- 1. Read carefully Calculators may NOT be used in this paper. Section A – Questions 1–20 (40 marks) Instructions for completion of Section A are given on page two. For this section of the examination you must use an HB pencil. Section B (30 marks) 1 Full credit will be given only where the solution contains appropriate working. 2 Answers obtained by readings from scale drawings will not receive any credit. MATHEMATICS HIGHER Paper 1 (Non-calculator) N A T I O N A L T U E S D A Y , 2 0 M A Y Q U A L I F I C A T I O N S 9 . 0 0 A M – 1 0 . 3 0 A M 2 0 0 8 X100/301 ©LI X100/301 6/27670 *X100/301*
- 2. Read carefully 1 Check that the answer sheet provided is for Mathematics Higher (Section A). 2 For this section of the examination you must use an HB pencil and, where necessary, an eraser. 3 Check that the answer sheet you have been given has your name, date of birth, SCN (Scottish Candidate Number) and Centre Name printed on it. Do not change any of these details. 4 If any of this information is wrong, tell the Invigilator immediately. 5 If this information is correct, print your name and seat number in the boxes provided. 6 The answer to each question is either A, B, C or D. Decide what your answer is, then, using your pencil, put a horizontal line in the space provided (see sample question below). 7 There is only one correct answer to each question. 8 Rough working should not be done on your answer sheet. 9 At the end of the exam, put the answer sheet for Section A inside the front cover of your answer book. Sample Question A curve has equation y = x3 – 4x. What is the gradient at the point where x = 2? A 8 B 1 C 0 D –4 The correct answer is A—8. The answer A has been clearly marked in pencil with a horizontal line (see below). Changing an answer If you decide to change your answer, carefully erase your first answer and using your pencil, fill in the answer you want. The answer below has been changed to D. A B C D A B C D Page two[X100/301]
- 3. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page three a1 a2 a3 b1 b2 b3 and b = . + −2 2 .g f c sinax cosa ax cosax sin−a ax ± ( )f x ( )′f x sinax 1 cos− +ax Ca cosax 1 sin +ax Ca ( )f x ( )f x dx∫ [X100/301] [Turn over
- 4. SECTION A ALL questions should be attempted. 1. A sequence is defined by the recurrence relation un+1 = 0.3un + 6 with u10 = 10. What is the value of u12? A 6.6 B 7.8 C 8.7 D 9.6 2. The x-axis is a tangent to a circle with centre (–7, 6) as shown in the diagram. What is the equation of the circle? A (x + 7)2 + (y – 6)2 = 1 B (x + 7)2 + (y – 6)2 = 49 C (x – 7)2 + (y + 6)2 = 36 D (x + 7)2 + (y – 6)2 = 36 3. The vectors u = and v = are perpendicular. What is the value of k? A 0 B 3 C 4 D 5 Page four[X100/301] y O x C(–7, 6) a1 a2 a3 − 1 k a1 a2 a3 0 4 k −1
- 5. Page five 4. 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 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? A y – 9 = (x – 7) B y + 9 = (x + 7) C y – 7 = (x – 9) D y + 9 = (x + 7) [X100/301] y O x (2, 5) (7, 9) − 5 4 4 5 − 4 5 5 4 [Turn over
- 6. Page six 6. What is the solution of the equation ? 7. The diagram shows a line L; the angle between L and the positive direction of the x-axis is 135°, as shown. What is the gradient of line L? [X100/301] π π π π A 6 2 B 3 3 C 4 5 D 6 π π− = ≤ ≤2sin 3 0 where 2 x x y O x 135° L − − − 1 A 2 3 B 2 C 1 1 D 2 − 1 2 −1 1 2
- 7. Page seven 8. The diagram shows part of the graph of a function with equation y = f(x). Which of the following diagrams shows the graph with equation y = –f(x – 2)? [X100/301] y O x (0, 4) (3, –3) A y O x (2, 4) (5, –3) C y O x (3, 5) (0, –2) B y O x (1, 3) (–2, –4) D y O x (5, 3) (2, –4) [Turn over
- 8. Page eight 9. Given that , find an expression for sin(x + a). 10. Here are two statements about the roots of the equation x2 + x + 1 = 0: (1) the roots are equal; (2) the roots are real. Which of the following is true? A Neither statement is correct. B Only statement (1) is correct. C Only statement (2) is correct. D Both statements are correct. 11. 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? A 1:1 B 1:2 C 1:4 D 1:6 3 A sin 5 4 3 B sin cos 5 5 3 4 C sin 5 5 2 3 D sin cos 5 5 x x x x x x + + − − [X100/301] π ≤ ≤ = 3 0 and sin 2 5 a a cosx
- 9. Page nine 12. In the diagram RSTU, VWXY represents a cuboid. represents vector f, represents vector g and represents vector h. Express in terms of f, g and h. A B C D 13. 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). What is the equation of the graph? A y = 3(x – 1)(x – 4) B y = 3(x + 1)(x + 4) C y = 12(x – 1)(x – 4) D y = 12(x + 1)(x + 4) [X100/301] SR ST SW VT W XV Y R U T S gf h = + +VT f g h = − +VT f g h = − + −VT f g h = − − +VT f g h y O x 12 1 4 y = f(x) [Turn over → → → → → → → →
- 10. Page ten 14. (2x + 3) dx. A –4cos (2x + 3) + c B –2cos (2x + 3) + c C 4cos (2x + 3) + c D 8cos (2x + 3) + c 15. What is the derivative of (x3 + 4)2 ? A (3x2 + 4)2 B (x3 + 4)3 C 6x2 (x3 + 4) D 2(3x2 + 4)–1 16. 2x2 + 4x + 7 is expressed in the form 2(x + p)2 + q. What is the value of q? A 5 B 7 C 9 D 11 17. A function f is given by f(x) = What is a suitable domain of f ? A x ≥ 3 B x ≤ 3 C –3 ≤ x ≤ 3 D –9 ≤ x ≤ 9 [X100/301] ∫Find 4sin 1 3 2 9 .x−
