AQA MFP2 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 MFP2 Unit Further Pure Tuesday 16 June 2015 1.30 pm to 3.00 pm 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 (JUN15MFP201) P88824/Jun15/E4 MFP2 Do not write outside the box Answer all questions Answer each question in the space provided for that question Express (a) A B þ in the form , where A and B are integers ðr þ 2Þr! ðr þ 1Þ! ðr þ 2Þ! [3 marks] Hence find (b) QUESTION PART REFERENCE n X ðr þ 2Þr! r¼1 [2 marks] Answer space for question (02) P/Jun15/MFP2 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (03) Turn over P/Jun15/MFP2 Do not write outside the box Sketch the graph of y ¼ x and state the equations of its asymptotes (a) [3 marks] Use the definitions of sinh x and cosh x in terms of e x and eÀx to show that (b) sech2 x þ tanh2 x ¼ [3 marks] Solve the equation sech2 x ¼ þ x , giving your answers in terms of natural logarithms [5 marks] (c) QUESTION PART REFERENCE Answer space for question (a) y~ ~ O x (04) P/Jun15/MFP2 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (05) Turn over P/Jun15/MFP2 Do not write outside the box A curve C is defined parametrically by t2 þ , y ¼ ln t t   2  2  dx dy Show that þ ¼ 1þ t dt dt x¼ (a) [4 marks] The arc of C from t ¼ to t ¼ is rotated through 2p radians about the x-axis Find the area of the surface generated, giving your answer in the form pðm ln þ nÞ , where m and n are integers [5 marks] (b) QUESTION PART REFERENCE Answer space for question (06) P/Jun15/MFP2 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (07) Turn over P/Jun15/MFP2 Do not write outside the box The expression f ðnÞ is given by f ðnÞ ¼ 24nþ3 þ 33nþ1 (a) Show that f ðk þ 1Þ À 16f ðkÞ can be expressed in the form A Â 33k , where A is an integer [3 marks] (b) Prove by induction that f ðnÞ is a multiple of 11 for all integers n [4 marks] QUESTION PART REFERENCE Answer space for question (08) P/Jun15/MFP2 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (09) Turn over P/Jun15/MFP2 Do not write outside the box 10 The locus of points, L , satisfies the equation jz À þ 4ij ¼ jzj Sketch L on the Argand diagram below (a) [3 marks] The locus L cuts the real axis at A and the imaginary axis at B (b) (i) Show that the complex number represented by C , the midpoint of AB, is 5 À i [4 marks] (ii) The point O is the origin of the Argand diagram Find the equation of the circle that passes through the points O, A and B, giving your answer in the form jz À aj ¼ k [2 marks] QUESTION PART REFERENCE Answer space for question (a) Im(z) ~ 5– – – – – ~ – – – – O – – – – – – À5 Re(z) – – – – À5 – (10) P/Jun15/MFP2 Do not write outside the box 11 QUESTION PART REFERENCE Answer space for question s (11) Turn over P/Jun15/MFP2 Do not write outside the box 12   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x À À1 Given that y ¼ ðx À 2Þ þ 4x À x þ sin , show that (a) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy ¼ k þ 4x À x dx where k is an integer [5 marks] Hence show that (b) ð pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi þ 4x À x dx ¼ p þ qp where p and q are rational numbers [3 marks] QUESTION PART REFERENCE Answer space for question (12) P/Jun15/MFP2 Do not write outside the box 13 QUESTION PART REFERENCE Answer space for question s (13) Turn over P/Jun15/MFP2 Do not write outside the box 14 The cubic equation 27z þ kz þ ¼ has roots a , b and g Write down the