C3 A Booster Course. Workbook. 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. (3) b) Hence, or otherwise, solve the equation
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1 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 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
3 BlueStar Mathematics Workshops (2011) 3
4 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 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
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6 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
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8 5. Figure 1 shows the graphs of y = x and y = x 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
9 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
10 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
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12 8. a) Solve the inequality 3x 4 < 7. b) Find, using algebra, the values of x for which x 2x = 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
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14 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
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16 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
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18 11. Find the solutions to the following equation to 3 decimal places. 2e x + 3e x = 7 (5) BlueStar Mathematics Workshops (2011) 18
19 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
20 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
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22 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
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24 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
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26 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
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28 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
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30 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
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32 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
33 BlueStar Mathematics Workshops (2011) 33
34 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
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36 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
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38 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
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40 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
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42 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
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44 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
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46 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
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48 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
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50 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
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52 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
53 BlueStar Mathematics Workshops (2011) 53
54 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
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56 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
57 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
58 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 ( ) dy dx = e x 2x 1 2x +1 3 ( ) 2 3 ex ( ( 2 x+1) + 5) 2 (4) BlueStar Mathematics Workshops (2011) 58
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60 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
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62 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
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64 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
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