- 11. 18. Vectors p and q are such that |p| = 3, |q| = 4 and p.q = 10. Find the value of q.(p + q). A 0 B 14 C 26 D 28 19. The diagram shows part of the graph whose equation is of the form y = 2mx . What is the value of m? A 2 B 3 C 8 D 18 20. The diagram shows part of the graph of y = log3(x – 4). The point (q, 2) lies on the graph. What is the value of q? A 6 B 7 C 8 D 13 [END OF SECTION A] Page eleven[X100/301] y x (3, 54) y O x y = log3(x – 4) (q, 2) (5, 0) [Turn over for Section B on Page twelve O
- 12. Page twelve ALL questions should be attempted. 21. A function f is defined on the set of real numbers by f(x) = x3 – 3x + 2. (a) Find the coordinates of the stationary points on the curve y = f(x) and determine their nature. (b) (i) Show that (x – 1) is a factor of x3 – 3x + 2. (ii) Hence or otherwise factorise x3 – 3x + 2 fully. (c) State the coordinates of the points where the curve with equation 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. (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. Marks 6 5 4 5 2 3 5 y = x3 – 6x2 + 8x y O x SECTION B [END OF SECTION B] [END OF QUESTION PAPER] [X100/301]
- 13. Read Carefully 1 Calculators may be used in this paper. 2 Full credit will be given only where the solution contains appropriate working. 3 Answers obtained by readings from scale drawings will not receive any credit. MATHEMATICS HIGHER Paper 2 N A T I O N A L T U E S D A Y , 2 0 M A Y Q U A L I F I C A T I O N S 1 0 . 5 0 A M – 1 2 . 0 0 N O O N 2 0 0 8 X100/302 ©LB X100/302 6/27670 *X100/302*
- 14. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page two a1 a2 a3 b1 b2 b3 and b = . + −2 2 .g f c sinax cosa ax cosax sin−a ax ± ( )f x ( )′f x sinax 1 cos− +ax Ca cosax 1 sin +ax Ca ( )f x ( )f x dx∫ [X100/302]
- 15. Page three ALL questions should be attempted. 1. The vertices of triangle ABC are A(7, 9), B(–3, –1) and C(5, –5) as shown in the diagram. The broken line represents the perpendicular bisector of BC. (a) Show that the equation of the perpendicular bisector of BC is y = 2x – 5. (b) Find the equation of the median from C. (c) Find the coordinates of the point of intersection of the perpendicular bisector of BC and the median from C. 2. The diagram shows a cuboid OABC, DEFG. F is the point (8, 4, 6). P divides AE in the ratio 2:1. Q is the midpoint of CG. (a) State the coordinates of P and Q. (b) Write down the components of (c) Find the size of angle QPA. Marks 4 3 3 2 2 5 A B O C x F(8, 4, 6) O Gz D A x Q [X100/302] [Turn over y PQ and PA. y C P E B → →
- 16. Page four 3. (a) (i) Diagram 1 shows part of the graph of y = f(x),where f(x) = pcos x. Write down the value of p. (ii) Diagram 2 shows part of the graph of y = g(x), where g(x) = qsin x. Write down the value of q. (b) Write f(x) + g(x) in the form kcos (x + a) where k > 0 and 0 < a < (c) Hence find as a single trigonometric expression. 4. (a) Write down the centre and calculate the radius of the circle with equation x2 + y2 + 8x + 4y – 38 = 0. (b) A second circle has equation (x – 4)2 + (y – 6)2 = 26. Find the distance between the centres of these two circles and hence show that the circles intersect. (c) The line with equation y = 4 – x is a common chord passing through the points of intersection of the two circles. Find the coordinates of the points of intersection of the two circles. 5. Solve the equation cos 2x ° + 2sin x ° = sin2 x ° in the interval 0 ≤ x < 360. Marks 2 4 2 2 4 5 5 y Diagram 1 O x [X100/302] 7 π 7− –π y Diagram 2 O xπ –3 2π 3 . 2 π ( ) ( )f x g x+′ ′
- 17. Page five [END OF QUESTION PAPER] 6. In the diagram, Q lies on the line joining (0, 6) and (3, 0). OPQR is a rectangle, where P and R lie on the axes and OR = t. (a) Show that QR = 6 – 2t. (b) Find the coordinates of Q for which the rectangle has a maximum area. 7. The parabola shown in the diagram has equation y = 32 – 2x2 . The shaded area lies between the lines y = 14 and y = 24. Calculate the shaded area. Marks 3 6 8 y Q 6 xtO P R 3 y y = 24 y = 14 xO [X100/302]
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- 21. Read carefully Calculators may NOT be used in this paper. Section A – Questions 1–20 (40 marks) Instructions for completion of Section A are given on page two. For this section of the examination you must use an HB pencil. Section B (30 marks) 1 Full credit will be given only where the solution contains appropriate working. 2 Answers obtained by readings from scale drawings will not receive any credit. MATHEMATICS HIGHER Paper 1 (Non-calculator) N A T I O N A L T H U R S D A Y , 2 1 M A Y Q U A L I F I C A T I O N S 9 . 0 0 A M – 1 0 . 3 0 A M 2 0 0 9 X100/301 ©LI X100/301 6/29520 *X100/301*
- 22. Read carefully 1 Check that the answer sheet provided is for Mathematics Higher (Section A). 2 For this section of the examination you must use an HB pencil and, where necessary, an eraser. 3 Check that the answer sheet you have been given has your name, date of birth, SCN (Scottish Candidate Number) and Centre Name printed on it. Do not change any of these details. 4 If any of this information is wrong, tell the Invigilator immediately. 5 If this information is correct, print your name and seat number in the boxes provided. 6 The answer to each question is either A, B, C or D. Decide what your answer is, then, using your pencil, put a horizontal line in the space provided (see sample question below). 7 There is only one correct answer to each question. 8 Rough working should not be done on your answer sheet. 9 At the end of the exam, put the answer sheet for Section A inside the front cover of your answer book. Sample Question A curve has equation y = x3 – 4x. What is the gradient at the point where x = 2? A 8 B 1 C 0 D –4 The correct answer is A—8. The answer A has been clearly marked in pencil with a horizontal line (see below). Changing an answer If you decide to change your answer, carefully erase your first answer and, using your pencil, fill in the answer you want. The answer below has been changed to D. A B C D A B C D Page two[X100/301]
