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9709


Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS Paper 1 Pure Mathematics 1 (P1)

9709/11
October/Nove

mber 2014 1 hour 45 minutes

Additional Materials:

Answer Booklet/Paper Graph Paper List of Formulae (MF9)

*1395233496*

READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction ?uid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 signi?cant ?gures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is speci?ed 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. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 75. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.

This document consists of 4 printed pages.
JC14 11_9709_11/RP ? UCLES 2014

[Turn over

2 1 In the expansion of 2 + ax 7 , the coef?cient of x is equal to the coef?cient of x2 . Find the value of [3] the non-zero constant a. Find the value of x satisfying the equation sin?1 x ? 1 = tan?1 3 . 13 sin2 2 + cos [3]

2

3

Solve the equation

+ cos

= 2 for 0 ≤

≤ 180 .

[4]

4

The line 4x + ky = 20 passes through the points A 8, ?4 and B b, 2b , where k and b are constants. (i) Find the values of k and b. (ii) Find the coordinates of the mid-point of AB. [4] [1]

5

Find the set of values of k for which the line y = 2x ? k meets the curve y = x2 + kx ? 2 at two distinct points. [5] Relative to an origin O, the position vector of A is 3i + 2j ? k and the position vector of B is 7i ? 3j + k. (i) Show that angle OAB is a right angle. (ii) Find the area of triangle OAB. [4] [3]

6

7

(i) A geometric progression has ?rst term a (a ≠ 0), common ratio r and sum to in?nity S. A second geometric progression has ?rst term a, common ratio 2r and sum to in?nity 3S. Find the value [3] of r. (ii) An arithmetic progression has ?rst term 7. The nth term is 84 and the (3n)th term is 245. Find [4] the value of n.

? UCLES 2014

9709/11/O/N/14

3 8

A C

O

rad 4 cm

D

B

In the diagram, AB is an arc of a circle with centre O and radius 4 cm. Angle AOB is radians. The point D on OB is such that AD is perpendicular to OB. The arc DC, with centre O, meets OA at C. (i) Find an expression in terms of for the perimeter of the shaded region ABDC. [4]

(ii) For the case where = 1 6 , ?nd the area of the shaded region ABDC, giving your answer in the form k , where k is a constant to be determined. [4] 2 . The curve y = f x passes through x2 [3] [4]

9

The function f is de?ned for x > 0 and is such that f x = 2x ? the point P 2, 6 . (i) Find the equation of the normal to the curve at P. (ii) Find the equation of the curve.

(iii) Find the x-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum. [4] (i) Express x2 ? 2x ? 15 in the form x + a
2

10

+ b.

[2]

The function f is de?ned for p ≤ x ≤ q, where p and q are positive constants, by f : x → x2 ? 2x ? 15. The range of f is given by c ≤ f x ≤ d, where c and d are constants. (ii) State the smallest possible value of c. For the case where c = 9 and d = 65, (iii) ?nd p and q, (iv) ?nd an expression for f ?1 x . [4] [3] [1]

[Question 11 is printed on the next page.]

? UCLES 2014

9709/11/O/N/14

[Turn over

4 11

y

Q 2, 3
y = 4x + 1
1 2

2 y= 1 2x + 1

P 0, 1

O
1

x

2 The diagram shows parts of the curves y = 4x + 1 2 and y = 1 2 x + 1 intersecting at points P 0, 1 and Q 2, 3 . The angle between the tangents to the two curves at Q is .

(i) Find , giving your answer in degrees correct to 3 signi?cant ?gures. (ii) Find by integration the area of the shaded region.

[6] [6]

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. ? UCLES 2014 9709/11/O/N/14


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