NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system.

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1 NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system. Duplicating, selling, or otherwise distributing this product is a violation of the law ; your license of the product will be terminated at any moment if you are selling or distributing the products. No parts of this book may be reproduced, stored in a retrieval system, of transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

2 mock papers 1 1. Given that 4 2 2x 3x + x+ 1 2 dx + e ( ax + bx + c) ( x 1) ( x 1), find the values of the constants a, b, c, d and e. (4)

3 Question 1 continued Q1 (Total 4 marks) Turn over

4 2. A curve C has equation x y = e 2 tan x, x ( 2n+ 1) π. 2 (a) Show that the turning points on C occur where tan x = 1. (b) Find an equation of the tangent to C at the point where x = 0. (6) (2)

5 Question 2 continued Q2 (Total 8 marks) Turn over

6 3. f( x) = ln( x+ 2) x+ 1, x> 2, x. (a) Show that there is a root of f( x ) = 0 in the interval 2< x < 3. (2) (b) Use the iterative formula to calculate the values of x1, x2 and x 3 giving your answers to 5 decimal places. (c) Show that x = is a root of f( x ) = 0 correct to 3 decimal places. (2)

7 Question 3 continued Q3 (Total 7 marks)

8 4. y A (5, 4) O x B ( 5, 4) Figure 1 Figure 1 shows a sketch of the curve with equation y = f( x). The curve passes through the origin O and the points A(5, 4) and B( 5, 4). In separate diagrams, sketch the graph with equation (a) y = f( x), (b) y = f( x), (c) y = 2f ( x+ 1). (4) On each sketch, show the coordinates of the points corresponding to A and B.

9 Question 4 continued Turn over

10 Question 4 continued

11 Question 4 continued Q4 (Total 10 marks) Turn over

12 5. The radioactive decay of a substance is given by ct R = 1000e, t 0. where R is the number of atoms at time t years and c is a positive constant. (a) Find the number of atoms when the substance started to decay. (1) It takes 5730 years for half of the substance to decay. (b) Find the value of c to 3 significant figures. (c) Calculate the number of atoms that will be left when t = (d) In the space provided on page 13, sketch the graph of R against t. (4) (2) (2)

13 Question 5 continued Q5 (Total 9 marks) Turn over

14 6. (a) Use the double angle formulae and the identity cos( A+ B) cos Acos B sin Asin B to obtain an expression for cos 3x in terms of powers of cos x only. (b) (i) Prove that (ii) Hence find, for 0 cos x 1 sin x + + π 2sec x, x ( 2n+ 1). 1 + sin x cos x 2 < x <2π, all the solutions of cos x 1 sin x + + = sin x cos x (4)

15 Question 6 continued Turn over

16 Question 6 continued

17 Question 6 continued Q6 (Total 11 marks)

18 7. A curve C has equation y = 3sin 2x+ 4cos 2x, -π x π. The point A(0, 4) lies on C. (a) Find an equation of the normal to the curve C at A. (b) Express y in the form Rsin( 2 x+α ), where R > 0 and 0 < α < π. 2 Give the value of α to 3 significant figures. (c) Find the coordinates of the points of intersection of the curve C with the x-axis. Give your answers to 2 decimal places. (5) (4) (4)

19 Question 7 continued Turn over

20 Question 7 continued

21 Question 7 continued Q7 (Total 13 marks) Turn over

22 8. The functions f and g are defined by 3 f: x 1 2x, x (a) Find the inverse function g: x -1 f. 3 4, x> 0, x x (2) (b) Show that the composite function gf is x gf : x x (4) (c) Solve gf ( x ) = 0. (2) (d) Use calculus to find the coordinates of the stationary point on the graph of y = gf(x). (5)

23 Question 8 continued Turn over

24 Question 8 continued Q8 END (Total 13 marks) TOTAL FOR PAPER: 75 MARKS

25 mock papers 2 1. The point P lies on the curve with equation The y-coordinate of P is 8. y = e x (a) Find, in terms of ln 2, the x-coordinate of P. (2) (b) Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are exact constants to be found. (4)