- 23. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page three a1 a2 a3 b1 b2 b3 and b = . + −2 2 .g f c sinax cosa ax cosax sin−a ax ± ( )f x ( )′f x sinax 1 cos− +ax Ca cosax 1 sin +ax Ca ( )f x ( )f x dx∫ [X100/301] [Turn over
- 24. SECTION A ALL questions should be attempted. 1. A sequence is defined by un + 1 = 3un + 4 with u1 = 2. What is the value of u3? A 34 B 21 C 18 D 13 2. A circle has equation x2 + y2 + 8x + 6y – 75 = 0. What is the radius of this circle? A 5 B 10 C D 3. Triangle PQR has vertices at P(–3, –2), Q(–1, 4) and R(3, 6). PS is a median. What is the gradient of PS? A – 2 B C 1 D 4. A curve has equation y = 5x3 – 12x. What is the gradient of the tangent at the point (1, –7)? A –7 B –5 C 3 D 5 Page four[X100/301] 75 175 7 4 − 7 4
- 25. 5. Here are two statements about the points S(2, 3) and T(5, –1): (1) The length of ST = 5 units; (2) The gradient of ST = Which of the following is true? A Neither statement is correct. B Only statement (1) is correct. C Only statement (2) is correct. D Both statements are correct. 6. A sequence is generated by the recurrence relation un+1 = 0.7un + 10. What is the limit of this sequence as n → ∞? 7. If the exact value of cos x is find the exact value of cos 2x. Page five[X100/301] 4 . 3 100 A 3 100 B 7 17 C 100 3 D 10 − − 3 A 5 2 B 5 2 C 5 3 D 5 1 , 5 [Turn over
- 26. Page six 8. What is the derivative of 9. The line with equation y = 2x intersects the circle with equation x2 + y2 = 5 at the points J and K. What are the x-coordinates of J and K? A xJ = 1, xK = –1 B xJ = 2, xK = –2 C xJ = 1, xK = –2 D xJ = –1, xK = 2 [X100/301] 3 1 , 0? 4 x x ≠ − − 2 2 4 4 1 A 12 1 B 12 4 C 3 D 4 x x x x
- 27. Page seven 10. Which of the following graphs has equation y = log5(x – 2)? [X100/301] A y O x (7, 1) (3, 0) C y O x (3, 1) (–1, 0) B y O x (1, 7) (0, 3) D y O x (1, 7) (7, 1) [Turn over
- 28. Page eight 11. How many solutions does the equation (4 sin x – )(sin x + 1) = 0 have in the interval 0 ≤ x < 2π? A 4 B 3 C 2 D 1 12. A function f is given by f(x) = 2x2 – x – 9. Which of the following describes the nature of the roots of f(x) = 0? A No real roots B Equal roots C Real distinct roots D Rational distinct roots 13. k and a are given by k sin a ° = 1 k cos a ° = where k > 0 and 0 ≤ a < 90. What are the values of k and a? k a A 2 60 B 2 30 C 60 D 30 14. If f(x) = 2sin 3x – + 5, what is the range of values of f(x)? A –1 ≤ f(x) ≤ 11 B 2 ≤ f(x) ≤ 8 C 3 ≤ f(x) ≤ 7 D –3 ≤ f(x) ≤ 7 [X100/301] 5 3 10 10 ( )π 2
- 29. Page nine 15. The line GH makes an angle of radians with the y-axis, as shown in the diagram. What is the gradient of GH? A B C D 16. The graph of y = 4x3 – 9x2 is shown in the diagram. Which of the following gives the area of the shaded section? [X100/301] 6 π 3 1 2 1 2 3 2 y O G x H 6 π y O 1 –5 x y = 4x3 – 9x2 04 3 5 14 3 0 02 5 12 0 A 3 B 3 C 12 18 D 12 18 x x x x x x x x − − ⎡ ⎤−⎣ ⎦ ⎡ ⎤− −⎣ ⎦ ⎡ ⎤−⎣ ⎦ ⎡ ⎤− −⎣ ⎦ [Turn over
- 30. Page ten 17. The vector u has components Which of the following is a unit vector parallel to u? A B –3i + 4k C D 18. Given that f(x) = (4 – 3x2 ) on a suitable domain, find f′(x). A –3x(4 – 3x2 ) B (4 – 6x) C 2(4 – 3x3 ) D 3x(4 – 3x2 ) 19. For what values of x is 6 + x – x2 < 0? A x > 3 only B x < –2 only C x < –2, x > 3 D –3 < x < 2 20. A = 2πr2 + 6πr. What is the rate of change of A with respect to r when r = 2? A 10π B 12π C 14π D 20π [END OF SECTION A] [X100/301] 3 0 . 4 −⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 3 4 5 5 − +i k 3 4 7 7 − +i k 1 1 3 4 − +i k 1 2 − 1 2 − 3 2 − 1 2 3 2 − 1 2 −
- 31. Page eleven ALL questions should be attempted. 21. Triangle PQR has vertex P on the x-axis, as shown in the diagram. Q and R are the points (4, 6) and (8, –2) respectively. The equation of PQ is 6x – 7y + 18 = 0. (a) State the coordinates of P. (b) Find the equation of the altitude of the triangle from P. (c) The altitude from P meets the line QR at T. Find the coordinates of T. 22. D, E and F have coordinates (10, –8, –15), (1, –2, –3) and (–2, 0, 1) respectively. (a) (i) Show that D, E and F are collinear. (ii) Find the ratio in which E divides DF. (b) G has coordinates (k, 1, 0). Given that DE is perpendicular to GE, find the value of k. 23. The diagram shows a sketch of the function y = f(x). (a) Copy the diagram and on it sketch the graph of y = f(2x). (b) On a separate diagram sketch the graph of y = 1 – f(2x). Marks 1 3 4 4 4 2 3 6x – 7y + 18 = 0 y O T P Q(4, 6) R(8, –2) x y = f(x) y O (2, 8) (–4, 8) x SECTION B [X100/301] [Turn over for Question 24 on Page twelve
