Book 4 June 2013 June 2014 June 2015 Name :
June 2013 1. Given that 4 3 2 2 ax bx c 2 2 3x 2x 5x 4 dxe x 4 x 4, x 2 find the values of the constants a, b, c, d and e. 2. Given that f(x) = ln x, x > 0 sketch on separate axes the graphs of (i) y = f(x), (ii) y = f(x), (iii) y = f(x 4). Show, on each diagram, the point where the graph meets or crosses the x-axis. In each case, state the equation of the asymptote. 3. Given that (7) (a) Show, without using a calculator, that 2 cos (x + 50) = sin (x + 40). tan x = 1 3 tan 40. (b) Hence solve, for 0 θ < 360, 2 cos (2θ + 50) = sin (2θ + 40), giving your answers to 1 decimal place.
4. f(x) = 25x 2 e 2x 16, x R. (a) Using calculus, find the exact coordinates of the turning points on the curve with equation y = f(x). (5) (b) Show that the equation f(x) = 0 can be written as x = 4 e x. 5 (1) The equation f(x) = 0 has a root α, where α = 0.5 to 1 decimal place. (c) Starting with x0 = 0.5, use the iteration formula xn+1 = 4 x e n 5 to calculate the values of x1, x2 and x3, giving your answers to 3 decimal places. (d) Give an accurate estimate for α to 2 decimal places, and justify your answer. 5. Given that x = sec 2 3y, 0 < y < 6 (a) find d x dy in terms of y. (b) Hence show that dy 1 dx 6 xx ( 1) 1 2 2 d y (c) Find an expression for in terms of x. Give your answer in its simplest form. 2 dx 6. Find algebraically the exact solutions to the equations (a) ln (4 2x) + ln (9 3x) = 2 ln (x + 1), 1 < x < 2, (b) 2 x e 3x+1 = 10. (5) Give your answer to (b) in the form a lnb c ln d where a, b, c and d are integers. (5)
7. The function f has domain 2 x 6 and is linear from ( 2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1. Figure 1 (a) Write down the range of f. (b) Find ff(0). (1) The function g is defined by g : 4 3x x, x R, x 5. 5 x (c) Find g 1 (x). (d) Solve the equation gf(x) = 16. (5)
8. Figure 2 Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s 1. Kate is 24 m ahead of John when she starts to cross the road from the fixed point A. John passes her as she reaches the other side of the road at a variable point B, as shown in Figure 2. Kate s speed is V m s 1 and she moves in a straight line, which makes an angle θ, 0 < θ < 150, with the edge of the road, as shown in Figure 2. You may assume that V is given by the formula V 21, 0 < θ < 150 24sin 7cos (a) Express 24sin θ + 7cos θ in the form R cos (θ α), where R and α are constants and where R > 0 and 0 < α < 90, giving the value of α to 2 decimal places. Given that θ varies, (b) find the minimum value of V. Given that Kate s speed has the value found in part (b), (c) find the distance AB. Given instead that Kate s speed is 1.68 m s 1, (d) find the two possible values of the angle θ, given that 0 < θ < 150. (6)
June 2014 1. The curve C has equation y = f (x) where 4x 1 f( x) x 2, x 2 (a) Show that f( x) 9 x 2 2 Given that P is a point on C such that f ʹ(x) = 1, (b) find the coordinates of P. 2. Find the exact solutions, in their simplest form, to the equations (a) 2 ln (2x + 1) 10 = 0 (b) 3 x e 4x = e 7 3. The curve C has equation x = 8y tan 2y. The point P has coordinates, 8. (a) Verify that P lies on C. (b) Find the equation of the tangent to C at P in the form ay = x + b, where the constants a and b are to be found in terms of π. (7) (1)
4. Figure 1 Figure 1 shows part of the graph with equation y = f (x), x. The graph consists of two line segments that meet at the point Q(6, 1). The graph crosses the y-axis at the point P(0, 11). Sketch, on separate diagrams, the graphs of (a) y = f (x) (b) y = 2f ( x) + 3 On each diagram, show the coordinates of the points corresponding to P and Q. Given that f (x) = a x b 1, where a and b are constants, (c) state the value of a and the value of b.
5. x 32x1 g( x) 2 x 3 x x 6, x > 3 x 1 (a) Show that g( x), x > 3 x 2 (b) Find the range of g. (c) Find the exact value of a for which g(a) = g 1 (a). 6. Figure 2 Figure 2 shows a sketch of part of the curve with equation 1 2 3 y 2cos x x 3x2 2 The curve crosses the x-axis at the point Q and has a minimum turning point at R. (a) Show that the x coordinate of Q lies between 2.1 and 2.2. (b) Show that the x coordinate of R is a solution of the equation 2 1 2 x 1 xsin x 3 2 Using the iterative formula 2 1 2 xn 1 1 xnsin xn, x0 = 1.3 3 2 (c) find the values of x1 and x2 to 3 decimal places.
