1. 3 2 Consider the function f(x) = x - 3x - 9x + 10, x (a) . Find the equation of the straight line passing through the maximum and minimum points of the graph y = f(x). (4) (b) Show that the point of inflexion of the graph y = f(x) lies on this straight line. (2) (Total 6 marks) 2. A jet plane travels horizontally along a straight path for one minute, starting at time t = 0, where -2 t is measured in seconds. The acceleration, a, measured in m s , of the jet plane is given by the straight line graph below. (a) Find an expression for the acceleration of the jet plane during this time, in terms of t. (1) (b) -1 Given that when t = 0 the jet plane is travelling at 125 m s , find its maximum velocity in -1 m s during the minute that follows. (4) IB Questionbank Mathematics Higher Level 3rd edition 1 (c) -1 Given that the jet plane breaks the sound barrier at 295 m s , find out for how long the jet plane is travelling greater than this speed. (3) (Total 8 marks) 3. 2 An open glass is created by rotating the curve y = x , defined in the domain x [0, 10], 2 radians about the y-axis. Units on the coordinate axes are defined to be in centimetres. (a) When the glass contains water to a height h cm, find the volume V of water in terms of h. (3) (b) 3 2 If the water in the glass evaporates at the rate of 3 cm per hour for each cm of exposed surface area of the water, show that, dV 3 2V dt , where t is measured in hours. (6) (c) If the glass is filled completely, how long will it take for all the water to evaporate? (7) (Total 16 marks) 4. A skydiver jumps from a stationary balloon at a height of 2000 m above the ground. -1 -0.2t Her velocity, v m s , t seconds after jumping, is given by v = 50(1 - e ). (a) Find her acceleration 10 seconds after jumping. (3) (b) How far above the ground is she 10 seconds after jumping? (3) (Total 6 marks) IB Questionbank Mathematics Higher Level 3rd edition 2 5. -1 A rocket is rising vertically at a speed of 300 m s when it is 800 m directly above the launch site. Calculate the rate of change of the distance between the rocket and an observer, who is 600 m from the launch site and on the same horizontal level as the launch site. diagram not to scale (Total 6 marks) 6. 1 x2 1 y2 1 a 2 The point P, with coordinates (p, q), lies on the graph of , a > 0. The tangent to the curve at P cuts the axes at (0, m) and (n, 0). Show that m + n = a. (Total 8 marks) 7. (a) e Using integration by parts, show that 2x 1 sin xdx e 2 x ( 2 sin x cos x) C 5 . (6) (b) dy 1 y 2 e2x d x Solve the differential equation sin x, given that y = 0 when x = 0, writing your answer in the form y = f(x). (5) IB Questionbank Mathematics Higher Level 3rd edition 3 (c) (i) Sketch the graph of y = f(x), found in part (b), for 0 x 1.5. Determine the coordinates of the point P, the first positive intercept on the x-axis, and mark it on your sketch. (ii) The region bounded by the graph of y = f(x) and the x-axis, between the origin and P, is rotated 360 about the x-axis to form a solid of revolution. Calculate the volume of this solid. (6) (Total 17 marks) 8. 3 x 3 The diagram below shows the graphs of y = 2 , y = 3 and a quadratic function, that all intersect in the same two points. Given that the minimum value of the quadratic function is -3, find an expression for the area of t (ax the shaded region in the form 2 bx c) dx, where the constants a, b, c and t are to be determined. (Note: The integral does not need to be evaluated.) 