Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Further Pure 4 Wednesday 30 January 2013 General Certificate of Education Advanced Level Examination January 2013 9.00 am to 10.30 am For this paper you must have: * the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator. Time allowed * 1 hour 30 minutes MFP4 Instructions * Use black ink or black ball-point pen. Pencil should only be used for drawing. * Fill in the es at the top of this page. * Answer all questions. * Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. * You must answer each question in the space provided for that question. If you require extra space, use an AQA supplementary answer book; do not use the space provided for a different question. * around each page. * Show all necessary working; otherwise marks for method may be lost. * Do all rough work in this book. Cross through any work that you do not want to be marked. Information * The marks for questions are shown in brackets. * The maximum mark for this paper is 75. Advice * Unless stated otherwise, you may quote formulae, without proof, from the booklet. * You do not necessarily need to use all the space provided. Question 1 2 3 4 5 6 7 8 TOTAL Mark (JAN13MFP401) 6/6/6/ MFP4
2 Answer all questions. Answer each question in the space provided for that question. 1 Two planes have equations x þ 2y þ 2z ¼ 5 and px þ 3y ¼ 10 where p is a non-zero constant. Given that the acute angle, y, between the planes is such that cos y ¼ 2, find the 3 value of p. (5 marks) Answer space for question 1 (02)
3 Answer space for question 1 Turn over s (03)
4 2 It is given that A and B are 3 3 matrices such that detðabþ ¼24 and detða 1 Þ¼ 3 (a) State the value of det A. (1 mark) (b) A three-dimensional shape S, with volume 20 cm 3, is transformed using matrix B. Find the volume of the image of S. (3 marks) Answer space for question 2 (04)
5 Answer space for question 2 Turn over s (05)
6 3 (a) Expand and simplify, as far as possible, ða 4bÞða þ 3bÞ where a and b are vectors. (3 marks) (b) Given that a and b are perpendicular, deduce that jða 4bÞða þ 3bÞj ¼ ljajjbj where l is an integer. (2 marks) Answer space for question 3 (06)
7 Answer space for question 3 Turn over s (07)
8 4 The matrix A is given by (a) 2 1 0 3 1 6 A ¼ 4 1 2 1 7 5 2 2 3 2 p 2 3 4 Given that A 2 6 ¼ 4 5 6 4 7 5, find the value of p and the value of q. (2 marks) 10 q 9 (b) Given that A 3 6A 2 þ 11A 6I ¼ 0, prove that A 1 ¼ 1 6 ða2 6A þ 11IÞ (2 marks) 2 3 r 2 2 (c) Given that A 1 ¼ 1 6 7 4 1 5 2 5, find the value of r and the value of s. 6 2 s 2 (2 marks) (d) Hence, or otherwise, find the solution of the system of equations x z ¼ k x þ 2y þ z ¼ 5 2x þ 2y þ 3z ¼ 7 giving your answers in terms of k. (3 marks) Answer space for question 4 (08)
9 Answer space for question 4 Turn over s (09)
10 Answer space for question 4 (10)
11 Answer space for question 4 Turn over s (11)
12 5 (a) By direct expansion, or otherwise, show that the value of is independent of k. 2 1 2k 1 1 k þ 1 2 k 1 1 (4 marks) (b) State, with a reason, whether the vectors 2 3 2 3 2 3 2 1 4 6 7 6 7 6 7 4 1 5, 4 1 5 and 4 3 5 2 1 1 are linearly dependent or linearly independent. (2 marks) (c) (i) State, with a reason, whether the equations 2x þ y þ 6z ¼ 1 x þ y þ 4z ¼ 0 2x þ 2y þ z ¼ 1 are consistent or inconsistent. (2 marks) (ii) The three equations given in part (c)(i) are the Cartesian equations of three planes. State the geometrical configuration of these three planes. (1 mark) Answer space for question 5 (12)
13 Answer space for question 5 Turn over s (13)
14 Answer space for question 5 (14)
15 Answer space for question 5 Turn over s (15)
16 6 The linear transformations T 1 and T 2 are represented by the matrices 2 pffiffi 3 2 3 1 1 0 0 0 3 2 2 6 7 M 1 ¼ 4 0 1 0 5 and M 2 ¼ 6 0 1 0 7 4 pffiffi 5 0 0 1 3 0 1 2 2 respectively. (a) Give a full geometrical description of the transformations: (i) T 1 ; (ii) T 2. (2 marks) (3 marks) (b) Find the matrix which represents the transformation T 1 followed by T 2. (2 marks) (c) The linear transformation T 3 is represented by the matrix 2 k 2 3 1 6 M 3 ¼ 4 1 1 1 7 5 3 4 1 where k is a constant. For one particular value of k, T 3 has a line L of invariant points. (i) Find k. (ii) Find the Cartesian equations of L in the form x p ¼ y q ¼ z. (7 marks) r Answer space for question 6 (16)
17 Answer space for question 6 Turn over s (17)
18 Answer space for question 6 (18)
19 Answer space for question 6 Turn over s (19)
20 7 The matrix M is defined by 2 a 0 3 a 6 M ¼ 4 0 6 7 05 a 0 2 (a) (b) where a is a real number. The distinct eigenvalues of M are l 1, l 2 and l 3 with corresponding eigenvectors v 1, v 2 and v 3. 2 3 0 6 7 Given that v 1 ¼ 4 1 5, find l 1. (2 marks) 0 2 3 1 6 7 Given that v 2 ¼ 4 0 5, find the value of a. (3 marks) 2 (c) Given that l 3 ¼ 6, find a possible eigenvector v 3. (3 marks) (d) The matrix M can be expressed as UDU 1, where D is a diagonal matrix. Write down possible matrices D and U. (3 marks) Answer space for question 7 (20)
21 Answer space for question 7 Turn over s (21)
22 Answer space for question 7 (22)
23 Answer space for question 7 Turn over s (23)
24 8 The four vertices of a parallelogram ABCD have coordinates (a) (i) Að1, 0, 2Þ, Bð3, 1, 5Þ, Cð7, 2, 4Þ and Dð5, 3, 1Þ ƒ! ƒ! Find AB AD. (3 marks) p (ii) Show that the area of the parallelogram is p ffiffiffiffiffi 10, where p is an integer to be found. (2 marks) (b) The diagonals AC and BD of the parallelogram meet at the point M. The line L passes through M and is perpendicular to the plane ABCD. Find an equation for the line L, giving your answer in the form ðr uþv ¼ 0. (4 marks) (c) The plane P is parallel to the plane ABCD and passes through the point Qð6, 5, 17Þ. (i) Find the coordinates of the point of intersection of the line L with the plane P. (6 marks) (ii) One face of a parallelepiped is ABCD and the opposite face lies in the plane P. Find the volume of the parallelepiped. (3 marks) Answer space for question 8 (24)
25 Answer space for question 8 Turn over s (25)
26 Answer space for question 8 (26)
27 Answer space for question 8 END OF S (27)
28 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Copyright ª 2013 AQA and its licensors. All rights reserved. (28)