C3 A Booster Course Workbook 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. b) Hence, or otherwise, solve the equation x = 2x 3 (3) (4) BlueStar Mathematics Workshops (2011) 1
2. The function f is defined by f : x x + 2 x 1, x R a) Show that for all values of x, ff (x) = x. b) Hence, write down an expression for f -1 (x). (3) (1) The function g is defined by g : x 2x 3, x R c) Solve the equation gf (x) = 0. (4) BlueStar Mathematics Workshops (2011) 2
BlueStar Mathematics Workshops (2011) 3
3. ( ) The diagram shows the graph of y = f (x) which meets the x-axis at the point 9 4, 0 ( ). and the y-axis at the point 0, 3 a) Sketch on separate diagrams the graphs of i) y = f x ii) y = f 1 ( ) ( x) (4) Given that f (x) is of the form f ( x) ax 1 2 + b, x 0, b) Find the values of the constants a and b. c) Find an expression for f 1 ( x). (3) (3) BlueStar Mathematics Workshops (2011) 4
BlueStar Mathematics Workshops (2011) 5
4. The functions f and g are defined by where k is a constant. a) Find expressions in terms of k for f : x kx + 2, x R g : x x 3k, x R ( ) ( ) i) f 1 x ii) fg x Given that fg (7) = 4, b) Find the values of k. (4) (1) BlueStar Mathematics Workshops (2011) 6
BlueStar Mathematics Workshops (2011) 7
5. Figure 1 shows the graphs of y = x and y = x 2 +1. The point P is the minimum point of y = x 2 +1, and Q is the point of intersection of the two graphs. Figure 1 a) Find the coordinates of P. b) Show that the y coordinate of Q is 3 2. (1) (4) BlueStar Mathematics Workshops (2011) 8
6. The function f is defined as f : x x +1 x 1, x R By considering ff (x), show that the function f has the line of symmetry y = x. (5) BlueStar Mathematics Workshops (2011) 9
7. The functions f is defined by f : x 3( x +1) 2x 2 + 7x 4 1 x + 4, x R a) Show that f ( x) = b) Find f 1 ( x) 1 2x 1 c) Find the domain of f 1 ( x) Given that the function g is defined by (4) (3) (1) g : x ln( x +1) d) Find the solution of fg( x) = 1 7. (4) (Taken from Jan 2012 paper) BlueStar Mathematics Workshops (2011) 10
BlueStar Mathematics Workshops (2011) 11
8. a) Solve the inequality 3x 4 < 7. b) Find, using algebra, the values of x for which x 2x + 5 3 = 0 (3) (3) c) Sketch the graphs of y = x + 3 and y = x 5. Use algebra the coordinates of where these lines meet. (3) BlueStar Mathematics Workshops (2011) 12
BlueStar Mathematics Workshops (2011) 13
9. The functions f is defined by f : x x 1 3, x R a) Solve the equation f ( x) = 4. The function g is defined by (2) g : x x 2 4x +18, x 0. b) Find the range of g. c) Evaluate gf (-4). (3) (3) BlueStar Mathematics Workshops (2011) 14
BlueStar Mathematics Workshops (2011) 15
10. The functions f and g are defined by f : x cos x, x R g : x x + π 2, x 0 a) State the range of f (x). b) Find the domain of fg (x). c) Determine the range of fg (x). (2) (3) (2) BlueStar Mathematics Workshops (2011) 16
BlueStar Mathematics Workshops (2011) 17
11. Find the solutions to the following equation to 3 decimal places. 2e x + 3e x = 7 (5) BlueStar Mathematics Workshops (2011) 18
12. Solve the following simultaneous equations, giving your values to 4 significant figures. e y + 5 9x = 0 y ln( x + 4) = 2 (7) BlueStar Mathematics Workshops (2011) 19
13. At time t = 0, there are 800 bacteria present in a culture. The number of bacteria present at time t hours is modeled by the continuous variable N and the relationship where a and b are constants. N = ae bt a) State the value of a. Given that when t = 2, N = 7200, (1) b) Find the value of b in the form ln k. (3) c) Find, to the nearest minute, the time taken for the number of bacteria present to double. (4) BlueStar Mathematics Workshops (2011) 20
BlueStar Mathematics Workshops (2011) 21
