IB Questionbank Mathematics Higher Level 3rd edition11.Consider the functions given below.f(x) = 2x + 3g(x) = , x 0 (a)(i)Find (g f)(x) and write down the domain of the function.(ii)Find (f g)(x) and write down the domain of the function.(2) (b)Find the coordinates of the point where the graph of y = f(x) and the graph ofy = (g1 f g)(x) intersect.(4)(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.The quadratic function f(x) = p + qx x2 has a maximum value of 5 when x = 3.(a)Find the value of p and the value of q.(4) (b)The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the equation of the new graph.(2)(Total 6 marks) 4.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) 5.A function is defined by h(x) = 2ex . Find an expression for h1(x).(Total 6 marks) 6.Given that Ax3 + Bx2 + x + 6 is exactly divisible by (x + 1)(x 2), find the value of Aand the value of B.(Total 5 marks)7.Shown below are the graphs of y = f(x) and y = g(x).If (f g)(x) = 3, find all possible values of x.(Total 4 marks) 8.Solve the equation 4x1 = 2x + 8.(Total 5 marks)9.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) 10.(a)Express the quadratic 3x2 6x + 5 in the form a(x + b)2 + c, where a, b, c .(3) (b)Describe a sequence of transformations that transforms the graph of y = x2 to the graph of y = 3x2 6x + 5.(3)(Total 6 marks) 11.A function f is defined by f(x) = , x 1.(a)Find an expression for f1(x).(3)(b)Solve the equation f1(x) = 1 + f1(x).(3)(Total 6 marks) 12.Find the set of values of x for which x 1>2x 1.(Total 4 marks) 13.Let g(x) = log52log3x. Find the product of the zeros of g.(Total 5 marks)14.(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)15.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) 16.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) 17.When the function q(x) = x3 + kx2 7x + 3 is divided by (x + 1) the remainder is seven times the remainder that is found when the function is divided by (x + 2).Find the value of k.(Total 5 marks) 18.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) 19.When 3x5 ax + b is divided by x 1 and x + 1 the remainders are equal. Given that a, b , find(a)the value of a;(4)(b)the set of values of b.(1)(Total 5 marks) 20.Consider the function f, where f(x) = arcsin (ln x).(a)Find the domain of f.(3) (b)Find f1(x).(3)(Total 6 marks) 21.The real root of the equation x3 x + 4 = 0 is 1.796 to three decimal places.Determine the real root for each of the following.(a)(x 1)3 (x 1) + 4 = 0(2)(b)8x3 2x + 4 = 0(3)(Total 5 marks) 22.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) 23.The functions f and g are defined as:f (x) = x 0g (x) = (a)Find h (x) where h (x) = g f (x).(2) (b)State the domain of h1 (x).(2) (c)Find h1 (x).(4)(Total 8 marks) 24.The polynomial P(x) = x3 + ax2 + bx + 2 is divisible by (x +1) and by (x 2).Find the value of a and of b, where a, b.(Total 6 marks) 25.Let f (x) = and g (x) = x 1.If h = g f, find(a)h (x);(2) (b)h1 (x), where h1 is the inverse of h.(4)(Total 6 marks) 26.When f(x) = x4 + 3x3 + px2 2x + q is divided by (x 2) the remainder is 15,and (x + 3) is a factor of f(x).Find the values of p and q.(Total 6 marks) 27.Find all values of x that satisfy the inequality .(Total 5 marks)28.The polynomial f(x) = x3 + 3x2 + ax + b leaves the same remainder when divided by (x 2) as when divided by (x +1). Find the value of a.(Total 6 marks) 29.The functions f and g are defined by f : x ex, g : x x + 2.Calculate(a)f1(3) g1(3);(3)(b)(f g)1(3).(3)(Total 6 marks) 30.(a)Show that p = 2 is a solution to the equation p3 + p2 5p 2 = 0.(2) (b)Find the values of a and b such that p3 + p2 5p 2 = (p 2)(p2 + ap + b).(4) (c)Hence find the other two roots to the equation p3 + p2 5p 2 = 0.(3) (d)An arithmetic sequence has p as its common difference. Also, a geometric sequence has p as its common ratio. Both sequences have 1 as their first term.(i)Write down, in terms of p, the first four terms of each sequence. (ii)If the sum of the third and fourth terms of the arithmetic sequence is equal to the sum of the third and fourth terms of the geometric sequence, find the three possible values of p. (iii)For which value of p found in (d)(ii) does the sum to infinity of the terms of the geometric sequence exist?(iv)For the same value p, find the sum of the first 20 terms of the arithmetic sequence, writing your answer in the form a + , where a, b, c .(13)(Total 22 marks)