- 32. [END OF SECTION B] [END OF QUESTION PAPER] Page twelve 24. (a) Using the fact that find the exact value of sin (b) Show that sin(A + B) + sin(A – B) = 2sin A cos B. (c) (i) Express in terms of (ii) Hence or otherwise find the exact value of sin + sin Marks 3 2 4 [X100/301] 7 , 12 3 4 π π π= + 12 π and . 3 4 π π 7 . 12 π ( ) 7 12 π ( ) . 12 π ( )
- 33. Read Carefully 1 Calculators may be used in this paper. 2 Full credit will be given only where the solution contains appropriate working. 3 Answers obtained by readings from scale drawings will not receive any credit. MATHEMATICS HIGHER Paper 2 N A T I O N A L T H U R S D A Y , 2 1 M A Y Q U A L I F I C A T I O N S 1 0 . 5 0 A M – 1 2 . 0 0 N O O N 2 0 0 9 X100/302 ©LI X100/302 6/29520 *X100/302*
- 34. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page two a1 a2 a3 b1 b2 b3 and b = . + −2 2 .g f c sinax cosa ax cosax sin−a ax ± ( )f x ( )′f x sinax 1 cos− +ax Ca cosax 1 sin +ax Ca ( )f x ( )f x dx∫ [X100/302]
- 35. Page three ALL questions should be attempted. 1. Find the coordinates of the turning points of the curve with equation y = x3 – 3x2 – 9x + 12 and determine their nature. 2. Functions f and g are given by f(x) = 3x + 1 and g(x) = x2 – 2. (a) (i) Find p(x) where p(x) = f(g(x)). (ii) Find q(x) where q(x) = g(f(x)). (b) Solve p′(x) = q′(x). 3. (a) (i) Show that x = 1 is a root of x3 + 8x2 + 11x – 20 = 0. (ii) Hence factorise x3 + 8x2 + 11x – 20 fully. (b) Solve log2 (x + 3) + log2 (x2 + 5x – 4) = 3. 4. (a) Show that the point P(5, 10) lies on circle C1 with equation (x + 1)2 + (y – 2)2 = 100. (b) PQ is a diameter of this circle as shown in the diagram. Find the equation of the tangent at Q. (c) Two circles, C2 and C3 , touch circle C1 at Q. The radius of each of these circles is twice the radius of circle C1 . Find the equations of circles C2 and C3 . Marks 8 3 3 4 5 1 5 4 [X100/302] [Turn over y P(5, 10) O Q x
- 36. Page four 5. The graphs of y = f(x) and y = g(x) are shown in the diagram. f(x) = –4 cos(2x) + 3 and g(x) is of the form g(x) = m cos(nx). (a) Write down the values of m and n. (b) Find, correct to one decimal place, the coordinates of the points of intersection of the two graphs in the interval (c) Calculate the shaded area. 6. The size of the human population, N, can be modelled using the equation N = N0ert where N0 is the population in 2006, t is the time in years since 2006, and r is the annual rate of increase in the population. (a) In 2006 the population of the United Kingdom was approximately 61 million, with an annual rate of increase of 1.6%. Assuming this growth rate remains constant, what would be the population in 2020? (b) In 2006 the population of Scotland was approximately 5.1 million, with an annual rate of increase of 0.43%. Assuming this growth rate remains constant, how long would it take for Scotland’s population to double in size? 7. Vectors p, q and r are represented on the diagram shown where angle ADC = 30 °. It is also given that |p| = 4 and|q| = 3. (a) Evaluate p.(q + r) and r.(p – q). (b) Find |q + r| and |p – q|. Marks 1 5 6 2 3 6 4 [X100/302] A pD 30 ° B r q C [END OF QUESTION PAPER] y xπ y = f(x) y = g(x) 7 3 0 –1 –30 .x π≤ ≤
- 37. Read carefully Calculators may NOT be used in this paper. Section A – Questions 1–20 (40 marks) Instructions for completion of Section A are given on page two. For this section of the examination you must use an HB pencil. Section B (30 marks) 1 Full credit will be given only where the solution contains appropriate working. 2 Answers obtained by readings from scale drawings will not receive any credit. MATHEMATICS HIGHER Paper 1 (Non-calculator) N A T I O N A L Q U A L I F I C A T I O N S 2 0 1 0 F R I D A Y , 2 1 M A Y 9 . 0 0 A M – 1 0 . 3 0 A M X100/301 © LI X100/301 6/31310 *X100/301*
- 38. Read carefully 1 Check that the answer sheet provided is for Mathematics Higher (Section A). 2 For this section of the examination you must use an HB pencil and, where necessary, an eraser. 3 Check that the answer sheet you have been given has your name, date of birth, SCN (Scottish Candidate Number) and Centre Name printed on it. Do not change any of these details. 4 If any of this information is wrong, tell the Invigilator immediately. 5 If this information is correct, print your name and seat number in the boxes provided. 6 The answer to each question is either A, B, C or D. Decide what your answer is, then, using your pencil, put a horizontal line in the space provided (see sample question below). 7 There is only one correct answer to each question. 8 Rough working should not be done on your answer sheet. 9 At the end of the exam, put the answer sheet for Section A inside the front cover of your answer book. Sample Question A curve has equation y = x3 – 4x. What is the gradient at the point where x = 2? A 8 B 1 C 0 D –4 The correct answer is A—8. The answer A has been clearly marked in pencil with a horizontal line (see below). Changing an answer If you decide to change your answer, carefully erase your first answer and, using your pencil, fill in the answer you want. The answer below has been changed to D. A B C D A B C D Page two[X100/301]
- 39. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page three + −2 2 .g f c sinax cosa ax cosax sin−a ax ± ( )f x ( )′f x sinax 1 cos− +ax Ca cosax 1 sin +ax Ca ( )f x ( )f x dx∫ [X100/301] [Turn over a1 a2 a3 b1 b2 b3 and b = .