0 (Total 8 marks) IB Questionbank Mathematics Higher Level 3rd edition 4 9. Points A, B and C are on the circumference of a circle, centre O and radius r. A trapezium OABC is formed such that AB is parallel to OC, and the angle AOC is , 2 < . diagram not to scale (a) Show that angle BOC is - . (3) (b) Show that the area, T, of the trapezium can be expressed as 1 2 1 r sin r 2 sin 2 2 T= 2 . (3) IB Questionbank Mathematics Higher Level 3rd edition 5 (c) (i) Show that when the area is maximum, the value of satisfies cos = 2 cos 2. (ii) Hence determine the maximum area of the trapezium when r = 1. (Note: It is not required to prove that it is a maximum.) (5) (Total 11 marks) 10. A body is moving through a liquid so that its acceleration can be expressed as v2 200 32 m s-2, -1 where v m s is the velocity of the body at time t seconds. -1 The initial velocity of the body was known to be 40 m s . (a) -1 Show that the time taken, T seconds, for the body to slow to V m s is given by T= 200 1 40 v V 2 80 2 dv. (4) (b) v dv ds , where s is displacement, in (i) Explain why acceleration can be expressed as metres, of the body at time t seconds. (ii) Hence find a similar integral to that shown in part (a) for the distance, S metres, -1 travelled as the body slows to V m s . (7) (c) Hence, using parts (a) and (b), find the distance travelled and the time taken until the body momentarily comes to rest. (3) (Total 14 marks) 11. A lighthouse L is located offshore, 500 metres from the nearest point P on a long straight shoreline. The narrow beam of light from the lighthouse rotates at a constant rate of 8 radians IB Questionbank Mathematics Higher Level 3rd edition 6 per minute, producing an illuminated spot S that moves along the shoreline. You may assume that the height of the lighthouse can be ignored and that the beam of light lies in the horizontal plane defined by sea level. When S is 2000 metres from P, (a) show that the speed of S, correct to three significant figures, is 214 000 metres per minute; (5) (b) find the acceleration of S. (3) (Total 8 marks) 12. The function f is defined by 3 2 f(x) = ( x 6 x 3x where D (a) 1 10) 2 , for x D, is the greatest possible domain of f. Find the roots of f(x) = 0. (2) (b) Hence specify the set D. (2) IB Questionbank Mathematics Higher Level 3rd edition 7 (c) Find the coordinates of the local maximum on the graph y = f(x). (2) (d) Solve the equation f(x) = 3. (2) (e) Sketch the graph of y= f(x), for x D. (3) (f) Find the area of the region completely enclosed by the graph of y = f(x). (3) (Total 14 marks) 13. The functions f, g and h are defined by x f(x) = 1 + e , for x , 1 g(x) = x , for x / {0}, 2n 1 , n h(x) = sec x, for x / 2 (a) . Determine the range of the composite function g f. (3) (b) Determine the inverse of the function g f, clearly stating the domain. (4) (c) (i) Show that the function y = (f g h)(x) satisfies the differential equation dy dx = (1 - y) sin x. y sin xdx , as a function of x. (ii) Hence, or otherwise, find (iii) You are given that the domain of y = (f g h)(x) can be extended to the whole IB Questionbank Mathematics Higher Level 3rd edition 8 real axis. That part of the graph of y = (f g h)(x), between its maximum at x = 0 and its first minimum for positive x, is rotated by 2 about the y-axis. Calculate the volume of the solid generated. (14) (Total 21 marks) 14. 3 3 Find the equation of the normal to the curve x y - xy = 0 at the point (1, 1). (Total 7 marks) 15. The line y = m(x - m) is a tangent to the curve (1 - x)y = 1. Determine m and the coordinates of the point where the tangent meets the curve. (Total 7 marks) a be x 16. x Let f(x) = ae b , where 0 < b < a. (b 2 a 2 ) e x (a) x 2 Show that f(x) = (ae b) . (3) (b) Hence justify that the graph of f has no local maxima or minima. (2) (c) Given that the graph of f has a point of inflexion, find its coordinates. (6) (d) Show that the graph of f has exactly two asymptotes. (3) IB Questionbank Mathematics Higher Level 3rd edition 9 (e) Let a = 4 and b = 1. Consider the region R enclosed by the graph of y = f(x), 1 the y-axis and the line with equation y = 2 . Find the volume V of the solid obtained when R is rotated through 2 about the x-axis. (5) (Total 19 marks) 17. 