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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/13
FURTHER MATHEMATICS
Paper 1
May/June 2013
3 hours
*7566713344*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10)
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 a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
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 a calculator is expected, where appropriate.
Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive
credit.
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.
This document consists of 4 printed pages.
JC13 06_9231_13/RP
© UCLES 2013
[Turn over
2
1
2
Let f r = r! r − 1. Simplify f r + 1 − f r and hence find
Ð
2n
r=n+1
r! r2 + 1.
[5]
The roots of the equation x4 − 4x2 + 3x − 2 = 0 are !, ", ' and $ ; the sum !n + "n + ' n + $ n is denoted
by Sn . By using the relation y = x2 , or otherwise, show that !2 , "2 , ' 2 and $ 2 are the roots of the
equation
y4 − 8y3 + 12y2 + 7y + 4 = 0.
3
State the value of S2 and hence show that
S8 = 8S6 − 12S4 − 72.
3
Prove by mathematical induction that, for every positive integer n,
dn x
e sin x = ï2n ex sin x + 14 n0.
dxn
4
3
7
dy
= − 43 at the point A 1, −2 on the curve with equation
dx
Show that
y3 − 3x2 y + 2 = 0,
and find the value of
1
5
d2 y
at A.
dx2
Show that Ó xe−x dx =
2
0
1
[8]
1
1
−
.
2 2e
[2]
Let In = Ó xn e−x dx. Show that I2n+1 = nI2n−1 −
2
0
1
for n ≥ 1.
2e
Find the exact value of I7 .
6
[3]
[3]
The linear transformation T : >4 → >4 is represented by the matrix M, where
€

−2
5
3
−1
−4
−2
0
1
M=
6 −14 −13
1
!
! −2! −11!
and ! is a constant. The null space of T is denoted by K1 when ! ≠ 0, and by K2 when ! = 0. Find a
basis for K1 and a basis for K2 .
[8]
© UCLES 2013
9231/13/M/J/13
3
7
Find the value of the constant , such that ,xe−x is a particular integral of the differential equation
dy
d2 y
+5
+ 4y = 6e−x .
2
dx
dx
4
Find the solution of the differential equation for which y = 2 and
8
9
dy
= 3 when x = 0.
dx
The curve C has parametric equations x = 32 t2 , y = t3 , for 0 ≤ t ≤ 2. Find the arc length of C.
[4]
Find the coordinates of the centroid of the region enclosed by C, the x-axis and the line x = 6.
[7]
The square matrix A has an eigenvalue , with corresponding eigenvector e. The non-singular matrix
M is of the same order as A. Show that Me is an eigenvector of the matrix B, where B = MAM−1 ,
and that , is the corresponding eigenvalue.
[3]
Let
A=
` −1
0
0
2
1
0
1a
4 .
2
Write down the eigenvalues of A and obtain corresponding eigenvectors.
Given that
M=
`1
0
0
0
1
0
[4]
1a
0 ,
1
find the eigenvalues and corresponding eigenvectors of B.
10
[6]
[4]
Use the identity 2 sin P cos Q  sin P + Q + sin P − Q to show that
2 sin 1 cos 1 − 14 0  cos 21 − 34 0 +
1
.
ï2
3
A curve has polar equation r = 2 sin 1 cos 1 − 14 0, for 0 ≤ 1 ≤ 34 0. Sketch the curve and state the polar
equation of its line of symmetry, justifying your answer.
[3]
Show that the area of the region enclosed by the curve is
3
8
0 + 1.
[6]
[Question 11 is printed on the next page.]
© UCLES 2013
9231/13/M/J/13
[Turn over
4
11
Answer only one of the following two alternatives.
EITHER
The line l1 passes through the point A whose position vector is 4i + 7j − k and is parallel to the vector
3i + 2j − k. The line l2 passes through the point B whose position vector is i + 7j + 11k and is parallel
to the vector i − 6j − 2k. The points P on l1 and Q on l2 are such that PQ is perpendicular to both l1
and l2 . Find the position vectors of P and Q.
[8]
Find the shortest distance between the line through A and B and the line through P and Q, giving your
answer correct to 3 significant figures.
[6]
OR
Show the cube roots of 1 on an Argand diagram.
[1]
Show that the two non-real cube roots can be expressed in the form 7 and 72 , and find these cube
[3]
roots in exact cartesian form x + iy.
Evaluate the determinant
~
1
37 2
27
37
2
72
~
27 2
7 .
3
It is given that Ï = 4ï3 cos 34 0 + i sin 34 0 − 4 cos
r cos 1 + i sin 1, giving exact values for r and 1.
11 0
6
Hence find the cube roots of Ï in the form r cos 1 + i sin 1.
3
+ i sin
11 0
6
.
Express Ï in the form
[5]
[2]
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University of 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 2013
9231/13/M/J/13