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Cambridge International Examinations

This document is the front cover and first two questions of a Cambridge International A-Level Mathematics exam from May/June 2017. The first question asks students to solve the equation ln x^2 + 1 = 1 + 2 ln x, giving their answer to 3 significant figures. The second question asks students to solve the inequality |x - 3| < 3x - 4. The document provides the standard instructions and information for candidates taking the exam, including specifying the number of marks allocated to each question.

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0% found this document useful (0 votes)

42 views20 pages

Cambridge International Examinations

Uploaded by

loycoy008

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Download as PDF, TXT or read online on Scribd

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Cambridge International Examinations

Cambridge International Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*9035132693*

NUMBER NUMBER

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 (P3) May/June 2017
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name in the spaces at the top of this page.
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 fluid.
DO NOT WRITE IN ANY BARCODES.

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.
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.

This document consists of 19 printed pages and 1 blank page.

JC17 06_9709_32/2R
© UCLES 2017 [Turn over
2

1 Solve the equation ln x2 + 1 = 1 + 2 ln x, giving your answer correct to 3 significant figures. [3]

................................................................................................................................................................

© UCLES 2017 9709/32/M/J/17

2 Solve the inequality x − 3 < 3x − 4. [4]

................................................................................................................................................................

© UCLES 2017 9709/32/M/J/17 [Turn over

3 (i) Express the equation cot 1 − 2 tan 1 = sin 21 in the form a cos4 1 + b cos2 1 + c = 0, where a, b
and c are constants to be determined. [3]

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17

(ii) Hence solve the equation cot 1 − 2 tan 1 = sin 21 for 90Å < 1 < 180Å. [2]

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17 [Turn over

4 The parametric equations of a curve are

x = t2 + 1, y = 4t + ln 2t − 1.

dy
(i) Express in terms of t. [3]
dx

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17

(ii) Find the equation of the normal to the curve at the point where t = 1. Give your answer in the
form ax + by + c = 0. [3]

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17 [Turn over

5 In a certain chemical process a substance A reacts with and reduces a substance B. The masses of A
= −0.2xy and
dy
and B at time t after the start of the process are x and y respectively. It is given that
dt
x= . At the beginning of the process y = 100.
10
1 + t2

(i) Form a differential equation in y and t, and solve this differential equation. [6]

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17

........................................................................................................................................................

(ii) Find the exact value approached by the mass of B as t becomes large. State what happens to the
mass of A as t becomes large. [2]

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17 [Turn over

6 Throughout this question the use of a calculator is not permitted.

The complex number 2 − i is denoted by u.

(i) It is given that u is a root of the equation x3 + ax2 − 3x + b = 0, where the constants a and b are
real. Find the values of a and b. [4]

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17

(ii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying both the inequalities z − u < 1 and z < z + i. [4]

© UCLES 2017 9709/32/M/J/17 [Turn over

(i) Prove that if y = = sec 1 tan 1.

1 dy
cos 1 d1
7 then [2]

........................................................................................................................................................

1 + sin 1
2 sec2 1 + 2 sec 1 tan 1 − 1.
1 − sin 1
(ii) Prove the identity [3]

........................................................................................................................................................

© UCLES 2017 9709/32/M/J/17

........................................................................................................................................................

10
1 + sin 1
(iii) Hence find the exact value of Ô d1.
4

1 − sin 1
[4]
0

........................................................................................................................................................

5x2 − 7x + 4
Let f x =
3x + 2 x2 + 5
8 .

(i) Express f x in partial fractions. [5]

........................................................................................................................................................

(ii) Hence obtain the expansion of f x in ascending powers of x, up to and including the term in x2 .
[5]

........................................................................................................................................................

16
−−→
9 Relative to the origin O, the point A has position vector given by OA = i + 2j + 4k. The line l has
equation r = 9i − j + 8k + - 3i − j + 2k.

(i) Find the position vector of the foot of the perpendicular from A to l. Hence find the position
vector of the reflection of A in l. [5]

........................................................................................................................................................

(ii) Find the equation of the plane through the origin which contains l. Give your answer in the form
ax + by + cz = d. [3]

........................................................................................................................................................

(iii) Find the exact value of the perpendicular distance of A from this plane. [3]

........................................................................................................................................................

10
y

x
O p 1
40

The diagram shows the curve y = x2 cos 2x for 0 ≤ x ≤ 14 0. The curve has a maximum point at M
where x = p.
@ A
(i) Show that p satisfies the equation p = 12 tan−1
1
. [3]
p

........................................................................................................................................................

A @
(ii) Use the iterative formula pn+1 = tan −1
1 1
to determine the value of p correct to 2 decimal
2
pn
places. Give the result of each iteration to 4 decimal places. [3]

........................................................................................................................................................

(iii) Find, showing all necessary working, the exact area of the region bounded by the curve and the
x-axis. [5]

........................................................................................................................................................

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.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
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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Cambridge International Examinations

Uploaded by

Cambridge International Examinations

Uploaded by

Cambridge International Examinations

Cambridge International Advanced Level

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 19 printed pages and 1 blank page.

© UCLES 2017 9709/32/M/J/17

2 Solve the inequality  x − 3 < 3x − 4. [4]

© UCLES 2017 9709/32/M/J/17 [Turn over

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17 [Turn over

4 The parametric equations of a curve are

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17 [Turn over

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17 [Turn over

6 Throughout this question the use of a calculator is not permitted.

The complex number 2 − i is denoted by u.

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17 [Turn over

(i) Prove that if y = = sec 1 tan 1.

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17 [Turn over

(i) Express f x in partial fractions. [5]

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17 [Turn over

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17 [Turn over

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17

© UCLES 2017 9709/32/M/J/17

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2 Solve the inequality x − 3 < 3x − 4. [4]