- 40. SECTION A ALL questions should be attempted. 1. A line L is perpendicular to the line with equation 2x – 3y – 6 = 0. What is the gradient of the line L? 2. A sequence is defined by the recurrence relation un + 1 = 2un + 3 and u0 = 1. What is the value of u2 ? A 7 B 10 C 13 D 16 3. Given that find 3u – 2v in component form. A B C D Page four[X100/301] 3 A 2 1 B 2 2 C 3 D 2 − − 2 0 1 – 1 2 4 and v = ,u = 4 – 1 – 5 4 – 4 11 8 – 1 5 8 – 4 – 5
- 41. 4. The diagram shows the graph with equation of the form y = acos bx for 0 ≤ x ≤ 2π. What is the equation of this graph? A y = 2cos3x B y = 2cos2x C y = 3cos2x D y = 4cos3x 5. When x2 + 8x + 3 is written in the form (x + p)2 + q, what is the value of q? A –19 B –13 C –5 D 19 Page five[X100/301] [Turn over y 2 O π 2π –2 x
- 42. Page six 6. The roots of the equation kx2 – 3x + 2 = 0 are equal. What is the value of k? 7. A sequence is generated by the recurrence relation un + 1 = + 7, with u0 = – 2. What is the limit of this sequence as n → ∞? [X100/301] 1 A 28 28 B 5 28 C 3 D 28 1 4 nu 9 A 8 8 B 9 8 C 9 9 D 8 − −
- 43. Page seven 8. The equation of the circle shown in the diagram is x2 + y2 – 6x – 10y + 9 = 0. The x-axis and the line l are parallel tangents to the circle. What is the equation of line l? A y = 5 B y = 10 C y = 18 D y = 20 9. Find 10. The vectors xi + 5j + 7k and –3i + 2j – k are perpendicular. What is the value of x? A 0 B 1 C D [X100/301] [Turn over y l O x − − − − − − + − + + − + + − − + 5 5 3 3 2 A 5sin5 5 2 1 B sin5 5 5 2 1 C sin5 3 5 2 D 5sin5 3 x x c x x c x x c x x c 4 (2 cos5 ) .x x dx− +∫ 4 3 10 3
- 44. 11. Functions f and g are defined on suitable domains by f(x) = cos x and g(x) = x + What is the value of 12. Page eight[X100/301] f g ( ) ? 6 π . 6 π 1 A 2 6 3 B 2 6 3 C 2 1 D 2 π π + + 5 1 If ( )= , 0, what is ( )?f x x f x x ′≠ 6 5 4 5 7 2 3 2 1 A 5 1 B 5 5 C 2 5 D 2 x x x x − − − − − − − − ( )
- 45. Page nine 13. Which of the following diagrams shows a parabola with equation y = ax2 + bx + c, where • a > 0 • b2 – 4ac > 0? [X100/301] [Turn over A y O x B y O x C y O x D y O x
- 46. Page ten 14. The diagram shows graphs with equations y = 14 – x2 and y = 2x2 + 2. Which of the following represents the shaded area? 15. The derivative of a function f is given by f ′(x) = x2 – 9. Here are two statements about f: (1) f is increasing at x = 1; (2) f is stationary at x = –3. Which of the following is true? A Neither statement is correct. B Only statement (1) is correct. C Only statement (2) is correct. D Both statements are correct. [X100/301] y 14 2 –2 2O y = 2x2 + 2 y = 14 – x2 x − − − − − − ∫ ∫ ∫ ∫ 14 2 2 14 2 2 2 2 2 2 2 2 A (12 3 ) B (3 12) C (12 3 ) D (3 12) x dx x dx x dx x dx
- 47. Page eleven 16. The diagram shows the graph with equation y = k(x – 1)2 (x + t). What are the values of k and t? k t A –2 –5 B –2 5 C 2 –5 D 2 5 17. If s(t) = t2 – 5t + 8, what is the rate of change of s with respect to t when t = 3? A –5 B 1 C 2 D 9 18. What is the solution of x2 + 4x > 0, where x is a real number? A –4 < x < 0 B x < –4, x > 0 C 0 < x < 4 D x < 0, x > 4 [X100/301] [Turn over y 10 1 5O x
- 48. Page twelve[X100/301] 19. The diagram shows the graph of y = f(x) where f is a logarithmic function. What is f(x)? A f(x) = log6(x – 3) B f(x) = log3(x + 3) C f(x) = log3(x – 3) D f(x) = log6(x + 3) 20. The diagram shows the graph of y = f(2x) – 3. What are the coordinates of the turning point on the graph of y = f(x)? A (12, 7) B (12, 1) C (3, 7) D (3, 1) y (6, 1) (4, 0)O x y (6, 4) y = f(2x) – 3 O x [END OF SECTION A] y = f(x)
- 49. Page thirteen SECTION B ALL questions should be attempted. 21. Triangle ABC has vertices A(4, 0), B(–4, 16) and C(18, 20), as shown in the diagram opposite. Medians AP and CR intersect at the point T(6, 12). (a) Find the equation of median BQ. (b) Verify that T lies on BQ. (c) Find the ratio in which T divides BQ. 22. (a) (i) Show that (x – 1) is a factor of f(x) = 2x3 + x2 – 8x + 5. (ii) Hence factorise f(x) fully. (b) Solve 2x3 + x2 – 8x + 5 = 0. (c) The line with equation y = 2x – 3 is a tangent to the curve with equation y = 2x3 + x2 – 6x + 2 at the point G. Find the coordinates of G. (d) This tangent meets the curve again at the point H. Write down the coordinates of H. [X100/301] [Turn over for Question 23 on Page fourteen Marks 3 1 2 5 1 5 1 y B P C T O A R Q x
