C3 PAPER JUNE 2014 *P43164A0232* 1. The curve C has equation y = f (x) where + 1. (a) Show that 9 f (x) = (3)

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1 PMT C3 papers from 2014 and 2013 C3 PAPER JUNE The curve C has equation y = f (x) where 4x + 1 f( x) =, x 2 x > 2 (a) Show that 9 f (x) = ( x ) 2 2 Given that P is a point on C such that f (x) = 1, (b) find the coordinates of P. 2 *P43164A0232*

2 PMT 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 4 *P43164A0432*

3 PMT 3. The curve C has equation x = 8y tan 2y π The point P has coordinates π, 8 (a) Verify that P lies on C. (1) (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. (7) 6 *P43164A0632*

4 PMT 4. y P(0, 11) y = f(x) O Q(6, 1) x 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. 8 *P43164A0832*

5 PMT 5. g( x ) = x + x ( x + 1), x 3 2 x + x 6 x + 1 (a) Show that g( x) =, x 2 (b) Find the range of g. x 3 (c) Find the exact value of a for which g(a) = g 1 (a). 12 *P43164A01232*

6 PMT 6. y O Q x R Figure 2 Figure 2 shows a sketch of part of the curve with equation y = 2cos x x x 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 x = 1 + xsin x 3 2 Using the iterative formula xn + 1 = 1 + xnsin xn x, 0 = (c) find the values of x 1 and x 2 to 3 decimal places. 16 *P43164A01632*

7 PMT 7. (a) Show that cosec 2x + cot 2x = cot x, x 90n, n (5) (b) Hence, or otherwise, solve, for 0 < 180, cosec ( ) + cot ( ) = 3 You must show your working. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) 20 *P43164A02032*

8 PMT 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 01. t 800e P =, t 0, t t 1 3e R (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 dt when t =10. Give your answer in its simplest form. (d) Explain why the population of primroses can never be 270 (1) 24 *P43164A02432*

9 PMT 9. (a) Express 2 sin 4 cos in the form R sin( ), where R and are constants, R 0 and 0 < α < π 2 Give the value of to 3 decimal places. H() = 4 + 5(2sin 3 4cos3) 2 Find (b) (i) the maximum value of H(), (ii) the smallest value of, for 0, at which this maximum value occurs. Find (c) (i) the minimum value of H(), (ii) the largest value of, for 0, at which this minimum value occurs. 28 *P43164A02832*

10 C3 JUNE 2014(R) PAPER 1. Express x + 3 2x 3 4x 9 as a single fraction in its simplest form. 2 *P43163A0228*

11 2. A curve C has equation y = e 4x + x 4 + 8x + 5 (a) Show that the x coordinate of any turning point of C satisfies the equation x 3 = 2 e 4x (b) On the axes given on page 5, sketch, on a single diagram, the curves with equations (i) y = x 3, (ii) y = 2 e 4x On your diagram give the coordinates of the points where each curve crosses the y-axis and state the equation of any asymptotes. (c) Explain how your diagram illustrates that the equation x 3 = 2 e 4x has only one root. (1) The iteration formula x x n 3 n+ 1 = ( 2 e 4 ), x0 = 1 can be used to find an approximate value for this root. 1 (d) Calculate the values of x 1 and x 2, giving your answers to 5 decimal places. (e) Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve C. 4 *P43163A0428*

12 Question 2 continued y O x *P43163A0528* 5 Turn over

13 3. (i) (a) Show that 2 tan x cot x = 5 cosec x may be written in the form a cos 2 x + b cos x + c = 0 stating the values of the constants a, b and c. (b) Hence solve, for 0 x 2, the equation 2 tan x cot x = 5 cosec x giving your answers to 3 significant figures. (ii) Show that nπ tan + cot cosec 2, θ, n Z 2 stating the value of the constant. 8 *P43163A0828*