26 2. f ( x) = 5cos x+ 12sin x Given that f ( x) = Rcos( x α), where R > 0 and 0 < α < π 2, (a) find the value of R and the value of α to 3 decimal places. (4) (b) Hence solve the equation for 0 x < 2π. 5cos x+ 12sin x= 6 (5) (c) (i) Write down the maximum value of 5cos x+ 12sin x. (1) (ii) Find the smallest positive value of x for which this maximum value occurs. (2)

27 3. y P Q 3 R x Figure 1 Figure 1 shows the graph of y = f( x), x. The graph consists of two line segments that meet at the point P. The graph cuts the y-axis at the point Q and the x-axis at the points ( 3, 0) and R. Sketch, on separate diagrams, the graphs of (a) y = f( x), (2) (b) y = f ( x). (2) Given that f( x) = 2 x+ 1, (c) find the coordinates of the points P, Q and R, (d) solve f( x) = 1 x. 2 (5)

28 4. The function f is defined by ( x ) f: x 2 1 1, x 3. 2 x 2x 3 > x 3 (a) Show that 1 f( x) =, x > 3. x + 1 (4) (b) Find the range of f. (c) Find f 1 (x). State the domain of this inverse function. (2) The function g is defined by g: x 2x 2 3, x. (d) Solve fg( x ) = 1. 8

29 5. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + cot 2 θ cosec 2 θ. (b) Solve, for 0 θ < 180, the equation 2 cot 2 θ 9 cosec θ = 3, giving your answers to 1 decimal place. (2) (6)

30 6. (a) Differentiate with respect to x, (i) e 3 x (sin x+ 2cos x), (ii) Given that 2 3x + 6x 7 y =, x 1, 2 ( x + 1) (b) show that (c) Hence find dy 20 = dx ( x+ 1). 3 d d 2 y x 2 and the real values of x for which d 2 y 15 =. dx 2 4 (5)

31 7. 3 f( x) = 3x 2x 6 (a) Show that f (x) = 0 has a root, α, between x = 1.4 and x = 1.45 (2) (b) Show that the equation f (x) = 0 can be written as 2 2 x = ( + x 0 x 3),. (c) Starting with x 0 =1.43, use the iteration x n ( xn 3) = + to calculate the values of x 1, x 2 and x 3, giving your answers to 4 decimal places. (d) By choosing a suitable interval, show that α = is correct to 3 decimal places.

32 Question 7 continued Q7 END (Total 11 marks) TOTAL FOR PAPER: 75 MARKS

33 mock papers 3 dy 1. (a) Find the value of at the point where x = 2 on the curve with equation dx y = x 2 (5x 1). sin 2x (b) Differentiate with respect to x. 2 x (6) (4)

34 2. f( x) = 2x + 2 x x x x 3 (a) Express f (x) as a single fraction in its simplest form. (4) (b) Hence show that 2 f() x = ( x ) 3 2

35 3. y 5 T (3, 5) 2 S (7, 2) O 3 7 x Figure 1 Figure 1 shows the graph of y = f (x), 1 < x < 9. The points T(3, 5) and S(7, 2) are turning points on the graph. Sketch, on separate diagrams, the graphs of (a) y = 2f (x) 4, (b) y = f( x). Indicate on each diagram the coordinates of any turning points on your sketch.

36 π 4. Find the equation of the tangent to the curve x= cos( 2y+ π ) at 0,. 4 Give your answer in the form y = ax + b, where a and b are constants to be found. (6)

37 5. The functions f and g are defined by (a) Write down the range of g. (b) Show that the composite function fg is defined by fg : x x 2 + 3e x 2, x R. (c) Write down the range of fg. d (d) Solve the equation fg x x xe x 2 ( ) ( + ). dx = 2 (1) (2) (1) (6)

38 6. (a) (i) By writing 3θ = (2θ + θ), show that (ii) Hence, or otherwise, for Give your answers in terms of π. sin 3θ = 3 sin θ 4 sin 3 θ. π 0 < θ < 3, solve 8 sin 3 θ 6 sin θ + 1 = 0. (4) (5) (b) Using sin( θ α) = sin θcosα cosθsin α, or otherwise, show that 1 sin15 = ( 6 2). 4 (4)

39 7. f( x) = 3xe x 1 The curve with equation y = f (x) has a turning point P. (a) Find the exact coordinates of P. (5) The equation f (x) = 0 has a root between x = 0.25 and x = 0.3 (b) Use the iterative formula x n = 3 e with x 0 = 0.25 to find, to 4 decimal places, the values of x 1, x 2 and x 3. x n (c) By choosing a suitable interval, show that a root of f (x) = 0 is x = correct to 4 decimal places.

