sin 2x = 2 + cos x in the interval 0 x 360. [1] 0606/21/O/N/12 [Turn over

54 6 4 It is given that f(x) = 4 + 8x x 2. (i) Find the value of a and of b for which f(x) = a (x + b) 2 and hence write down the coordinates of the stationary point of the curve y = f(x). [3] (ii) On the axes below, sketch the graph of y = f(x), showing the coordinates of the point where your graph intersects the y-axis. [2] y O x 0606/21/O/N/12

55 7 5 It is given that A = , B = and C = 2 (i) Calculate ABC. [4] 3. (ii) Calculate A 1 B. [4] 0606/21/O/N/12 [Turn over

56 8 6 The normal to the curve y = x 3 + 6x 2 34x + 44 at the point P (2, 8) cuts the x-axis at A and the y-axis at B. Show that the mid-point of the line AB lies on the line 4y = x + 9. [8] 0606/21/O/N/12

57 9 7 In this question 1 0 is a unit vector due east and 0 1 is a unit vector due north. At 1200 a coastguard, at point O, observes a ship with position vector km relative to O. The ship is moving at a steady speed of 10 kmh 1 on a bearing of 330. (i) Find the value of p such that 5 p kmh 1 represents the velocity of the ship. [2] (ii) Write down, in terms of t, the position vector of the ship, relative to O, t hours after [2] (iii) Find the time when the ship is due north of O. [2] (iv) Find the distance of the ship from O at this time. [2] 0606/21/O/N/12 [Turn over

58 8 10 S Q y cm O 1 rad x cm P R In the diagram PQ and RS are arcs of concentric circles with centre O and angle POQ = 1 radian. The radius of the larger circle is x cm and the radius of the smaller circle is y cm. (i) Given that the perimeter of the shaded region is 20 cm, express y in terms of x. [2] (ii) Given that the area of the shaded region is 16 cm 2, express y 2 in terms of x 2. [2] 0606/21/O/N/12

59 11 (iii) Find the value of x and of y. [4] 0606/21/O/N/12 [Turn over

60 12 9 (a) An art gallery displays 10 paintings in a row. Of these paintings, 5 are by Picasso, 4 by Monet and 1 by Turner. (i) Find the number of different ways the paintings can be displayed if there are no restrictions. [1] (ii) Find the number of different ways the paintings can be displayed if the paintings by each of the artists are kept together. [3] (b) A committee of 4 senior students and 2 junior students is to be selected from a group of 6 senior students and 5 junior students. (i) Calculate the number of different committees which can be selected. [3] 0606/21/O/N/12

61 13 One of the 6 senior students is a cousin of one of the 5 junior students. (ii) Calculate the number of different committees which can be selected if at most one of these cousins is included. [3] 0606/21/O/N/12 [Turn over

62 14 10 (i) The remainder when the expression x 3 + 9x 2 + bx + c is divided by x 2 is twice the remainder when the expression is divided by x 1. Show that c = 24. [5] (ii) Given that x + 8 is a factor of x 3 + 9x 2 + bx + 24, show that the equation x 3 + 9x 2 + bx + 24 = 0 has only one real root. [4] 0606/21/O/N/12

63 15 BLANK PAGE QUESTION 11 IS PRINTED ON THE NEXT PAGE. 0606/21/O/N/12 [Turn over

64 16 11 Answer only one of the following alternatives. EITHER A particle travels in a straight line so that, t s after passing through a fixed point O, its displacement, s m, from O is given by s = t 2 10t + 10ln(l + t), where t > 0. (i) Find the distance travelled in the twelfth second. [2] (ii) Find the value of t when the particle is at instantaneous rest. [5] (iii) Find the acceleration of the particle when t = 9. [3] OR A particle travels in a straight line so that, t s after passing through a fixed point O, its velocity, v cms 1, is given by v = 4e 2t 24t. (i) Find the velocity of the particle as it passes through O. [1] (ii) Find the distance travelled by the particle in the third second. [4] (iii) Find an expression for the acceleration of the particle and hence find the stationary value of the velocity. [5] Start your answer to Question 11 here. Indicate which question you are answering. EITHER OR /21/O/N/12

65 17 Continue your answer here /21/O/N/12 [Turn over

66 18 Continue your answer here if necessary /21/O/N/12

67 19 BLANK PAGE 0606/21/O/N/12

68 20 BLANK PAGE 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. University of 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. 0606/21/O/N/12