7. (a) Show that cosec 2x + cot 2x = cot x, x 90n, n (b) Hence, or otherwise, solve, for 0 θ < 180, (5) You must show your working. cosec (4θ + 10 ) + cot (4θ + 10 ) = 3 (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) 8. A rare species of primrose is being studied. The population, P, of primroses at time t years after the study started is modelled by the equation 0.1t 800e P, t 0, t 0.1t 1 3e (a) Calculate the number of primroses at the start of the study. (b) Find the exact value of t when P = 250, giving your answer in the form a ln(b) where a and b are integers. (c) Find the exact value of d P when t =10. Give your answer in its simplest form. dt (d) Explain why the population of primroses can never be 270. (1)
9. (a) Express 2 sin θ 4 cos θ in the form R sin(θ α), where R and α are constants, R > 0 and 0 < α < 2. Give the value of α to 3 decimal places. H(θ) = 4 + 5(2sin 3θ 4cos3θ) 2 Find (b) (i) the maximum value of H(θ), (ii) the smallest value of θ, for 0 θ π, at which this maximum value occurs. Find (c) (i) the minimum value of H(θ), (ii) the largest value of θ, for 0 θ π, at which this minimum value occurs.
June 2015 1. Given that tan = p, where p is a constant, p 1, use standard trigonometric identities, to find in terms of p, (a) tan 2, (b) cos, (c) cot ( 45). Write each answer in its simplest form. 2. Given that f(x) = 2e x 5, x R, (a) sketch, on separate diagrams, the curve with equation (i) y = f (x), (ii) y = f (x). On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote. (b) Deduce the set of values of x for which f (x) = f (x). (c) Find the exact solutions of the equation f (x) = 2. (6) (1)
3. g( ) = 4 cos 2 + 2 sin 2. Given that g( ) = R cos (2 ), where R > 0 and 0 < < 90, (a) find the exact value of R and the value of to 2 decimal places. (b) Hence solve, for 90 < < 90, 4 cos 2 + 2 sin 2 = 1, giving your answers to one decimal place. (5) Given that k is a constant and the equation g( ) = k has no solutions, (c) state the range of possible values of k. 4. Water is being heated in an electric kettle. The temperature, C, of the water t seconds after the kettle is switched on, is modelled by the equation = 120 100e t, 0 t T. (a) State the value of when t = 0. (1) Given that the temperature of the water in the kettle is 70 C when t = 40, (b) find the exact value of, giving your answer in the form ln a, where a and b are integers. b When t = T, the temperature of the water reaches 100 C and the kettle switches off. (c) Calculate the value of T to the nearest whole number.
5. The point P lies on the curve with equation Given that P has (x, y) coordinates x = (4y sin 2y) 2. p,, where p is a constant, 2 (a) find the exact value of p. (1) The tangent to the curve at P cuts the y-axis at the point A. (b) Use calculus to find the coordinates of A. (6)
6. Figure 1 Figure 1 is a sketch showing part of the curve with equation y = 2 x + 1 3 and part of the line with equation y = 17 x. The curve and the line intersect at the point A. (a) Show that the x-coordinate of A satisfies the equation ln(20 x) x = ln2 1. (b) Use the iterative formula xn + 1 = ln(20 xn ) ln2 1, x0 = 3, to calculate the values of x1, x2 and x3, giving your answers to 3 decimal places. (c) Use your answer to part (b) to deduce the coordinates of the point A, giving your answers to one decimal place.
7. Figure 2 Figure 2 shows a sketch of part of the curve with equation g(x) = x 2 (1 x)e 2x, x 0. (a) Show that g' (x) = f(x)e 2x, where f(x) is a cubic function to be found. (b) Hence find the range of g. (6) (c) State a reason why the function g 1 (x) does not exist. (1) 8. (a) Prove that sec 2A + tan 2A cos A sin A, A cos A sin A ( 2n 1), n Z. 4 (5) (b) Hence solve, for 0 < 2, sec 2 + tan 2 = 2 1. Give your answers to 3 decimal places.
9. Given that k is a negative constant and that the function f(x) is defined by ( x 5k)( x k) f (x) = 2, x 0, 2 2 x 3kx 2k (a) show that f (x) = x k. x 2k (b) Hence find f ' (x), giving your answer in its simplest form. (c) State, with a reason, whether f (x) is an increasing or a decreasing function. Justify your answer.