- 50. [END OF SECTION B] [END OF QUESTION PAPER] Page fourteen 23. (a) Diagram 1 shows a right angled triangle, where the line OA has equation 3x – 2y = 0. (i) Show that (ii) Find the value of sina. (b) A second right angled triangle is added as shown in Diagram 2. The line OB has equation 3x – 4y = 0. Find the values of sinb and cosb. (c) (i) Find the value of sin(a – b). (ii) State the value of sin(b – a). Marks 4 4 4 [X100/301] y O x A a Diagram 1 = 3 tan . 2 a y O x A a Diagram 2 b B
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- 53. Read Carefully 1 Calculators may be used in this paper. 2 Full credit will be given only where the solution contains appropriate working. 3 Answers obtained by readings from scale drawings will not receive any credit. © LI X100/302 6/31310 *X100/302* MATHEMATICS HIGHER Paper 2 N A T I O N A L Q U A L I F I C A T I O N S 2 0 1 0 F R I D A Y , 2 1 M A Y 1 0 . 5 0 A M – 1 2 . 0 0 N O O N X100/302
- 54. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page two + −2 2 .g f c sinax cosa ax cosax sin−a ax ± ( )f x ( )′f x sinax 1 cos− +ax Ca cosax 1 sin +ax Ca ( )f x ( )f x dx∫ [X100/302] a1 a2 a3 b1 b2 b3 and b = .
- 55. Page three ALL questions should be attempted. 1. The diagram shows a cuboid OPQR,STUV relative to the coordinate axes. P is the point (4, 0, 0), Q is (4, 2, 0) and U is (4, 2, 3). M is the midpoint of OR. N is the point on UQ such that (a) State the coordinates of M and N. (b) Express VM and VN in component form. (c) Calculate the size of angle MVN. 2. (a) 12 cos x ° – 5 sin x ° can be expressed in the form k cos(x + a)°, where k > 0 and 0 ≤ a < 360. Calculate the values of k and a. (b) (i) Hence state the maximum and minimum values of 12 cos x ° – 5 sin x °. (ii) Determine the values of x, in the interval 0 ≤ x < 360, at which these maximum and minimum values occur. Marks 2 2 5 4 3 [X100/302] [Turn over z y S M T N R O P (4, 0, 0) Q (4, 2, 0) x = 1 3 UN UQ. → → U (4, 2, 3) V
- 56. Page four 3. (a) (i) Show that the line with equation y = 3 – x is a tangent to the circle with equation x2 + y2 + 14x + 4y – 19 = 0. (ii) Find the coordinates of the point of contact, P. (b) Relative to a suitable set of coordinate axes, the diagram below shows the circle from (a) and a second smaller circle with centre C. The line y = 3 – x is a common tangent at the point P. The radius of the larger circle is three times the radius of the smaller circle. Find the equation of the smaller circle. 4. Solve 2 cos 2x – 5 cos x – 4 = 0 for 0 ≤ x < 2π. Marks 5 6 5 [X100/302] P C
- 57. Page five 5. The parabolas with equations y = 10 – x2 and are shown in the diagram below. A rectangle PQRS is placed between the two parabolas as shown, so that: • Q and R lie on the upper parabola; • RQ and SP are parallel to the x-axis; • T, the turning point of the lower parabola, lies on SP. (a) (i) If TP = x units, find an expression for the length of PQ. (ii) Hence show that the area, A, of rectangle PQRS is given by A(x) = 12x – 2x3 . (b) Find the maximum area of this rectangle. Marks 3 6 [X100/302] 22 5 (10 )y x= − y = 10 – x2 y O R S T Q P x 22 5 (10 )y x= − [Turn over for Questions 6 and 7 on Page six
- 58. Page six 6. (a) A curve has equation Show that the equation of the tangent to this curve at the point where x = 9 is (b) Diagram 1 shows part of the curve and the tangent. The curve cuts the x-axis at the point A. Find the coordinates of point A. (c) Calculate the shaded area shown in diagram 2. 7. (a) Given that log4 x = P, show that log16 x = (b) Solve log3x + log9x = 12. Marks 5 1 7 3 3 [X100/302] [END OF QUESTION PAPER] 1 2(2 9) .y x= − 1 3 .y x= y O x9 1 3 y x= 1 2(2 9)y x= − y O A A x9 1 3 y x= 1 2(2 9)y x= − Diagram 1 Diagram 2 1 2 P.
