AQA MFP1 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 Subsidiary Examination June 2015 Mark Mathematics MFP1 Unit Further Pure Friday June 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 (JUN15MFP101) P88830/Jun15/E4 MFP1 Do not write outside the box Answer all questions Answer each question in the space provided for that question The quadratic equation 2x þ 6x þ ¼ has roots a and b Write down the value of a þ b and the value of ab (a) [2 marks] (b) Find a quadratic equation, with integer coefficients, which has roots a À and b2 À [5 marks] (c) Hence find the values of a and b [2 marks] QUESTION PART REFERENCE Answer space for question (02) P/Jun15/MFP1 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (03) Turn over P/Jun15/MFP1 Do not write outside the box ð4 Explain why (a) xÀ4 dx is an improper integral x1:5 [1 mark] ð4 Either find the value of the integral (b) xÀ4 dx or explain why it does not have a x1:5 finite value [4 marks] QUESTION PART REFERENCE Answer space for question (04) P/Jun15/MFP1 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (05) Turn over P/Jun15/MFP1 Do not write outside the box Show that ð2 þ iÞ can be expressed in the form þ bi , where b is an integer [3 marks] (a) It is given that þ i is a root of the equation (b) z þ pz þ q ¼ where p and q are real numbers (i) Show that p ¼ À11 and find the value of q [4 marks] (ii) Given that À i is also a root of z þ pz þ q ¼ , find a quadratic factor of z þ pz þ q with real coefficients [2 marks] (iii) Find the real root of the equation z þ pz þ q ¼ [2 marks] QUESTION PART REFERENCE Answer space for question (06) P/Jun15/MFP1 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (07) Turn over P/Jun15/MFP1 Do not write outside the box Find the general solution, in degrees, of the equation (a) sinð3x þ 45°Þ ¼ [5 marks] Use your general solution to find the solution of sinð3x þ 45°Þ ¼ that is closest to (b) 200° [1 mark] QUESTION PART REFERENCE Answer space for question (08) P/Jun15/MFP1 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (09) Turn over P/Jun15/MFP1 Do not write outside the box 10   À2 c The matrix A is defined by A ¼ d (a) Given that the image of the point ð5, 2Þ under the transformation represented by A is ðÀ2, 1Þ, find the value of c and the value of d [4 marks]  pffiffiffi pffiffiffi  2 pffiffiffi pffiffiffi The matrix B is defined by B ¼ À 2 (b) (i) Show that B4 ¼ kI , where k is an integer and I is the  identity matrix [2 marks] (ii) Describe the transformation represented by the matrix B as a combination of two geometrical transformations [5 marks] (iii) Find the matrix B17 [2 marks] QUESTION PART REFERENCE Answer space for question (10) P/Jun15/MFP1 Do not write outside the box 11 QUESTION PART REFERENCE Answer space for question s (11) Turn over P/Jun15/MFP1 Do not write outside the box 12 QUESTION PART REFERENCE Answer space for question (12) P/Jun15/MFP1 Do not write outside the box 13 QUESTION PART REFERENCE Answer space for question s (13) Turn over P/Jun15/MFP1 Do not write outside the box 14 A curve C1 has equation x2 y2 À ¼1 16 Sketch the curve C1 , stating the values of its intercepts with the coordinate axes [2 marks] (a)   k , where k < , to give a curve C2 The curve C1 is translated by the vector (b) Given that C2 passes through the origin ð0, 0Þ, find the equations of the asymptotes of C2 [3 marks] QUESTION PART REFERENCE Answer space for question (14) P/Jun15/MFP1 Do not write outside the box 15 QUESTION PART REFERENCE Answer space for question s (15) Turn over P/Jun15/MFP1 Do not write outside the box 16 The equation 2x þ 5x þ 3x À 132 000 ¼ has exactly one real root a (a) (i) Show that a lies in the interval 39 < a < 40 [2 marks] (ii) Taking