# UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/01

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2 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, b b ac x = ± 4 2a 2. Binomial Theorem ) ( ) (a + b) n = a n n + a n 1 n b + a n 2 b 2 n (1 (2 + + r a n r b r + + b n, n where n is a positive integer and = n! r. (n r)!r! ) ( ) 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1. sec 2 A = 1 + tan 2 A. cosec 2 A = 1 + cot 2 A. Formulae for ABC a b c = =. sin A sin B sin C a 2 = b 2 + c 2 2bc cos A. 1 = bc sin A. 2

3 Given that A =, find (A 2 ) 1. [4] 2 A student has a collection of 9 CDs, of which 4 are by the Beatles, 3 are by Abba and 2 are by the Rolling Stones. She selects 4 of the CDs from her collection. Calculate the number of ways in which she can make her selection if (i) her selection must contain her favourite Beatles CD, [2] (ii) her selection must contain 2 CDs by one group and 2 CDs by another. [3] 1 3 Given that θ is acute and that sinθ =, express, without using a calculator, sinθ in the 3 cosθ sinθ form a + b, where a and b are integers. [5] 4 The position vectors of points A and B relative to an origin O are 3i j and i + 2j respectively. The point C lies on AB and is such that AC = 3 5 AB. Find the position vector of C and show that it is a unit vector. [6] 5 The function f is defined, for 0 x 180, by f(x) = A + 5 cos Bx, where A and B are constants. (i) Given that the maximum value of f is 3, state the value of A. [1] (ii) State the amplitude of f. [1] (iii) Given that the period of f is 120, state the value of B. [1] (iv) Sketch the graph of f. [3] 6 Given that each of the following functions is defined for the domain 2 x 3, find the range of (i) f : x 2 3x, [1] (ii) g : x 2 3x, [2] (iii) h : x 2 3x. [2] State which of the functions f, g and h has an inverse. [2] [Turn over

4 4 7 (a) Variables l and t are related by the equation l = l 0 (1 + α) t where l 0 and α are constants. Given that l 0 = 0.64 and α = , find the value of t for which l = [3] (b) Solve the equation 1 + lg(8 x) = lg(3x + 2). [4] 8 x y The table above shows experimental values of the variables x and y which are related by an equation of the form y = kx n, where k and n are constants. (i) Using graph paper, draw the graph of lg y against lg x. [3] (ii) Use your graph to estimate the value of k and of n. [4] 9 (i) Determine the set of values of k for which the equation x 2 + 2x + k = 3kx 1 has no real roots. [5] (ii) Hence state, giving a reason, what can be deduced about the curve y = (x + 1) 2 and the line y = 3x 1. [2] 10 The remainder when 2x 3 + 2x 2 13x + 12 is divided by x + a is three times the remainder when it is divided by x a. (i) Show that 2a 3 + a 2 13a + 6 = 0. [3] (ii) Solve this equation completely. [5] 11 A particle travels in a straight line so that, t seconds after passing a fixed point A on the line, its acceleration, a ms 2, is given by a = 2 2t. It comes to rest at a point B when t = 4. (i) Find the velocity of the particle at A. [4] (ii) Find the distance AB. [3] (iii) Sketch the velocity-time graph for the motion from A to B. [1]

5 12 Answer only one of the following two alternatives. EITHER 5 y A O B x The diagram, which is not drawn to scale, shows part of the graph of y = 8 e 2x, crossing the y-axis at A. The tangent to the curve at A crosses the x-axis at B. Find the area of the shaded region bounded by the curve, the tangent and the x-axis. [10] OR A piece of wire, of length 2 m, is divided into two pieces. One piece is bent to form a square of side x m and the other is bent to form a circle of radius r m. (i) Express r in terms of x and show that the total area, A m 2, of the two shapes is given by (π + 4)x A = 2 4x + 1. π [4] Given that x can vary, find (ii) the stationary value of A, [4] (iii) the nature of this stationary value. [2]

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