AQA MFP4 p QP JUN15

Centre Number For Examiner’s Use Candidate Number Surname Other Names Examiner’s Initials Candidate Signature Question General Certificate of Education Advanced Level Examination June 2015 Mark Mathematics MFP4 Unit Further Pure Wednesday 20 May 2015 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 TOTAL Time allowed * hour 30 minutes Instructions * Use black ink or black ball-point pen Pencil should only be used for drawing * Fill in the boxes 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; not use the space provided for a different question * Do not write outside the box 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 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 not necessarily need to use all the space provided (JUN15MFP401) P89112/Jun15/E3 MFP4 Do not write outside the box Answer all questions Answer each question in the space provided for that question The points U , V and W have position vectors u , v and w respectively relative to an origin O, where 3 3 a u ¼ 5, v ¼ À4 and w ¼ 2 À2 Find ðu  vÞ:w in terms of a (a) [2 marks] Given that u , v and w are linearly dependent: (b) (i) find the value of a; [1 mark] (ii) express u as a linear combination of v and w [3 marks] QUESTION PART REFERENCE Answer space for question (02) P/Jun15/MFP4 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (03) Turn over P/Jun15/MFP4 Do not write outside the box The vectors a , b and c are such that c  a ¼ 2i and b  a ¼ 3j Simplify ða þ 2b À 6cÞ Â ða À b þ 3cÞ , giving your answer in the form li þ mj [5 marks] QUESTION PART REFERENCE Answer space for question (04) P/Jun15/MFP4 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (05) Turn over P/Jun15/MFP4 Do not write outside the box Factorise completely the determinant (a) a b À c Àbc b a À c Àca Àc a þ b ab [6 marks] Hence, or otherwise, find the values of a for which the equations (b) ax þ y À 6z ¼ 3x þ ða À 2Þy À 2az ¼ À2x þ ða þ 3Þy þ 3az ¼ not have a unique solution [3 marks] QUESTION PART REFERENCE Answer space for question (06) P/Jun15/MFP4 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (07) Turn over P/Jun15/MFP4 Do not write outside the box Find the eigenvalues and corresponding eigenvectors of the matrix (a)  À1 M¼  [6 marks]  b Given that U ¼ a À2 (b)   and U MU ¼ , find the values of a and b À1  [3 marks] QUESTION PART REFERENCE Answer space for question (08) P/Jun15/MFP4 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (09) Turn over P/Jun15/MFP4 Do not write outside the box 10 A system of equations is given by 2x À 11y À 3z ¼ 5x À 10y À 4z ¼ 9x À 17y À 7z ¼ 11 Find the solution of this system of equations, showing all your working (a) [5 marks] Interpret this solution geometrically (b) [1 mark] QUESTION PART REFERENCE Answer space for question (10) P/Jun15/MFP4 Do not write outside the box 11 QUESTION PART REFERENCE Answer space for question s (11) Turn over P/Jun15/MFP4 Do not write outside the box 12 31 3 The line L has equation @r À 5A  À1 ¼ The plane P contains the line L and the point Að4, 1, À2Þ Show that A does not lie on the line L (a) [1 mark] Find an equation of the plane P, giving your answer in the form r:n ¼ c (b) [5 marks] The point D has coordinates ð8, À2, 6Þ Find the coordinates of the image of D after reflection in the plane P [5 marks] (c) QUESTION PART REFERENCE Answer space for question (12) P/Jun15/MFP4 Do not write outside the box 13 QUESTION PART REFERENCE Answer space for question s (13) Turn over P/Jun15/MFP4 Do not write outside the box 14 QUESTION PART REFERENCE Answer space for question (14) P/Jun15/MFP4 Do not write outside the box 15 QUESTION PART REFERENCE Answer space for question s (15) Turn over P/Jun15/MFP4 Do not write outside the box 16  3:4 The matrix A ¼ 1:2 transformation T  represents a transformation that is a shear S followed by a (a) The shear S is such that the image of the point ð1, 1Þ is ð5, À3Þ and the line y ¼ Àx is a line of invariant points Find the matrix that represents S [4 marks] (b) (i) Hence find the matrix that represents the transformation T [4 marks] (ii) Give a full description of the transformation T [2 marks] QUESTION PART REFERENCE Answer space for question (16) P/Jun15/MFP4 Do not write outside the box 17 QUESTION PART REFERENCE Answer space for question s (17) Turn over P/Jun15/MFP4 Do not write outside the box 18 k The linear transformation T is represented by the matrix M ¼ 4 À1 À1 In the case when M is a non-singular matrix: (a) (i) find the possible values of k ; [3 marks] (ii) find MÀ1 in terms of k [5 marks] (b) In the case when k ¼ , the matrix MÀ1 is applied to a solid shape of volume cm3 Find the volume of the image [3 marks] (c) In the case when k ¼ , verify that the image of every point under T lies in the plane x À y þ z ¼ [3 marks] (d) Find the value of k for which T has a line of invariant points and obtain the Cartesian equations of this line [5 marks] QUESTION PART REFERENCE Answer space for question (18) P/Jun15/MFP4 Do not write outside the box 19 QUESTION PART REFERENCE Answer space for question s (19) Turn over P/Jun15/MFP4 Do not write outside the box 20 QUESTION PART REFERENCE Answer space for question (20) P/Jun15/MFP4 Do not write outside the box 21 QUESTION PART REFERENCE Answer space for question END OF QUESTIONS (21) P/Jun15/MFP4 Do not write outside the box 22 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (22) P/Jun15/MFP4 Do not write outside the box 23 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (23) P/Jun15/MFP4 Do not write outside the box 24 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Copyright ª 2015 AQA and its licensors All rights reserved (24) P/Jun15/MFP4 [...]... QUESTIONS (21) P/ Jun15 /MFP4 Do not write outside the box 22 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (22) P/ Jun15 /MFP4 Do not write outside the box 23 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (23) P/ Jun15 /MFP4 Do not write outside the box 24 There are no questions printed on this page DO NOT... s (11) Turn over P/ Jun15 /MFP4 Do not write outside the box 12 0 2 31 2 3 2 3 2 3 0 The line L has equation @r À 4 1 5A  4 À1 5 ¼ 4 0 5 0 2 0 6 The plane P contains the line L and the point Að4, 1, À2Þ Show that A does not lie on the line L (a) [1 mark] Find an equation of the plane P, giving your answer in the form r:n ¼ c (b) [5 marks] The point D has coordinates ð8, À2, 6Þ Find... s (17) Turn over P/ Jun15 /MFP4 Do not write outside the box 18 2 3 1 2 k The linear transformation T is represented by the matrix M ¼ 4 0 3 4 5 À1 1 À1 8 In the case when M is a non-singular matrix: (a) (i) find the possible values of k ; [3 marks] (ii) find MÀ1 in terms of k [5 marks] (b) In the case when k ¼ 1 , the matrix MÀ1 is applied to a solid shape of volume 6 cm3 Find the volume... s (15) Turn over P/ Jun15 /MFP4 Do not write outside the box 16  3:4 2 The matrix A ¼ 1:2 1 transformation T 7  represents a transformation that is a shear S followed by a (a) The shear S is such that the image of the point ð1, 1Þ is ð5, À3Þ and the line y ¼ Àx is a line of invariant points Find the matrix that represents S [4 marks] (b) (i) Hence find the matrix that represents the transformation... PAGE ANSWER IN THE SPACES PROVIDED (23) P/ Jun15 /MFP4 Do not write outside the box 24 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Copyright ª 2015 AQA and its licensors All rights reserved (24) P/ Jun15 /MFP4 ... s (09) Turn over P/ Jun15 /MFP4 Do not write outside the box 10 A system of equations is given by 5 2x À 11y À 3z ¼ 1 5x À 10y À 4z ¼ 6 9x À 17y À 7z ¼ 11 Find the solution of this system of equations, showing all your working (a) [5 marks] Interpret this solution geometrically (b) [1 mark] QUESTION PART REFERENCE Answer space for question 5 ... s (07) Turn over P/ Jun15 /MFP4 Do not write outside the box 8 Find the eigenvalues and corresponding eigenvectors of the matrix 4 (a)  1 À1 M¼ 2 4  [6 marks]  4 b Given that U ¼ a À2 (b)   3 0 and U MU ¼ , find the values of a and b 0 2 À1  [3 marks] QUESTION PART REFERENCE Answer space for question 4 ... s (13) Turn over P/ Jun15 /MFP4 Do not write outside the box 14 QUESTION PART REFERENCE Answer space for question 6 ... (14) P/ Jun15 /MFP4 Do not write outside the box 15 QUESTION PART REFERENCE Answer space for question 6 ... (06) P/ Jun15 /MFP4 Do not write outside the box 7 QUESTION PART REFERENCE Answer space for question 3