69 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * * ADDITIONAL MATHEMATICS 0606/23 Paper 2 October/November hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is Total This document consists of 16 printed pages. DC (LEO/JG) 50029/6 [Turn over

70 2 Mathematical mulae 1. ALGEBRA Quadratic Equation the equation ax 2 + bx + c = 0, b b ac x = 4 2a 2 Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r! 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A mulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 bc sin A /23/O/N/12

71 3 1 Solve the equation 5x + 7 = 13. [3] 2 (i) Given that A = , find the inverse matrix, A 1. [2] (ii) your answer to part (i) to solve the simultaneous equations 7x + 8y = 39, 4x + 6y = 23. [2] 0606/23/O/N/12 [Turn over

72 4 3 Without using a calculator, simplify (3 3 1)2, giving your answer in the form a 3 + b, where a and b are integers. [4] 0606/23/O/N/12

73 5 4 The points X, Y and Z are such that XY = 3 YZ. The position vectors of X and Z, relative to an 4 origin O, are 27 and 20 7 respectively. Find the unit vector in the direction OY. [5] 0606/23/O/N/12 [Turn over

74 6 5 Find the set of values of m for which the line y = mx + 2 does not meet the curve y = mx 2 + 7x [6] 0606/23/O/N/12

75 7 6 (a) Given that cos x = p, find an expression, in terms of p, for tan 2 x. [3] (b) Prove that (cot θ + tan θ) 2 = sec 2 θ + cosec 2 θ. [3] 0606/23/O/N/12 [Turn over

76 8 7 (a) Find (x + 3) x dx. [3] (b) Find 20 (2x + 5) 2 dx and hence evaluate (2x + 5) 2 dx. [4] 0606/23/O/N/12

77 9 8 Solutions to this question by accurate drawing will not be accepted. The points A (4, 5), B( 2, 3), C(1, 9) and D are the vertices of a trapezium in which BC is parallel to AD and angle BCD is 90. Find the area of the trapezium. [8] 0606/23/O/N/12 [Turn over

78 10 9 The table shows experimental values of two variables x and y. x y It is known that x and y are related by the equation y = a x 2 + bx, where a and b are constants. (i) A straight line graph is to be drawn to represent this information. Given that x 2 y is plotted on the vertical axis, state the variable to be plotted on the horizontal axis. [1] (ii) On the grid opposite, draw this straight line graph. [3] (iii) your graph to estimate the value of a and of b. [3] (iv) Estimate the value of y when x is 3.7. [2] 0606/23/O/N/12

79 /23/O/N/12 [Turn over

80 12 10 C 80 m A 200 m D x m B A track runs due east from A to B, a distance of 200 m. The point C is 80 m due north of B. A cyclist travels on the track from A to D, where D is x m due west of B. The cyclist then travels in a straight line across rough ground from D to C. The cyclist travels at 10 m s 1 on the track and at 6 m s 1 across rough ground. (i) Show that the time taken, T s, for the cyclist to travel from A to C is given by T = 200 x 10 + (x ) 6 [2] (ii) Given that x can vary, find the value of x for which T has a stationary value and the corresponding value of T. [6] 0606/23/O/N/12

81 (a) Solve (2 x 2 2 ) = 100, giving your answer to 1 decimal place. [3] (b) Solve log y 2 = 3 log y 256. [3] (c) Solve 65z 2 36 z = 216z z. [4] 0606/23/O/N/12 [Turn over

82 14 12 Answer only one of the following alternatives. EITHER (i) Express 4x x + 55 in the form (ax + b) 2 + c, where a, b and c are constants and a is positive. [3] The functions f and g are defined by f : x 4x x + 55 for x > 4, g : x 1 x for x > 0. (ii) Find f 1 (x). [3] (iii) Solve the equation fg(x) = 135. [4] OR The functions h and k are defined by h : x 2x 7 for x c, k : x 3x 4 for x > 2. x 2 (i) State the least possible value of c. [1] (ii) Find h 1 (x). [2] (iii) Solve the equation k(x) = x. [3] (iv) Find an expression for the function k 2, in the form k 2 : x a + b where a and b are x constants. [4] 0606/23/O/N/12

83 Start your answer to Question 12 here. Indicate which question you are answering. 15 EITHER OR 0606/23/O/N/12 [Turn over

84 16 Continue your answer here if necessary. 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. University of 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. 0606/23/O/N/12