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- 61. Read carefully Calculators may NOT be used in this paper. Section A – Questions 1–20 (40 marks) Instructions for completion of Section A are given on page two. For this section of the examination you must use an HB pencil. Section B (30 marks) 1 Full credit will be given only where the solution contains appropriate working. 2 Answers obtained by readings from scale drawings will not receive any credit. MATHEMATICS HIGHER Paper 1 (Non-calculator) N A T I O N A L Q U A L I F I C A T I O N S 2 0 1 1 W E D N E S D A Y , 1 8 M A Y 9 . 0 0 A M – 1 0 . 3 0 A M X100/301 © LI X100/301 6/32010 *X100/301*
- 62. Read carefully 1 Check that the answer sheet provided is for Mathematics Higher (Section A). 2 For this section of the examination you must use an HB pencil and, where necessary, an eraser. 3 Check that the answer sheet you have been given has your name, date of birth, SCN (Scottish Candidate Number) and Centre Name printed on it. Do not change any of these details. 4 If any of this information is wrong, tell the Invigilator immediately. 5 If this information is correct, print your name and seat number in the boxes provided. 6 The answer to each question is either A, B, C or D. Decide what your answer is, then, using your pencil, put a horizontal line in the space provided (see sample question below). 7 There is only one correct answer to each question. 8 Rough working should not be done on your answer sheet. 9 At the end of the exam, put the answer sheet for Section A inside the front cover of your answer book. Sample Question A curve has equation y = x3 – 4x. What is the gradient at the point where x = 2? A 8 B 1 C 0 D –4 The correct answer is A—8. The answer A has been clearly marked in pencil with a horizontal line (see below). Changing an answer If you decide to change your answer, carefully erase your first answer and, using your pencil, fill in the answer you want. The answer below has been changed to D. A B C D A B C D Page two[X100/301]
- 63. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page three + −2 2 .g f c sinax cosa ax cosax sin−a ax ± ( )f x ( )′f x sinax 1 cos− +ax Ca cosax 1 sin +ax Ca ( )f x ( )f x dx∫ [X100/301] [Turn over a1 a2 a3 b1 b2 b3 and b = .
- 64. SECTION A ALL questions should be attempted. 1. Given that p = , q = and r = , express 2p – q – in component form. 2. A line l has equation 3y + 2x = 6. What is the gradient of any line parallel to l? Page four[X100/301] A 2 2 B 3 3 C 2 D 2 − − 1 A 9 15 1 B 11 13 5 C 9 13 5 D 11 15 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ 1 2 r 2 5 7 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ −⎝ ⎠ 1 0 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ −⎝ ⎠ 4 2 0 −⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
- 65. 3. The diagram shows the graph of y = f(x). Which of the following shows the graph of y = f(x + 2) – 1? Page five[X100/301] [Turn over • • y (1, 2) O (–2, –3) x A B C D • • y (–4, 3) O (–1, –2) x • • y 3 O (3, –2) x • • y (3, 1) O –4 x • • y (–1, 1) O (–4, –4) x
- 66. Page six 4. A tangent to the curve with equation y = x3 – 2x is drawn at the point (2, 4). What is the gradient of this tangent? A 2 B 3 C 4 D 10 5. If x2 – 8x + 7 is written in the form (x – p)2 + q, what is the value of q? A –9 B –1 C 7 D 23 6. The point P(2, –3) lies on the circle with centre C as shown. The gradient of CP is –2. What is the equation of the tangent at P? A y + 3 = –2(x – 2) B y – 3 = –2(x + 2) C y + 3 = (x – 2) D y – 3 = (x + 2) 7. A function f is defined on the set of real numbers by f(x) = x3 – x2 + x + 3. What is the remainder when f(x) is divided by (x – 1)? A 0 B 2 C 3 D 4 [X100/301] y C O P(2, –3) x1 2 1 2
- 67. Page seven 8. A line makes an angle of 30° with the positive direction of the x-axis as shown. What is the gradient of the line? 9. The discriminant of a quadratic equation is 23. Here are two statements about this quadratic equation: (1) the roots are real; (2) the roots are rational. Which of the following is true? A Neither statement is correct. B Only statement (1) is correct. C Only statement (2) is correct. D Both statements are correct. [X100/301] [Turn over y O 30° x 1 A 3 1 B 2 1 C 2 3 D 2
- 68. 10. Solve 2 cos x = for x, where 0 ≤ x < 2π. 11. Find Page eight[X100/301] π π π π π π π π 5 A and 3 3 2 B and 3 3 5 C and 6 6 11 D and 6 6 − − − − − − − + − + − + − + 1 42 1 22 3 42 3 22 A 2 3 1 B 2 2 8 C 3 3 8 1 D 3 2 x x c x x c x x c x x c 3 1 32 4 , where 0.x x dx x− ⎛ ⎞ + >⎜ ⎟ ⎝ ⎠ ⌠ ⎮⎮ ⌡
- 69. Page nine 12. The diagram shows two right-angled triangles with sides and angles as given. What is the value of sin( p + q )? 13. Given that f(x) = 4 sin 3x, find f′(0). A 0 B 1 C 12 D 36 [X100/301] [Turn over 2 3 2 p q 1 5 2 2 A 35 2 5 B 35 2 2 C 3 3 5 4 1 D 33 5 + + + +
- 70. Page ten 14. An equilateral triangle of side 3 units is shown. The vectors p and q are as represented in the diagram. What is the value of p.q? A 9 B C D 0 15. Given that the points S(–4, 5, 1), T(–16, –4, 16) and U(–24, –10, 26) are collinear, calculate the ratio in which T divides SU. A 2 : 3 B 3 : 2 C 2 : 5 D 3 : 5 16. Find [X100/301] p q 9 2 9 2 3 3 3 3 1 A 9 1 B 1 C 1 D 12 c x c x c x c x − + − + + + 4 1 , where 0. 3 dx x x ≠⌠ ⎮ ⌡