13. A bead is projected vertically upwards in a jar of liquid with a velocity of 13 ms -1. Its velocity, v ms -1, at time t seconds after projection, is given by v = ce kt 2 a) Find the value of c. Given that the bead has a velocity of 7 ms -1 after 5.1 seconds, b) Find the value of k correct to 4 decimal places. c) Find the time taken for its velocity to decrease from 10 ms -1 to 4 ms -1. (1) (3) (4) BlueStar Mathematics Workshops (2011) 22
BlueStar Mathematics Workshops (2011) 23
14. f ( x) e 5 2 x x 5 Show that the equation f (x) = 0 a) has a root in the interval (1.4, 1.5), b) can be written as x = e 1 kx, stating the value of k. c) Using the iteration formula x n+1 = e 1 kx n, with x 0 = 1.5 and the value of k found in b), find x 1, x 2 and x 3. Give the value of x 3 correct to 3 decimal places. (4) (2) (2) BlueStar Mathematics Workshops (2011) 24
BlueStar Mathematics Workshops (2011) 25
15. The diagram shows part of the curve with the equation y = 3x + ln x x 2 and the line y = x. Given that the curve and the line intersect at the points A and B, show that a) The x coordinates of A and B are the solutions of the equation x = e x2 2 x b) The x coordinate of A lies in the interval (0.4, 0.5), c) The x coordinate of B lies in the interval (2.3, 2.4). d) Use the iteration formula x n+1 = e x n 2 2 x n, with x 0 = 0.5, to find the x coordinate of A correct to decimal places. (3) e) Justify your answer of part d). (2) (2) (1) (1) BlueStar Mathematics Workshops (2011) 26
BlueStar Mathematics Workshops (2011) 27
16. a) Prove that, for cos x 0, sin2x tan x tan x cos2x (5) b) Hence, or otherwise, solve the equation. sin2x tan x = 2cos2x, for x in the interval 0 x 180. (4) BlueStar Mathematics Workshops (2011) 28
BlueStar Mathematics Workshops (2011) 29
17. a) Use the identities of sin( A + B) and sin( A B) to prove that sin P sinq 2cos P + Q 2 sin P Q 2 (4) b) Hence, or otherwise, solve the equation. sin 4x = sin2x, for x in the interval 0 x 180. (6) BlueStar Mathematics Workshops (2011) 30
BlueStar Mathematics Workshops (2011) 31
18. a) Express 2cos x + 5sin x in the form Rcos( x α ) where R > 0 and 0 < α < 90 giving your values to 3 significant figures. (4) b) Hence, or otherwise, solve the equation. 2cos x + 5sin x = 3, for x in the interval 0 x 360, giving your answers to 1 decimal place. (4) BlueStar Mathematics Workshops (2011) 32
BlueStar Mathematics Workshops (2011) 33
19. a) Find the exact values of R and α, where R > 0 and 0 < α < π, for which 2 cos x sin x Rcos( x + α ). (4) b) Use the identity cos X + cosy = 2cos X + Y 2 cos X Y 2 or otherwise, find in terms of π, the values of x in the interval 0 < x < 2π, for which cos x + 2 cos 3x π 4 = sin x (8) BlueStar Mathematics Workshops (2011) 34
BlueStar Mathematics Workshops (2011) 35
20. a) Prove that for all values of x cos( x + 30) + sin x cos( x 30) (4) b) Hence, find the exact value of cos75 cos15, giving your answer in the form k 2. (3) c) Solve the equation 3cos( x + 30) + sin x = 3cos( x 30) +1, for x in the interval 180 x 180. (6) BlueStar Mathematics Workshops (2011) 36
BlueStar Mathematics Workshops (2011) 37
21. a) Express 4sin x cos x in the form Rsin( x α ), where R > 0 and 0 < α < 90. Give the values of R and α to 3 significant figures. (4) b) Show that the equation 2cosec x cot x + 4 = 0 can be written in the form 4sin x cos x + 2 = 0. (2) c) Hence, or otherwise, solve the equation 2cosec x cot x + 4 = 0 for the values of x in the interval 0 < x < 360. (4) BlueStar Mathematics Workshops (2011) 38
BlueStar Mathematics Workshops (2011) 39
22. a) Express 3cosθ + 4sinθ in the form Rcos( x α ), where R > 0 and 0 < α < π 2. (4) b) Given that the function f is defined by f ( θ) 1 3cos2θ 4sin2θ, 0 θ π, state the range of f ( θ) and solve the equation f ( θ) = 0. (6) c) Fine the coordinates of the turning points of the curve with the equation y = 2 3cos x + 4sin x for the values of x in the interval 0 < x < 2π. (3) BlueStar Mathematics Workshops (2011) 40
