A level Mathematics Question paper Pure Core 3 June 2015

1 Centre NumberCandidate NumberSurnameOther NamesCandidate SignatureGeneral Certificate of EducationAdvanced level ExaminationJune 2015 MathematicsMPC3 Unit Pure Core 3 Friday 5 June 2015 am to amFor this paper you must have:*the blue AQA booklet of formulae and statistical may use a graphics allowed*1 hour 30 minutesInstructions*Use black ink or black ball-point pen. Pencil should only be used fordrawing.*Fill in the boxes at the top of this page.*Answerallquestions.*Write the Question part reference (eg (a), (b)(i) etc) in the left-handmargin.

2 *You must answer each Question in the space provided for thatquestion. If you require extra space, use an AQA supplementaryanswer book; donotuse 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 donot want to be *The marks for questions are shown in brackets.*The maximum mark for this paper is *Unless stated otherwise, you may quote formulae, without proof, fromthe booklet.

3 *You do not necessarily need to use all the space Examiner s UseExaminer s InitialsQuestionMark12345678 TOTALP88756/Jun15/E4 MPC3(JUN15 MPC301) each Question in the space provided for that (a)Use the mid-ordinate rule with four strips to find an estimate for 5:51:5e2 xln 3x 2 dx,giving your answer to three decimal places.[4 marks](b)Find the exact value of the gradient of the curvey e2 xln 3x 2 at the point onthe curve wherex 2.[4 marks]Answer space for Question not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(02)3 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(03)QUESTIONPARTREFERENCE4~~yxO2 (a)Sketch, on the axes below, the curve with equationy 4 j2x 1j, indicating thecoordinates where the curve crosses the axes.

4 [4 marks](b)Solve the equationx 4 j2x 1j.[3 marks](c)Solve the inequalityx<4 j2x 1j.[2 marks](d)Describe a sequence of two geometrical transformations that maps the graph ofy j2x 1jonto the graph ofy 4 j2x 1j.[4 marks]Answer space for Question 2(a)..Do not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(04)5 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(05)QUESTIONPARTREFERENCE63 (a)It is given that the curves with equationsy 6lnxandy 8x x2 3intersect ata single point wherex a.

5 (i)Show thatalies between5and6.[2 marks](ii)Show that the equationx 4 ffiffiffiffiffiffiffiffiffiffiffiffiffif fiffiffiffiffiffiffiffi13 6lnxpcan be rearranged into the form6lnx x2 8x 3 0[3 marks](iii)Use the iterative formulaxn 1 4 ffiffiffiffiffiffiffiffiffiffiffiffiffif fiffiffiffiffiffiffiffiffiffi13 6lnxnpwithx1 5to find the values ofx2andx3, giving your answers to three decimalplaces.[2 marks](b)A curve has equationy f x wheref x 6lnx x2 8x 3.(i)Find the exact values of the coordinates of the stationary points of the curve.

6 [5 marks](ii)Hence, or otherwise, find the exact values of the coordinates of the stationary pointsof the curve with equationy 2f x 4 [2 marks]Answer space for Question not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(06)7 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(07)QUESTIONPARTREFERENCE8 Answer space for Question not writeoutside theboxP/Jun15/MPC3(08)QUESTIONPARTREFERE NCE9 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(09)QUESTIONPARTREFERENCE104 The functionsfandgare defined byf x 5 e3x,for all real values ofxg x 12x 3,forx6 1:5(a)Find the range off.

7 [2 marks](b)The inverse offisf 1.(i)Findf 1 x .[3 marks](ii)Solve the equationf 1 x 0.[1 mark](c)Find an expression forgg x , giving your answer in the formax bcx d, wherea,b,canddare integers.[3 marks]Answer space for Question not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(10)11 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn oversQUESTIONPARTREFERENCE(11)125 (a)By writingtanxassinxcosx, use the quotient rule to show thatddx tanx sec2x.[2 marks](b)Use integration by parts to find xsec2xdx.

8 [4 marks](c)The region bounded by the curvey 5ffiffiffixp secx, thex-axis from0to1and the linex 1is rotated through2pradians about thex-axis to form a the value of the volume of the solid generated, giving your answer to twosignificant figures.[3 marks]Answer space for Question not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(12)13 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(13)QUESTIONPARTREFERENCE14~~yxO6 (a)Sketch, on the axes below, the curve with equationy sin 1 3x , whereyis the exact values of the coordinates of the end points of the graph.

9 [3 marks](b)Given thatx 13siny, write downdxdyand hence finddydxin terms ofy.[2 marks]Answer space for Question 6(a)..Do not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(14)15 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(15)QUESTIONPARTREFERENCE167 Use the substitutionu 6 x2to find the value of 21x3ffiffiffiffiffiffiffiffiffiffiffiffi ffi6 x2pdx, giving youranswer in the formpffiffiffi5p qffiffiffi2p, wherepandqare rational numbers.[7 marks]Answer space for Question not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(16)17 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(17)QUESTIONPARTREFERENCE188 (a)Show that the equation4 cosec2y cot2y k, wherek6 4, can be written in theformsec2y k 1k 4[5 marks](b)Hence, or otherwise, solve the equation4 cosec2 2x 75 cot2 2x 75 5giving all values ofxin the interval0 <x<180.

10 [5 marks]Answer space for Question not writeoutside theboxP/Jun15/MPC3 QUESTIONPARTREFERENCE(18)19 Answer space for Question not writeoutside theboxP/Jun15/MPC3 Turn overs(19)QUESTIONPARTREFERENCE20 Answer space for Question OF QUESTIONSC opyright 2015 AQA and its licensors. All rights not writeoutside theboxP/Jun15/MPC3(20)QUESTIONPARTREFERE NCE