14 4. (i) Given that x = sec 2 2y, 0 < y < 4 π show that (ii) Given that dy 1 = dx 4x ( x 1) y = (x 2 + x 3 )ln2x find the exact value of d y d at e x =, giving your answer in its simplest form. x 2 (5) (iii) Given that f( x) = 3cos x ( x + 1) 1 3, x 1 show that f '( x) = g( x) ( x + 1) 4 3, x 1 where g(x) is an expression to be found. 12 *P43163A01228*

15 5. (a) Sketch the graph with equation y = 4x 3 stating the coordinates of any points where the graph cuts or meets the axes. Find the complete set of values of x for which (b) (c) 4x 3 2 2x 4x x 16 *P43163A01628*

16 6. The function f is defined by f : x e 2x + k 2, x, k is a positive constant. (a) State the range of f. (b) Find f 1 and state its domain. (1) The function g is defined by g : x ln(2x), x 0 (c) Solve the equation g(x) + g(x 2 ) + g(x 3 ) = 6 giving your answer in its simplest form. (d) Find fg(x), giving your answer in its simplest form. (e) Find, in terms of the constant k, the solution of the equation fg(x) = 2k 2 20 *P43163A02028*

17 7. y C O 2 x Figure 1 Figure 1 shows the curve C, with equation y = 6 cos x sin x for 0 x 2 (a) Express 6 cos x sin x in the form R cos(x ), where R and are constants with R 0 and 0 2 π. Give your value of to 3 decimal places. (b) Find the coordinates of the points on the graph where the curve C crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by H 2πt 2πt = cos + 2.5sin, 0 t where H is the number of hours of daylight and t is the number of weeks since her first recording. Use this function to find (c) the maximum and minimum values of H predicted by the model, (d) the values for t when H = 16, giving your answers to the nearest whole number. [You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.] (6) 24 *P43163A02428*

18 physicsandmathstutor.com June 2013 C3 JUNE 2013 PAPER 1. Given that x 2x 5x 4 2 dx + e ax + bx + c +, 2 2 x 4 x 4 x ± 2 find the values of the constants a, b, c, d and e. 2 *P43016A0232*

19 2. Given that physicsandmathstutor.com June 2013 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. (7) 4 *P43016A0432*

20 physicsandmathstutor.com June Given that (a) Show, without using a calculator, that 2cos(x + 50) = sin(x + 40) tan x = 1 3 tan (b) Hence solve, for 0 < 360, 2cos(2 + 50) = sin(2 + 40) giving your answers to 1 decimal place. 8 *P43016A0832*

21 physicsandmathstutor.com June f(x) = 25x 2 e 2x 16, x (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 5 e x (1) The equation f(x) = 0 has a root, where = 0.5 to 1 decimal place. (c) Starting with x 0 = 0.5, use the iteration formula x n+1 = 4 5 e x n to calculate the values of x 1, x 2 and x 3, giving your answers to 3 decimal places. (d) Give an accurate estimate for to 2 decimal places, and justify your answer. 12 *P43016A01232*

22 physicsandmathstutor.com June Given that x = sec 2 3y, 0 < y < 6 π (a) find d x dy in terms of y. (b) Hence show that dy dx = 1 6xx ( ) (c) Find an expression for d 2 y 2 dx in terms of x. Give your answer in its simplest form. 16 *P43016A01632*

23 physicsandmathstutor.com June Find algebraically the exact solutions to the equations (a) ln(4 2x) + ln(9 3x) = 2ln(x + 1), 1 < x < 2 (5) (b) 2 x e 3x+1 = 10 Give your answer to (b) in the form a c + + ln b where a, b, c and d are integers. ln d (5) 20 *P43016A02032*

24 physicsandmathstutor.com June 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. y 10 2 O 2 6 x Figure 1 (a) Write down the range of f. (b) Find ff(0). (1) The function g is defined by g : x x, 5 x x, x (c) Find g 1 (x) (d) Solve the equation gf(x) = 16 (5) 24 *P43016A02432*