x1 ¼ 40 as a first approximation to a , use the Newton–Raphson method to find a second approximation, x2 , to a Give your answer to two decimal places [3 marks] Use the formulae for (b) n X r and r¼1 n X n X r to show that r¼1 2rð3r þ 2Þ ¼ nðn þ pÞð2n þ qÞ r¼1 where p and q are integers [5 marks] (c) (i) Express log8 4r in the form lr , where l is a rational number [1 mark] (ii) By first finding a suitable cubic inequality for k , find the greatest value of k for which 60 X ð3r þ 2Þ log8 4r is greater than 106 060 r¼kþ1 [4 marks] QUESTION PART REFERENCE Answer space for question (16) P/Jun15/MFP1 Do not write outside the box 17 QUESTION PART REFERENCE Answer space for question s (17) Turn over P/Jun15/MFP1 Do not write outside the box 18 QUESTION PART REFERENCE Answer space for question (18) P/Jun15/MFP1 Do not write outside the box 19 QUESTION PART REFERENCE Answer space for question s (19) Turn over P/Jun15/MFP1 Do not write outside the box 20 A curve C has equation y¼ xðx À 3Þ x2 þ State the equation of the asymptote of C (a) [1 mark] The line y ¼ k intersects the curve C Show that 4k À 4k À (b) [5 marks] Hence find the coordinates of the stationary points of the curve C (c) (No credit will be given for solutions based on differentiation.) [5 marks] QUESTION PART REFERENCE Answer space for question (20) P/Jun15/MFP1 Do not write outside the box 21 QUESTION PART REFERENCE Answer space for question s (21) Turn over P/Jun15/MFP1 Do not write outside the box 22 QUESTION PART REFERENCE Answer space for question (22) P/Jun15/MFP1 Do not write outside the box 23 QUESTION PART REFERENCE Answer space for question END OF QUESTIONS (23) P/Jun15/MFP1 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/MFP1 [...]... END OF QUESTIONS (23) P/ Jun15 /MFP1 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 /MFP1 ... s (15) Turn over P/ Jun15 /MFP1 Do not write outside the box 16 The equation 2x 3 þ 5x 2 þ 3x À 132 000 ¼ 0 has exactly one real root a 7 (a) (i) Show that a lies in the interval 39 < a < 40 [2 marks] (ii) Taking x1 ¼ 40 as a first approximation to a , use the Newton–Raphson method to find a second approximation, x2 , to a Give your answer to two decimal places [3 marks] Use the formulae... Turn over P/ Jun15 /MFP1 Do not write outside the box 14 A curve C1 has equation 6 x2 y2 À ¼1 9 16 Sketch the curve C1 , stating the values of its intercepts with the coordinate axes [2 marks] (a)   k , where k < 0 , to give a curve C2 The curve C1 is translated by the vector 0 (b) Given that C2 passes through the origin ð0, 0Þ, find the equations of the asymptotes of C2 [3 marks] QUESTION PART REFERENCE... Turn over P/ Jun15 /MFP1 Do not write outside the box 20 A curve C has equation 8 y¼ xðx À 3Þ x2 þ 3 State the equation of the asymptote of C (a) [1 mark] The line y ¼ k intersects the curve C Show that 4k 2 À 4k À 3 4 0 (b) [5 marks] Hence find the coordinates of the stationary points of the curve C (c) (No credit will be given for solutions based on differentiation.) [5 marks] QUESTION PART REFERENCE... (12) P/ Jun15 /MFP1 Do not write outside the box 13 QUESTION PART REFERENCE Answer space for question 5 ... (16) P/ Jun15 /MFP1 Do not write outside the box 17 QUESTION PART REFERENCE Answer space for question 7 ... s (17) Turn over P/ Jun15 /MFP1 Do not write outside the box 18 QUESTION PART REFERENCE Answer space for question 7 ... (18) P/ Jun15 /MFP1 Do not write outside the box 19 QUESTION PART REFERENCE Answer space for question 7 ... s (11) Turn over P/ Jun15 /MFP1 Do not write outside the box 12 QUESTION PART REFERENCE Answer space for question 5 ... (20) P/ Jun15 /MFP1 Do not write outside the box 21 QUESTION PART REFERENCE Answer space for question 8