1 x Let f(x) = 1 x and g(x) = x 1 , x > - 1. Find the set of values of x for which f (x) f(x) g(x). (Total 7 marks) sin 18. (a) d Integrate 1 cos . (3) a (b) Given that sin 1 1 cos d 2 2 and 2 < a < , find the value of a. (2) (Total 5 marks) cos 2 x 19. (a) Solve the differential equation e y ee y dy dx = 0, given that y = 0 when x = . (7) (b) Find the value of y when x = 2 . (1) (Total 8 marks) IB Questionbank Mathematics Higher Level 3rd edition 10 20. The diagram below shows a circle with centre at the origin O and radius r > 0. A point P(x, y), (x > 0, y > 0) is moving round the circumference of the circle. y tan arcsin r . Let m = 3 (a) dm r dy dt 10 r 2 y 2 . Given that dt = 0.001r, show that (6) (b) dm State the geometrical meaning of dt . (1) (Total 7 marks) IB Questionbank Mathematics Higher Level 3rd edition 11 21. Let f be a function defined by f(x) = x + 2 cos x, x [0, 2]. The diagram below shows a region S bound by the graph of f and the line y = x. A and C are the points of intersection of the line y = x and the graph of f, and B is the minimum point of f. (a) If A, B and C have x-coordinates a, b and c. a , b and c 2 6 2 , where a, b, c , find the values of (4) (b) Find the range of f. (3) (c) Find the equation of the normal to the graph of f at the point C, giving your answer in the form y = px + q. (5) IB Questionbank Mathematics Higher Level 3rd edition 12 (d) The region S is rotated through 2 about the x-axis to generate a solid. (i) Write down an integral that represents the volume V of this solid. (ii) Show that V = 6 . 2 (7) (Total 19 marks) 22. (a) 2 Differentiate f(x) = arcsin x + 2 1 x , x [-1, 1]. (3) (b) Find the coordinates of the point on the graph of y = f(x) in [-1, 1], where the gradient of the tangent to the curve is zero. (3) (Total 6 marks) 23. -2 The acceleration in m s of a particle moving in a straight line at time t seconds, t 0, is given 1 v -1 by the formula a = 2 . When t = 0, the velocity is 40 m s . Find an expression for v in terms of t. (Total 6 marks) 24. 3 2 The cubic curve y = 8x + bx + cx + d has two distinct points P and Q, where the gradient is zero. (a) 2 Show that b > 24c. (4) (b) 1 3 , 12 and , 20 2 2 , respectively, find the Given that the coordinates of P and Q are values of b, c and d. (4) (Total 8 marks) IB Questionbank Mathematics Higher Level 3rd edition 13 25. Using the substitution x = 2 sin , show that 4 x 2 x dx Ax 4 x 2 B arcsin 2 + constant, where A and B are constants whose values you are required to find. (Total 8 marks) 26. -1 A helicopter H is moving vertically upwards with a speed of 10 m s . The helicopter is h m directly above the point Q, which is situated on level ground. The helicopter is observed from the point P, which is also at ground level, and PQ = 40 m. This information is represented in the diagram below. diagram not to scale When h = 30, (a) show that the rate of change of HPQ is 0.16 radians per second; (3) (b) find the rate of change of PH. (4) (Total 7 marks) IB Questionbank Mathematics Higher Level 3rd edition 14 27. 5 Consider the graphs y = e and y = e sin 4x, for 0 x 4 . -x (a) -x 5 On the same set of axes draw, on graph paper, the graphs, for 0 x 4 . Use a scale of 1 cm to 8 on your x-axis and 5 cm to 1 unit on your y-axis. (3) (b) n Show that the x-intercepts of the graph y = e sin 4x are 4 , n = 0, 1, 2, 3, 4, 5. -x (3) (c) -x Find the x-coordinates of the points at which the graph of y = e sin 4x meets the graph -x of y = e . Give your answers in terms of . (3) (d) -x -x (i) Show that when the graph of y = e sin 4x meets the graph of y = e , their gradients are equal. (ii) Hence explain why these three meeting points are not local maxima of the -x graph y = e sin 4x. (6) (e) (i) Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima 5 -x of y = e sin 4x for 0 x 4 . You do not need to show that they are maximum values, but the values should be simplified. (ii) Show that y1, y2 and y3 form a geometric sequence and determine the common ratio r. (7) (Total 22 marks) IB Questionbank Mathematics Higher Level 3rd edition 15 28. xy 2 y Find the gradient of the curve e + ln(y ) + e = 1 + e at the point (0, 1). (Total 7 marks) 29. Consider the function f, defined by f(x) = x - a x , where x 0, a (a) + . Find in terms of a (i) the zeros of f; (ii) the values of x for which f is decreasing; (iii) the values of