IB Questionbank Mathematics Higher Level 3rd edition11.(a)Sketch the curve f (x) = |1 + 3 sin (2x)|, for 0 x p. Write down on the graph the values of the x and y intercepts.(4) (b)By adding one suitable line to your sketch, find the number of solutions to the equation f (x) = 4( x).(2)(Total 6 marks)2.The diagram below shows the graph of the function y = f(x), defined for all x ,where b > a > 0.Consider the function g(x) = . (a)Find the largest possible domain of the function g.(2)(b)On the axes below, sketch the graph of y = g(x). On the graph, indicate any asymptotes and local maxima or minima, and write down their equations and coordinates.(6)(Total 8 marks) 3.(a)Sketch the curve y = ln x cos x 0.1, 0 < x < 4 showing clearly the coordinates of the points of intersection with the x-axis and the coordinates of any local maxima and minima.(5) (b)Find the values of x for which ln x > cos x + 0.1, 0 < x < 4.(2)(Total 7 marks) 4.The diagram below shows a solid with volume V, obtained from a cube with edge a > 1 when a smaller cube with edge is removed.diagram not to scale Let x = .(a)Find V in terms of x.(4)(b)Hence or otherwise, show that the only value of a for which V = 4x is a = .(4)(Total 8 marks) 5.Consider the function g, where g(x) = .(a)Given that the domain of g is x a, find the least value of a such that g has an inverse function.(1)(b)On the same set of axes, sketch(i)the graph of g for this value of a;(ii)the corresponding inverse, g1.(4) (c)Find an expression for g1(x).(3)(Total 8 marks) 6.Consider the graphs y = ex and y = ex sin 4x, for 0 x .(a)On the same set of axes draw, on graph paper, the graphs, for 0 x .Use a scale of 1 cm to on your x-axis and 5 cm to 1 unit on your y-axis.(3) (b)Show that the x-intercepts of the graph y = ex sin 4x are , n = 0, 1, 2, 3, 4, 5.(3) (c)Find the x-coordinates of the points at which the graph of y = ex sin 4x meets the graph of y = ex. Give your answers in terms of .(3) (d)(i)Show that when the graph of y = ex sin 4x meets the graph of y = ex, their gradients are equal. (ii)Hence explain why these three meeting points are not local maxima of thegraph y = ex sin 4x.(6)(e)(i)Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima of y = ex sin 4x for 0 x . 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) 7.A tangent to the graph of y = ln x passes through the origin.(a)Sketch the graphs of y = ln x and the tangent on the same set of axes, and hence find the equation of the tangent.(11) (b)Use your sketch to explain why ln x for x > 0.(1) (c)Show that xe ex for x > 0.(3) (d)Determine which is larger, e or e.(2)(Total 17 marks) 8.Let f be a function defined by f(x) = x arctan x, x .(a)Find f(1) and f().(2) (b)Show that f(x) = f(x), for x .(2) (c)Show that x , for x .(2) (d)Find expressions for f(x) and f(x). Hence describe the behaviour of the graph of f at the origin and justify your answer.(8) (e)Sketch a graph of f, showing clearly the asymptotes.(3) (f)Justify that the inverse of f is defined for all x and sketch its graph.(3)(Total 20 marks) 9.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 , where a, b, c , find the values of a, b and c.(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) (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 = 62.(7)(Total 19 marks) 10.The diagram shows the graph of y = f(x). The graph has a horizontal asymptote at y = 2. (a)Sketch the graph of y = .(3) (b)Sketch the graph of y = x f(x).(3)(Total 6 marks) 11.(a)Let a > 0. Draw the graph of y = for a x a on the grid below.(2) (b)Find k such that .(5)(Total 7 marks) 12.A function is defined as f(x) = , with k > 0 and x 0.(a)Sketch the graph of y = f(x).(1) (b)Show that f is a one-to-one function.(1) (c)Find the inverse function, f1(x) and state its domain.(3) (d)If the graphs of y = f(x) and y = f1(x) intersect at the point (4, 4) find the value of k.(2) (e)Consider the graphs of y = f(x) and y = f1(x) using the value of k found in part (d).(i)Find the area enclosed by the two graphs. (ii)The line x = c cuts the graphs of y = f(x) and y = f1(x) at the points P and Q respectively. Given that the tangent to y = f(x) at point P is parallel to the tangent to y = f1(x) at point Q find the value of c.(9)(Total 16 marks)13.The diagram shows the graphs of a linear function f and a quadratic function g.On the same axes sketch the graph of . Indicate clearly where the x-intercept and the asymptotes occur.(Total 5 marks) 14.(a)(i)Sketch the graphs of y = sin x and y = sin 2x, on the same set of axes,for 0 x .(ii)Find the x-coordinates of the points of intersection of the graphs in thedomain 0 x .(iii)Find the area enclosed by the graphs.(9) (b)Find the value of using the substitution x = 4 sin2 .(8)(c)The increasing function f satisfies f(0) = 0 and f(a) = b, where a > 0 and b > 0.(i)By reference to a sketch, show that .(ii)Hence find the value of .(8)(Total 25 marks) 15.The cumulative frequency graph below represents the weight in grams of 80 apples picked from a particular tree. (a)Estimate the(i)median weight of the apples;(ii)30th percentile of the weight of the apples.(2) (b)Estimate the number of apples that weigh more than 110 grams.(2)(Total 4 marks) 16.Sketch the graph of f(x) = x + . Clearly mark the coordinates of the two maximum points and the two minimum points. Clearly mark and state the equations of the vertical asymptotes and the oblique asymptote.(Total 7 marks) 17.The functions f, g and h are defined byf(x) = 1 + ex, for x ,g(x) = , for x / {0},h(x) = sec x, for x /. (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 = (1 y) sin x. (ii)Hence, or otherwise, find , as a function of x. (iii)You are given that the domain of y = (f g h)(x) can be extended to the whole 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) 18.The graph of y = is drawn below. (a)Find the value of a, the value of b and the value of c.(4)(b)Using the values of a, b and c found in part (a), sketch the graph of y = on the axes below, showing clearly all intercepts and asymptotes.(4)(Total 8 marks)