40 8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ α), where R and α are constants, R > 0 and 0 < α < 90. (4) (b) Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs. The temperature, f (t), of a warehouse is modelled using the equation f (t) = cos(15t) + 4 sin(15t), where t is the time in hours from midday and 0 t < 24. (c) Calculate the minimum temperature of the warehouse as given by this model. (d) Find the value of t when this minimum temperature occurs. (2)

41 Question 8 continued Q8 END (Total 12 marks) TOTAL FOR PAPER: 75 MARKS

42 mock papers 4 1. y A 2 1 O x Figure 1 Figure 1 shows part of the curve with equation x-axis at the point A where x = α. y x x 3 2 = , which intersects the To find an approximation to α, the iterative formula is used. 2 x + = + 2 n 1 2 ( xn ) (a) Taking x 0 = 2.5, find the values of x 1, x 2, x 3 and x 4. Give your answers to 3 decimal places where appropriate. (b) Show that α = correct to 3 decimal places.

43 2. (a) Use the identity cos 2 θ + sin 2 θ = 1 to prove that tan 2 θ = sec 2 θ 1. (2) (b) Solve, for 0 θ < 360, the equation 2 tan 2 θ + 4 sec θ + sec 2 θ = 2 (6)

44 3. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation 5 P = 80e 1 t, t, t 0 (a) Write down the number of rabbits that were introduced to the island. (1) (b) Find the number of years it would take for the number of rabbits to first exceed (2) (c) Find d P. dt (2) (d) Find P when d P = 50. dt

45 4. (i) Differentiate with respect to x (a) (b) x 2 cos3x 2 ln( x + 1) x (4) (ii) A curve C has the equation y= ( 4x+ 1 ), x >, y > 0 The point P on the curve has x-coordinate 2. Find an equation of the tangent to C at P in the form ax + by + c = 0, where a, b and c are integers. (6) 1 4

46 5. y O B x A Figure 2 Figure 2 shows a sketch of part of the curve with equation y = f(x), x. 1 The curve meets the coordinate axes at the points A(0,1 k) and B ( ln k,0 ), where k is a constant and k > 1, as shown in Figure 2. On separate diagrams, sketch the curve with equation 2 (a) y = f( x), (b) y = 1 f ( x). (2) Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes. Given that 2x f( x) = e k, (c) state the range of f, (1) (d) find 1 f ( x), (e) write down the domain of 1 f. (1)

47 6. (a) Use the identity cos( A+ B) = cos Acos B sin Asin B, to show that 2 cos 2A = 1 2sin A (2) The curves C 1 and C 2 have equations C 1 : y = 3sin2x C 2 : y 2 = 4sin x 2 cos 2x (b) Show that the x-coordinates of the points where C 1 and C 2 intersect satisfy the equation 4cos 2x + 3sin 2x = 2 (c) Express 4cos2x + 3sin2x in the form R cos (2x α), where R > 0 and 0 < α < 90, giving the value of α to 2 decimal places. (d) Hence find, for 0 x < 180, all the solutions of 4cos 2x + 3sin 2x = 2 giving your answers to 1 decimal place. (4)

48 7. The function f is defined by 2 x 8 f( x) = 1 +, ( x + 4) ( x 2)( x + 4) x x 4, x 2 x 3 (a) Show that f( x) = x 2 (5) The function g is defined by x e 3 g( x) =, x e 2 x, x ln 2 x e (b) Differentiate g( x) to show that g( x) = x 2 (e 2) (c) Find the exact values of x for which g( x) = 1 (4)

49 8. (a) Write down sin 2x in terms of sin x and cos x. (1) (b) Find, for 0 < x < π, all the solutions of the equation cosec x 8cos x = 0 giving your answers to 2 decimal places. (5)

50 Question 8 continued Q8 END (Total 6 marks) TOTAL FOR PAPER: 75 MARKS

C3 papers June 2007 to 2008

C3 papers June 2007 to 2008 physicsandmathstutor.com June 007 C3 papers June 007 to 008 1. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04* physicsandmathstutor.com June 007 x + 3 9+