x for which f is increasing; (iv) the range of f. (10) (b) State the concavity of the graph of f. (1) (Total 11 marks) 30. Below is a sketch of a Ferris wheel, an amusement park device carrying passengers around the rim of the wheel. IB Questionbank Mathematics Higher Level 3rd edition 16 (a) The circular Ferris wheel has a radius of 10 metres and is revolving at a rate of 3 radians per minute. Determine how fast a passenger on the wheel is going vertically upwards when the passenger is at point A, 6 metres higher than the centre of the wheel, and is rising. (7) (b) The operator of the Ferris wheel stands directly below the centre such that the bottom of the Ferris wheel is level with his eyeline. As he watches the passenger his line of sight makes an angle with the horizontal. Find the rate of change of at point A. (3) (Total 10 marks) 31. 3 2 Find the gradient of the tangent to the curve x y = cos (y) at the point (1, 1). (Total 6 marks) 32. By using an appropriate substitution find tan ln y dy, y 0. y (Total 6 marks) 33. 2 3 2 A family of cubic functions is defined as fk (x) = k x kx + x, k (a) + . Express in terms of k (i) f k (x) and f k (x); (ii) the coordinates of the points of inflexion Pk on the graphs of fk. (6) (b) Show that all Pk lie on a straight line and state its equation. (2) IB Questionbank Mathematics Higher Level 3rd edition 17 (c) Show that for all values of k, the tangents to the graphs of fk at Pk are parallel, and find the equation of the tangent lines. (5) (Total 13 marks) 34. x The curve y = e x + 1 intersects the x-axis at P. (a) Find the x-coordinate of P. (2) (b) Find the area of the region completely enclosed by the curve and the coordinate axes. (3) (Total 5 marks) 35. 2 2x Consider the curve with equation f (x) = e for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks) 36. A particle moves in a straight line in a positive direction from a fixed point O. 1 The velocity v m s , at time t seconds, where t 0, satisfies the differential equation dv v 1 v 2 . dt 50 1 The particle starts from O with an initial velocity of 10 m s . (a) (i) Express as a definite integral, the time taken for the particle's velocity to decrease 1 1 from 10 m s to 5 m s . (ii) Hence calculate the time taken for the particle's velocity to decrease from 10 m s 1 to 5 m s . 1 (5) IB Questionbank Mathematics Higher Level 3rd edition 18 (b) (i) Show that, when v 0, the motion of this particle can also be described by the dv 1 v 2 50 differential equation dx where x metres is the displacement from O. (ii) Given that v = 10 when x = 0, solve the differential equation expressing x in terms of v. x 50 . x 1 10 tan 50 Hence show that v = 10 tan (iii) (14) (Total 19 marks) 37. 1 dy 2 ln (1 e 2 x ) , show that dx 3 (e-y - 3). If y = 3 (Total 7 marks) 38. The population of mosquitoes in a specific area around a lake is controlled by pesticide. The rate of decrease of the number of mosquitoes is proportional to the number of mosquitoes at any time t. Given that the population decreases from 500 000 to 400 000 in a five year period, find the time it takes in years for the population of mosquitoes to decrease by half. (Total 8 marks) 39. x x 9 x 2 2 arcsin 3. The function f is defined by f(x) = (a) Write down the largest possible domain, for each of the two terms of the function, f, and hence state the largest possible domain, D, for f. (2) (b) Find the volume generated when the region bounded by the curve y = f(x), the x-axis, the y-axis and the line x = 2.8 is rotated through 2 radians about the x-axis. (3) (c) Find f(x) in simplified form. (5) IB Questionbank Mathematics Higher Level 3rd edition 19 p (d) Hence show that p 11 2 x 2 p dx 2 p 9 p 2 4 arcsin 3 9 x , where p D. 2 (2) (e) Find the value of p that maximises the value of the integral in (d). (2) x (2 x 2 25) 3 (f) (i) (ii) 2 2 Show that f(x) = (9 x ) . Hence justify that f(x) has a point of inflexion at x = 0, but not at x = 25 2 . (7) (Total 21 marks) IB Questionbank Mathematics Higher Level 3rd edition 20