> Monday 4 October 2021 – Afternoon A Level Further Mathematics B (MEI) Y420/01 Core Pure

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Section A (31 marks) Answer all the questions. 1 (a) Express ( ) 2 1 r r ( ) 2 1 1 - + in partial fractions. [3] (b) Hence find = ( ) 2 1 r r ( ) 2 1 n 1 1 - + /r , expressing the result ... as a single fraction. [4] 2 In this question you must show detailed reasoning. Find the gradient of the curve y x = 6 2 arcsin( ) at the point with x-coordinate 4 1 . Express the result in the form m n, where m and n are integers. [4] 3 In this question you must show detailed reasoning. The complex numbers z1 and z2 are given by z1 =-2 2 + i and z2 = + 2 i acos s 6 1 r r in 6 1 k. (a) Find the modulus and argument of z1. [2] (b) Hence express zz 1 2 in exact modulus-argument form. [4] 4 In this question you must show detailed reasoning. Determine the mean value of 1 4x 1 2 + between x =-1 and x = 1. Give your answer to 3 significant figures. [4] 5 (a) Use a Maclaurin series to find a quadratic approximation for ln( ) 1 2 + x . [1] (b) Find the percentage error in using the approximation in part (a) to calculate ln( . 1 2). [3] (c) Jane uses the Maclaurin series in part (a) to try to calculate an approximation for ln3. Explain whether her method is valid. [2] 6 Given that y m = x is an invariant line of the transformation with matrix 1 2 22 - JKKL NOOP , determine the possible values of m. [4]3 © OCR 2021 Y420/01 Oct21 Turn over Section B (113 marks) Answer all the questions. 7 Prove that = r n 2 4 2 2 r n n 1 1 1 = - + - - /r for all n H 1. [6] 8 The equation 4 4 x x 4 3 2 - + + px qx- = 9 0, where p and q are constants, has roots a, -a, b and 1 b . (a) Determine the exact roots of the equation. [5] (b) Determine the values of p and q. [4] 9 The transformation T of the plane has associated matrix M, where M 1 2 01 = - - JKKL NOOP . (a) On the grid in the Printed Answer Booklet, plot the image OAʹBʹCʹ of the unit square OABC under the transformation T. [2] (b) (i) Calculate the value of detM. [1] (ii) Explain the significance of the value of detM in relation to the image OAʹBʹCʹ. [2] (c) T is equivalent to a sequence of two transformations of the plane. (i) Specify fully two transformations equivalent to T. [3] (ii) Use matrices to verify your answer. [3] 10 (a) Show on an Argand diagram the points representing the three cube roots of unity. [2] (b) (i) Find the exact roots of the equation z3 - = 1 3i, expressing them in the form reii, where r 2 0 and -r i 1 1 r. [5] (ii) The points representing the cube roots of unity form a triangle D1. The points representing the roots of the equation z3 - = 1 3i form a triangle D2. State a sequence of two transformations that maps D1 onto D2. [2] (iii) The three roots in part (b)(i) are z1, z2 and z3. By simplifying z z 1 2 3 + +z , verify that the sum of these roots is zero. [2] (iv) Hence show that sin 20° + sin 140° = sin100°. [2]4 © OCR 2021 Y420/01 Oct21 11 (a) Given that u = mi + j - 3k and v = i + 2j - 2k, find the following, giving your answers in terms of m. (i) u.v [1] (ii) u # v [2] (b) Hence determine (i) the acute angle between the planes 2 3 x y + - z = 10 and x y z + - 2 2 = 10, [3] (ii) the shortest distance between the lines x y z 3 3 1 3 - 2 = = - - and x y z 1 2 4 22 = - = + - , giving your answer as a multiple of 2. [3] 12 Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers z and w respectively. [Show More]

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