25 physicsandmathstutor.com June B 3ms 1 V ms 1 7 m A 24 m 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 ms 1 and she moves in a straight line, which makes an angle, 0 < < 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 < < sin θ + 7cosθ (a) Express 24sin + 7cos in the form Rcos( ), where R and are constants and where R > 0 and 0 < < 90, giving the value of to 2 decimal places. Given that 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, given that 0 < < 150. (6) 28 *P43016A02832*

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34 PhysicsAndMathsTutor.com C3 JUNE 2013 (W/D) PAPER 6x g( x) = 2, x 0 2 x + 3x + 2 June 2013 (Withdrawn) 4 2x (a) Show that g( x) =, x + 1 (b) y (0, 4) x 0 O (2, 0) x Figure 1 Figure 1 shows a sketch of the curve with equation y = g(x), x 0 The curve meets the y-axis at (0, 4) and crosses the x-axis at (2, 0). On separate diagrams sketch the graph with equation (i) y = 2g(2x), (ii) y = g 1 (x). Show on each sketch the coordinates of each point at which the graph meets or crosses the axes. (5) 2 *P41826A0232*

35 PhysicsAndMathsTutor.com 2. Given that tan 40 = p, find in terms of p (a) cot 40 (b) sec 40 June 2013 (Withdrawn) (1) (c) tan 85 6 *P41826A0632*

36 PhysicsAndMathsTutor.com June 2013 (Withdrawn) 3. y y = 2 x 5 O P x Q Figure 2 Figure 2 shows a sketch of the graph with equation y = 2 x 5. The graph intersects the positive x-axis at the point P and the negative y-axis at the point Q. (a) State the coordinates of P and the coordinates of Q. (b) Solve the equation 2 x 5 = 3 x 8 *P41826A0832*

37 PhysicsAndMathsTutor.com June 2013 (Withdrawn) 4. (a) On the same diagram, sketch and clearly label the graphs with equations y = e x and y = 10 x Show on your sketch the coordinates of each point at which the graphs cut the axes. (b) Explain why the equation e x 10 + x = 0 has only one solution. (1) (c) Show that the solution of the equation e x 10 + x = 0 lies between x = 2 and x = 3 (d) Use the iterative formula x n + 1 = ln(10 x n ), x 1 = 2 to calculate the values of x 2, x 3 and x 4. Give your answers to 4 decimal places. 10 *P41826A01032*

38 PhysicsAndMathsTutor.com June 2013 (Withdrawn) 5. (i) (a) Show that d dx x 1 2 x ln x ln = 2 x + 1 x 1 2 The curve with equation y = x ln x, x > 0 has one turning point at the point P. (b) Find the exact coordinates of P. Give your answer in its simplest form. (ii) A curve C has equation y x = x Find d y, and show that C has no turning points. dx + k, where k is a positive constant. k 14 *P41826A01432*

39 PhysicsAndMathsTutor.com June 2013 (Withdrawn) 6. y O 4 x Figure 3 Figure 3 shows a sketch of the graph of y = f(x) where f ( x) = e 5 2x, x 4 2x 8 4, x > 4 (a) State the range of f(x). (1) (b) Determine the exact value of ff (0). (c) Solve f(x) = 21 Give each answer as an exact answer. (5) (d) Explain why the function f does not have an inverse. (1) 18 *P41826A01832*

40 PhysicsAndMathsTutor.com June 2013 (Withdrawn) 7. (a) Prove that cosx 1 sin x π + = 2sec x, x (2n + 1), n 1 sin x cosx 2 π (b) Hence find, for 0 < x <, the exact solution of 4 cos x 1 sin x 1 sin x + = 8sin x cos x 22 *P41826A02232*