- 71. Page eleven[X100/301] 17. The diagram shows the graph of a cubic. What is the equation of this cubic? A y = –x(x + 1)(x – 2) B y = –x(x – 1)(x + 2) C y = x(x + 1)(x – 2) D y = x(x – 1)(x + 2) 18. If f(x) = (x – 3)(x + 5), for what values of x is the graph of y = f(x) above the x-axis? A –5 < x < 3 B –3 < x < 5 C x < –5, x > 3 D x < –3, x > 5 • y (1, 2) O–1 2 x [Turn over
- 72. Page twelve 19. Which of the following diagrams represents the graph with equation log3 y = x? [X100/301] y (1, 3) O x A y (1, 1) O x B y (1, 3) O 1 x C y (3, 1) O 1 x D
- 73. Page thirteen[X100/301] 20. On a suitable domain, D, a function g is defined by g(x) = sin2 Which of the following gives the real values of x in D and the corresponding values of g(x)? A x ≥ 0 and –1 ≤ g(x) ≤ 1 B x ≥ 0 and 0 ≤ g(x) ≤ 1 C x ≥ 2 and –1 ≤ g(x) ≤ 1 D x ≥ 2 and 0 ≤ g(x) ≤ 1 2.x − [END OF SECTION A] [Turn over for SECTION B
- 74. Page fourteen SECTION B ALL questions should be attempted. 21. A quadrilateral has vertices A(–1, 8), B(7, 12), C(8, 5) and D(2, –3) as shown in the diagram. (a) Find the equation of diagonal BD. (b) The equation of diagonal AC is x + 3y = 23. Find the coordinates of E, the point of intersection of the diagonals. (c) (i) Find the equation of the perpendicular bisector of AB. (ii) Show that this line passes through E. [X100/301] Marks 2 3 5 y A B E O D C x
- 75. [END OF SECTION B] [END OF QUESTION PAPER] Page fifteen 22. A function f is defined on the set of real numbers by f(x) = (x – 2)(x2 + 1). (a) Find where the graph of y = f(x) cuts: (i) the x-axis; (ii) the y-axis. (b) Find the coordinates of the stationary points on the curve with equation y = f(x) and determine their nature. (c) On separate diagrams sketch the graphs of: (i) y = f(x); (ii) y = –f(x). 23. (a) Solve cos 2x° – 3cosx° + 2 = 0 for 0 ≤ x < 360. (b) Hence solve cos 4x° – 3cos2x° + 2 = 0 for 0 ≤ x < 360. Marks 2 8 3 5 2 [X100/301]
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- 77. Read Carefully 1 Calculators may be used in this paper. 2 Full credit will be given only where the solution contains appropriate working. 3 Answers obtained by readings from scale drawings will not receive any credit. © LI X100/302 6/32010 *X100/302* MATHEMATICS HIGHER Paper 2 N A T I O N A L Q U A L I F I C A T I O N S 2 0 1 1 W E D N E S D A Y , 1 8 M A Y 1 0 . 5 0 A M – 1 2 . 0 0 N O O N X100/302
- 78. FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B sin 2A = 2sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2sin2 A Table of standard derivatives: Table of standard integrals: Page two 2 2 .g f c+ − sinax cosa ax cosax sina ax− ± ( )f x ( )f x′ [X100/302] sinax 1 cosax Ca− + cosax 1 sinax Ca + ( )f x ( )f x dx∫
- 79. ALL questions should be attempted. 1. D,OABC is a square based pyramid as shown in the diagram below. O is the origin, D is the point (2, 2, 6) and OA = 4 units. M is the mid-point of OA. (a) State the coordinates of B. (b) Express DB and DM in component form. (c) Find the size of angle BDM. 2. Functions f, g and h are defined on the set of real numbers by • f(x) = x3 – 1 • g(x) = 3x + 1 • h(x) = 4x – 5. (a) Find g(f(x)). (b) Show that g(f(x)) + xh(x) = 3x3 + 4x2 – 5x – 2. (c) (i) Show that (x – 1) is a factor of 3x3 + 4x2 – 5x – 2. (ii) Factorise 3x3 + 4x2 – 5x – 2 fully. (d) Hence solve g(f(x)) + xh(x) = 0. y M BC A D(2, 2, 6) Page three Marks 1 3 5 2 1 5 1 [X100/302] [Turn over z x O → →
- 80. 3. (a) A sequence is defined by with u0 = –16. Write down the values of u1 and u2. (b) A second sequence is given by 4, 5, 7, 11, . . . . It is generated by the recurrence relation vn + 1 = pvn + q with v1 = 4. Find the values of p and q. (c) Either the sequence in (a) or the sequence in (b) has a limit. (i) Calculate this limit. (ii) Why does the other sequence not have a limit? 4. The diagram shows the curve with equation y = x3 – x2 – 4x + 4 and the line with equation y = 2x + 4. The curve and the line intersect at the points (–2, 0), (0, 4) and (3, 10). Calculate the total shaded area. 1 1 2n nu u+ = − Marks 1 3 3 10 y y = 2x + 4 y = x3 – x2 – 4x + 4 –2 O 3 x Page four[X100/302]
- 81. Page five 5. Variables x and y are related by the equation y = kxn . The graph of log2 y against log2 x is a straight line through the points (0, 5) and (4, 7), as shown in the diagram. Find the values of k and n. 6. (a) The expression 3sinx – 5cosx can be written in the form R sin(x +a) where R > 0 and 0 ≤ a < 2π. Calculate the values of R and a. (b) Hence find the value of t, where 0 ≤ t ≤ 2, for which 7. Circle C1 has equation (x + 1)2 + (y – 1)2 = 121. A circle C2 with equation x2 + y2 – 4x + 6y + p = 0 is drawn inside C1. The circles have no points of contact. What is the range of values of p? Marks 5 4 7 9 [X100/302] log2 y O (0, 5) (4, 7) log2 x [END OF QUESTION PAPER] 0 (3cos 5sin ) 3. t x x dx+ =∫
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