values of ab þ bg þ ga and abg (a) [2 marks] (b) (i) In the case where b ¼ g , find the roots of the equation [5 marks] (ii) Find the value of k in this case [1 mark] (c) (i) In the case where a ¼ À i , find a and a [2 marks] (ii) Hence find the value of k in this case [2 marks] In the case where k ¼ À12 , find a cubic equation with integer coefficients which has (d) roots 1 þ , þ and þ a b g [5 marks] QUESTION PART REFERENCE Answer space for question (14) P/Jun15/MFP2 Do not write outside the box 15 QUESTION PART REFERENCE Answer space for question s (15) Turn over P/Jun15/MFP2 Do not write outside the box 16 QUESTION PART REFERENCE Answer space for question (16) P/Jun15/MFP2 Do not write outside the box 17 QUESTION PART REFERENCE Answer space for question s (17) Turn over P/Jun15/MFP2 Do not write outside the box 18 The complex number o is given by o ¼ cos (a) (i) 2p 2p þ i sin 5 Verify that o is a root of the equation z5 ¼ [1 mark] (ii) Write down the three other non-real roots of z5 ¼ , in terms of o [1 mark] (b) (i) Show that þ o þ o þ o þ o ¼ [1 mark]  (ii) Hence show that oþ o 2   À ¼ þ oþ o [2 marks] 2p ¼ Hence show that cos (c) pffiffiffi 5À1 [4 marks] QUESTION PART REFERENCE Answer space for question (18) P/Jun15/MFP2 Do not write outside the box 19 QUESTION PART REFERENCE Answer space for question s (19) Turn over P/Jun15/MFP2 Do not write outside the box 20 QUESTION PART REFERENCE Answer space for question END OF QUESTIONS Copyright ª 2015 AQA and its licensors All rights reserved (20) P/Jun15/MFP2 [...]... over P/ Jun15 /MFP2 Do not write outside the box 12   p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x À 2 À1 Given that y ¼ ðx À 2Þ 5 þ 4x À x 2 þ 9 sin , show that 3 6 (a) p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy ¼ k 5 þ 4x À x 2 dx where k is an integer [5 marks] Hence show that (b) ð 7 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffi 2 5 þ 4x À x 2 dx ¼ p 3 þ qp 2 where p and q are rational numbers [3 marks] QUESTION PART REFERENCE Answer space... (17) Turn over P/ Jun15 /MFP2 Do not write outside the box 18 The complex number o is given by o ¼ cos 8 (a) (i) 2p 2p þ i sin 5 5 Verify that o is a root of the equation z5 ¼ 1 [1 mark] (ii) Write down the three other non-real roots of z5 ¼ 1 , in terms of o [1 mark] (b) (i) Show that 1 þ o þ o 2 þ o 3 þ o 4 ¼ 0 [1 mark]  (ii) Hence show that 1 oþ o 2   1 À 1 ¼ 0 þ oþ o [2 marks] 2p ¼ Hence show... [5 marks] QUESTION PART REFERENCE Answer space for question 7 (14) P/ Jun15 /MFP2 Do not write outside... s (15) Turn over P/ Jun15 /MFP2 Do not write outside the box 16 QUESTION PART REFERENCE Answer space for question 7 ... (16) P/ Jun15 /MFP2 Do not write outside the box 17 QUESTION PART REFERENCE Answer space for question 7 ... (12) P/ Jun15 /MFP2 Do not write outside the box 13 QUESTION PART REFERENCE Answer space for question 6 ... (18) P/ Jun15 /MFP2 Do not write outside the box 19 QUESTION PART REFERENCE Answer space for question 8 ... s (19) Turn over P/ Jun15 /MFP2 Do not write outside the box 20 QUESTION PART REFERENCE Answer space for question 8 ... END OF QUESTIONS Copyright ª 2015 AQA and its licensors All rights reserved (20) P/ Jun15 /MFP2 ... s (13) Turn over P/ Jun15 /MFP2 Do not write outside the box 14 The cubic equation 27z 3 þ kz 2 þ 4 ¼ 0 has roots a , b and g 7 Write down the values of ab þ bg þ ga and abg (a) [2 marks] (b) (i) In the case where b ¼ g , find