41 PhysicsAndMathsTutor.com June 2013 (Withdrawn) π 8. (a) Express 9cos θ 2sin θ in the form Rcos(θ + α), where R > 0 and 0 < α <. 2 Give the exact value of R and give the value of α to 4 decimal places. (b) (i) State the maximum value of 9cos θ 2sin θ (ii) Find the value of θ, for 0<θ <2π, at which this maximum occurs. Ruth models the height H above the ground of a passenger on a Ferris wheel by the equation πt πt H = 10 9cos + 2sin 5 5 where H is measured in metres and t is the time in minutes after the wheel starts turning. H (c) Calculate the maximum value of H predicted by this model, and the value of t, when this maximum first occurs. Give your answers to 2 decimal places. (d) Determine the time for the Ferris wheel to complete two revolutions. 26 *P41826A02632*

42 PhysicsAndMathsTutor.com June 2013 (Withdrawn) 9. y O A x Figure 4 Figure 4 shows a sketch of the curve with equation x = (9 + 16y 2y 2 ) The curve crosses the x-axis at the point A. (a) State the coordinates of A. (b) Find an expression for d x dy, in terms of y. (c) Find an equation of the tangent to the curve at A (1) 30 *P41826A03032*

43 C3 JANUARY 2013 PAPER 1. The curve C has equation y = ( 2x 3) 5 The point P lies on C and has coordinates (w, 32). Find (a) the value of w, (b) the equation of the tangent to C at the point P in the form y = mx+ c, where m and c are constants. (5) 2 *P41486A0228*

44 x 1 2. g( x) = e + x 6 (a) Show that the equation g( x ) = 0 can be written as x= ln( 6 x) + 1, x <6 The root of g( x ) = 0 is. The iterative formula x = n+ x + 1 ln( 6 n) 1, x 0 = 2 is used to find an approximate value for. (b) Calculate the values of x 1, x 2 and x 3 to 4 decimal places. (c) By choosing a suitable interval, show that = correct to 3 decimal places. 4 *P41486A0428*

45 3. y y = f(x) Q(0,2) P( 3,0) O x Figure 1 Figure 1 shows part of the curve with equation y = f( x), x. The curve passes through the points Q( 02, ) and P( 3, 0 ) as shown. (a) Find the value of ff ( 3 ). On separate diagrams, sketch the curve with equation (b) y (c) y = f 1 ( x), = f( x) 2, 1 (d) y = 2f ( x 2 ). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes. 6 *P41486A0628*

46 4. (a) Express 6 cos+8sin in the form Rcos(), where R > 0 and 0 < α < π. 2 Give the value of to 3 decimal places. (b) p( ) = 4, 0θ 2π cos+ 8sin Calculate (i) the maximum value of p(), (ii) the value of at which the maximum occurs. 10 *P41486A01028*

47 5. (i) Differentiate with respect to x (a) y = x 3 ln 2x (b) y = ( x+ sin x) 2 3 (6) Given that x= cot y, (ii) show that d y = 1 dx 1+ x 2 (5) 14 *P41486A01428*

48 6. (i) Without using a calculator, find the exact value of (sin cos 22.5 ) 2 You must show each stage of your working. (5) (ii) (a) Show that cos 2 + sin = 1 may be written in the form k sin 2 sin = 0, stating the value of k. (b) Hence solve, for 0 < 360, the equation cos 2 + sin = 1 18 *P41486A01828*

49 7. h( x) = x + x + 18, x ( x + 5)( x+ 2) (a) Show that h( x) = 2x 2 x + 5 (b) Hence, or otherwise, find h( x ) in its simplest form. y y = h(x) O Figure 2 x Figure 2 shows a graph of the curve with equation y (c) Calculate the range of h( x ). = h( x). (5) 22 *P41486A02228*

50 8. The value of Bob s car can be calculated from the formula V t t = 025. e + e where V is the value of the car in pounds ( ) and t is the age in years. (a) Find the value of the car when t = 0 (b) Calculate the exact value of t when V = 9500 (1) (c) Find the rate at which the value of the car is decreasing at the instant when t = 8. Give your answer in pounds per year to the nearest pound. 26 *P41486A02628*