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Cambridge International Examinations

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Cambridge International Examinations

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Cambridge International Examinations

Cambridge International Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*8927566197*

NUMBER NUMBER

MATHEMATICS 9709/31
Paper 3 Pure Mathematics 3 (P3) October/November 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 11_9709_31/FP
© UCLES 2017 [Turn over
2

1 Find the quotient and remainder when x4 is divided by x2 + 2x − 1. [3]

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

© UCLES 2017 9709/31/O/N/17

2 Two variable quantities x and y are believed to satisfy an equation of the form y = C ax , where C and
a are constants. An experiment produced four pairs of values of x and y. The table below gives the
corresponding values of x and ln y.

x 0.9 1.6 2.4 3.2

ln y 1.7 1.9 2.3 2.6

By plotting ln y against x for these four pairs of values and drawing a suitable straight line, estimate
the values of C and a. Give your answers correct to 2 significant figures. [5]

ln y

x
0 1 2 3 4

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

© UCLES 2017 9709/31/O/N/17 [Turn over

3 The equation x3 = 3x + 7 has one real root, denoted by !.

(i) Show by calculation that ! lies between 2 and 3. [2]

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

© UCLES 2017 9709/31/O/N/17

Two iterative formulae, A and B, derived from this equation are as follows:

xn+1 = 3xn + 7 3 ,
1
(A)
xn3 − 7
xn+1 = . (B)
3
Each formula is used with initial value x1 = 2.5.

(ii) Show that one of these formulae produces a sequence which fails to converge, and use the other
formula to calculate ! correct to 2 decimal places. Give the result of each iteration to 4 decimal
places. [4]

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

© UCLES 2017 9709/31/O/N/17 [Turn over

4 (i) Prove the identity tan 45Å + x + tan 45Å − x 2 sec 2x. [4]

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

© UCLES 2017 9709/31/O/N/17

(ii) Hence sketch the graph of y = tan 45Å + x + tan 45Å − x for 0Å ≤ x ≤ 90Å. [3]

© UCLES 2017 9709/31/O/N/17 [Turn over

5 The equation of a curve is 2x4 + xy3 + y4 = 10.

dy 8x3 + y3
(i) Show that =− . [4]
dx 3xy2 + 4y3

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

© UCLES 2017 9709/31/O/N/17

(ii) Hence show that there are two points on the curve at which the tangent is parallel to the x-axis
and find the coordinates of these points. [4]

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

© UCLES 2017 9709/31/O/N/17 [Turn over

6 The variables x and y satisfy the differential equation

dy
= 4 cos2 y tan x,
dx
for 0 ≤ x < 12 0, and x = 0 when y = 14 0. Solve this differential equation and find the value of x when
y = 13 0. [8]

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

© UCLES 2017 9709/31/O/N/17

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

© UCLES 2017 9709/31/O/N/17 [Turn over

7 (a) The complex number u is given by u = 8 − 15i. Showing all necessary working, find the two
square roots of u. Give answers in the form a + ib, where the numbers a and b are real and exact.
[5]

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

© UCLES 2017 9709/31/O/N/17

(b) On an Argand diagram, shade the region whose points represent complex numbers satisfying
both the inequalities z − 2 − i ≤ 2 and 0 ≤ arg z − i ≤ 14 0. [4]

4x2 + 9x − 8
8 Let f x = .
x + 2 2x − 1

B C
(i) Express f x in the form A + + . [4]
x + 2 2x − 1

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

15
4
(ii) Hence show that Ó f x dx = 6 + 12 ln 16
7
. [5]
1

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

9
y

R
x
O 2

− 12 x
The diagram shows the curve y = 1 + x2 e for x ≥ 0. The shaded region R is enclosed by the curve,
the x-axis and the lines x = 0 and x = 2.

(i) Find the exact values of the x-coordinates of the stationary points of the curve. [4]

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

42
(ii) Show that the exact value of the area of R is 18 − . [5]
e

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

10 The equations of two lines l and m are r = 3i − j − 2k + , −i + j + 4k and r = 4i + 4j − 3k + - 2i + j − 2k

respectively.

(i) Show that the lines do not intersect. [3]

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

(ii) Calculate the acute angle between the directions of the lines. [3]

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

(iii) Find the equation of the plane which passes through the point 3, −2, −1 and which is parallel
to both l and m. Give your answer in the form ax + by + cz = d . [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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