Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level

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1 Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level CANDIDATE NAME * * CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/01 Paper 1 Pure Mathematics 1 (P1) For Examination from 2017 SPECIMEN PAPER Candidates answer on the Question Paper. Additional Materials: List of Formulae(MF9) 1 hour 45 minutes READ THESE INSTRUCTIONS FIRST WriteyourCentrenumber,candidatenumberandnameinthespacesatthetopofthispage. Writeindarkblueorblackpen. YoumayuseanHBpencilforanydiagramsorgraphs. Do not use staples, paper clips, glue or correction fluid. DONOTWRITEINANYBARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion. Thetotalnumberofmarksforthispaperis75. Thisdocumentconsistsof19printedpagesand1blankpage. UCLES2016 [Turn over

2 2 1 In the expansion of 1 2x a +x 5, where a is a non-zero constant, show that the coefficient of x 2 a is zero. [3] Thefunctionfis such thatf x = 3x 2 7 andf 3 = 5. Findf x. [3]

3 3 3 Solvetheequation sin 1 4x 4 +x 2 = 1. [4] [Turn over

4 4 4 (i) Show thattheequation 4cos tan +15 = 0canbeexpressed as 4sin 2 15sin 4 = 0. [3]

5 (ii) Hencesolvethe equation 4cos tan = 0for [3] [Turn over

6 6 5 Acurvehas equationy= 8 x +2x. (i) Find dy dx and d2 y dx2. [3]

7 7 (ii) Find the coordinates of the stationary points and state, with a reason, the nature of each stationary point. [5] [Turn over

8 6 Acurvehas equationy=x 2 x+3 and alinehas equationy=3x +a,whereais aconstant. 8 (i) Show that the x-coordinates of the points of intersection of the line and the curve are given by theequationx 2 4x + 3 a = 0. [1] (ii) For the case where the line intersects the curve at two points, it is given that thex-coordinate of one of the points of intersection is 1. Find the x-coordinate of the other point of intersection. [2]

9 9 (iii) For the case where the line is a tangent to the curve at a point P, find the value of a and the coordinates of P. [4] [Turn over

10 10 7 C r B A E D The diagram shows a circle with centre A and radius r. Diameters CAD and BAE are perpendicular to each other. Alargercirclehas centrebandpassesthroughcandd. (i) Show thattheradius ofthelargercircleisr 2. [1] (ii) Findthearea oftheshadedregioninterms ofr. [6]

11 11 [Turn over

12 8 Thefirst termofaprogression is 4xand thesecond term isx (i) For the case where the progression is arithmetic with a common difference of 12, find the possible values of x and the corresponding values of the third term. [4]

13 13 (ii) Forthecasewheretheprogression is geometricwith asum toinfinity of8,find thethird term. [4] [Turn over

14 14 9 (i) Express x 2 +6x 5intheforma x +b 2 +c,wherea,bandcareconstants. [3]

15 15 Thefunctionf:x x 2 +6x 5 is defined forx m, wheremisaconstant. (ii) Statethesmallest valueofmforwhich fis one-one. [1] (iii) Forthecasewherem=5,findanexpressionforf 1 x andstatethedomain off 1. [4] [Turn over

16 10 16 S R P Q T 8 cm 10 cm C M B k j 6 cm O i a cm A The diagram shows a cuboid OABCPQRS with a horizontal base OABC in which AB = 6cm and OA = acm,whereais a constant. The heightopof thecuboid is 10cm. The pointt onbr is such thatbt = 8cm,andM isthemid-pointofat. Unitvectorsi,jandkareparalleltoOA,OC andop respectively. (i) Forthecasewherea=2,findtheunit vectorin thedirection of PM. [4]

17 17 (ii) ForthecasewhereangleATP = cos 1 2 7,find thevalueofa. [5] [Turn over

18 11 y 18 P 6, 5 y = 1 + 4x 1 2 O Q 8, 0 x The diagram shows part of the curve y = 1 +4x 1 2 and a point P 6, 5 lying on the curve. The line PQintersectsthex-axis atq 8, 0. (i) Show thatpq is anormal to thecurve. [5]

19 19 (ii) Find, showing all necessary working, the exact volume of revolution obtained when the shaded region is rotated through360 about thex-axis. [7] [Inpart(ii)youmayfinditusefultoapplythefactthatthevolume,V,ofaconeofbaseradiusr andvertical heighth,is given byv = 1 3 r2 h.]

20 20 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher(ucles) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate(UCLES), which is itself a department of the University of Cambridge.

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