BlueStar Mathematics Workshops (2011) 41
23. a) Prove the identity 1 cos x 1+ cos x x tan2 2 π b) Use the above identity to find the value of tan 2 12 and b are integers. c) Hence, or otherwise, solve the equation 1 cos x 1+ cos x = 1 sec x 2, (4) in the form a + b 3, where a (3) for the values of x in the interval 0 < x < 2π, giving your values in terms of π. (5) BlueStar Mathematics Workshops (2011) 42
BlueStar Mathematics Workshops (2011) 43
24. a) Use the identities of cos( A + B) and cos( A B) to prove that sin Asin B 1 2 cos A B ( ) cos( A + B) (3) b) Hence, or otherwise, find the values of x in the interval 0 x π for which 4sin x + π 3 = cosec x π 6 giving your answers as exact multiplies of π. (7) BlueStar Mathematics Workshops (2011) 44
BlueStar Mathematics Workshops (2011) 45
25. a) For values of θ in the interval 0 θ 360, solve the equation. 2sin( θ + 30 ) = sin( θ 30 ) (6) BlueStar Mathematics Workshops (2011) 46
BlueStar Mathematics Workshops (2011) 47
26. a) Use the identity to prove cos( A + B) cos AcosB sin Asin B cos x 2cos 2 x 2 1 b) Solve the equation (3) sin x 1+ cos x = 3cot x 2, for the values of x in the interval 0 x 360. (7) BlueStar Mathematics Workshops (2011) 48
BlueStar Mathematics Workshops (2011) 49
27. a) Prove the identity cosec θ sinθ cosθ cotθ b) Find the values of x in the interval 0 x 2π for which (3) 2sec x + tan x = 2cos x, giving your answers in terms of π. (6) BlueStar Mathematics Workshops (2011) 50
BlueStar Mathematics Workshops (2011) 51
28. a) Use the identities of cos( A + B) and cos( A B) to prove that cosp + cosq = 2cos P + Q 2 cos P Q 2 (4) b) Hence, or otherwise, find the values of x in the interval 0 x 2π for which cos x + cos2x + cos3x = 0 (7) BlueStar Mathematics Workshops (2011) 52
BlueStar Mathematics Workshops (2011) 53
29. a) By writing 3θ = ( 2θ +θ), show that sin3θ = 3sinθ 4sin 3 θ. b) Hence, or otherwise, solve For 0 < θ < π. 28sin 3 θ 21sinθ + 5 = 0, (4) (5) BlueStar Mathematics Workshops (2011) 54
BlueStar Mathematics Workshops (2011) 55
30. A curve has the equation x = tan 2 y. a) Show that dy dx = 1 2 x x +1 ( ). b) Find the equation of the normal to the curve when y = π 4. (3) (5) BlueStar Mathematics Workshops (2011) 56
31. Differentiate the following with respect to x a) ( 4x 1) 5 b) e 3x c) Hence, or otherwise, find dy dx given that the curve y, (2) (1) y = e 3( 4 x 1)5 (3) BlueStar Mathematics Workshops (2011) 57
32. Differentiate the following with respect to x. 3 a) 3x + 5 b) e x 2x +1 (3) (4) c) Hence, or otherwise, show that the curve differentiates to e x y = 3 2x +1 + 5 1 3 ( ) dy dx = e x 2x 1 2x +1 3 ( ) 2 3 ex ( ( 2 x+1) + 5) 2 (4) BlueStar Mathematics Workshops (2011) 58
BlueStar Mathematics Workshops (2011) 59
33. The curve with the equation y = 1 2 x2 3ln x, x > 0, has a stationary point at A. a) Find the exact x coordinate of A. b) Determine the nature of this stationary point. (3) (2) c) Show that the y coordinate of A is 3 2 ( 1 ln3 ) (2) d) Find the equation of the tangent to the curve at the point where x = 1, giving your answer in the form ax + by = c, where a, b and c are integers. (3) BlueStar Mathematics Workshops (2011) 60
BlueStar Mathematics Workshops (2011) 61
34. a) Use the derivatives of sin x and cos x to prove that d dx cot x ( ) = cosec2 x (4) b) Show that the curve with the equation y = e x cot x has no turning points. (5) BlueStar Mathematics Workshops (2011) 62
BlueStar Mathematics Workshops (2011) 63
35. Given that show that f ( x) = e 2 x cos3x, f '( x) = Re 2 x cos( 3x + α ) where R and α are constants to be found. (5) END BlueStar Mathematics Workshops (2011) 64