A Level Further Mathematics B (MEI) Y420/01

A Level Further Mathematics B (MEI) Y420/01 Core Pure Practice Paper – Set 1 Time allowed: 2 hours 40 minutes You must have: • Printed Answer Booklet • Formulae Further Mathematics B (MEI) You may use: • a scientific or graphical calculator OCR is an exempt Charity This study source was downloaded by from CourseH on :12:31 GMT -05:00 2 © OCR 2018 Practice paper Y420/01 Section A (33 marks) Answer all the questions. 1 Using standard summation formulae, find 1 r r 2 n + r= / ^ h, giving your answer in a fully factorised form. [5] 2 (i) Describe the transformation of the plane represented by the matrix M 1 k 0 1 = c m, where k is a non-zero constant. [2] (ii) The image of the point (2, 3) under the transformation in part (i) lies on the y-axis. Find k. [3] 3 In this question you must show detailed reasoning. Given that z = 2 is a root of the equation z kz z7 6 0 3 2 + + - = , find, in exact form, the other two roots. [6] 4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve y 3 x 1 2 = + , the x-axis, the y-axis and the line x = 1. y = 1 3 x2 + O x y Fig. 4 This region is rotated through 2r radians about the x-axis. Find, in an exact form, the volume of the solid of revolution generated. [4] 5 In this question you must show detailed reasoning. Show that d e e sinh 2x x 4 1 1 0 1 2 y = - a k . [3] 6 Verify that the lines x 2 y z 2 3 3 2 - = + = and x y z 3 6 1 4 4 6 - + = + = - intersect, stating clearly the coordinates of the point of intersection. [6] 7 In this question you must show detailed reasoning. Evaluate 3 d x x 1 2 1 2 0 ^ + h y . [4] This study source was downloaded by from CourseH on :12:31 GMT -05:00 3 © OCR 2018 Practice paper Y420/01 Turn over Section B (111 marks) Answer all the questions. 8 In this question you must show detailed reasoning. Find 10 r r 1 2 100 r= + / , expressing your answer as an exact fraction. [5] 9 (i) On an Argand diagram, draw the locus of points defined by z- -3 4i = 3. [2] (ii) Find the complex number z which lies on the locus in part (i) and has the smallest possible argument. Give your answer in the form a + bi. [7] 10 Given that the points A (1, 0, 2), B (−2, −6, 2), C (0, 6, 3) and D (4, m, 1) are coplanar, find m. [8] 11 Matrix M is given by M c 1 c 1 = - c m. (i) Given that det 0 M = , find the possible values of c. [2] (ii) Prove by induction that M M 2 n n 1 = - , for all positive integers n. [7] 12 (i) Determine, from first principles, the Maclaurin series up to the x3 term for the function arcsin x2 1 . [6] (ii) Use the series in part (i) to find a rational approximation for r. [3] 13 (i) Prove, using exponential functions, that cosh 2 1 x x 2 sinh2 = + . [3] (ii) In this question you must show detailed reasoning. Use the result in part (i) to solve the equation 2 2 cosh x x - = sinh 5, giving the answers in exact logarithmic form. [6] 14 (i) Show that the planes with equations x y z a x y z b x y z c 2 2 m m + + = + + = - + = where m, a, b and c are real constants, always meet at a point. [5] (ii) You are now given that a 2 2 = + m , b = 0 and c = 0. Show that, for different values of m, the point of intersection of the planes always lies on a fixed line. [7]