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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/13
FURTHER MATHEMATICS
Paper 1
October/November 2013
3 hours
*7188280838*
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 11_9231_13/RP
© UCLES 2013
[Turn over
2
1
Express
1
in partial fractions.
r r + 1 r − 1
Find
[1]
Ð r r + 1 r − 1 .
4
Ð r r + 1 r − 1 .
1
n
1
r=2
State the value of
∞
1
r=2
`1
2
Show that the matrix
3
3
4
0
−3
2a
−2 has no inverse.
−4
Solve the system of equations
3
[2]
x + 4y + 2Ï = 0,
3x
− 2Ï = 4,
3x − 3y − 4Ï = 5.
[4]
Find the general solution of the differential equation
dy
d2 y
+2
+ 4y = 4x2 + 8.
2
dx
dx
4
7
A curve has parametric equations
x = 21 − sin 21,
y = 1 − cos 21,
for −30 ≤ 1 ≤ 30.
Show that
dy
= cot 1,
dx
except for certain values of 1, which should be stated.
Find the value of
5
d2 y
when 1 = 14 0.
dx2
[4]
[3]
The equation
8x3 + 36x2 + kx − 21 = 0,
where k is a constant, has roots a − d, a, a + d. Find the numerical values of the roots and determine
the value of k.
[8]
© UCLES 2013
9231/13/O/N/13
3
6
7
[In this question you may use, without proof, the formula Ó sec x dx = ln sec x + tan x + const.]
(a) Let y = sec x. Find the mean value of y with respect to x over the interval 61 0 ≤ x ≤ 13 0.
[4]
(b) The curve C has equation y = − ln cos x, for 0 ≤ x ≤ 13 0. Find the arc length of C.
[4]
The curve C has equation
2x2 + 5x − 1
.
x+2
y=
Find the equations of the asymptotes of C.
Show that
[3]
dy
> 2 at all points on C.
dx
[3]
Sketch C.
8
[3]
The points A, B, C have position vectors
4i + 5j + 6k,
5i + 7j + 8k,
2i + 6j + 4k,
respectively, relative to the origin O. Find a cartesian equation of the plane ABC.
[4]
The point D has position vector 6i + 3j + 6k. Find the coordinates of E, the point of intersection of
the line OD with the plane ABC.
[4]
Find the acute angle between the line ED and the plane ABC.
9
[3]
Prove by mathematical induction that, for every positive integer n,
cos 1 + i sin 1n = cos n1 + i sin n1.
5
Express sin5 1 in the form p sin 51 + q sin 31 + r sin 1, where p, q and r are rational numbers to be
determined.
[6]
10
The curve C has polar equation r = 2 sin 1 1 − cos 1, for 0 ≤ 1 ≤ 0. Find
coordinates of the point of C that is furthest from the pole.
dr
and hence find the polar
d1
[5]
Sketch C.
[2]
Find the exact area of the sector from 1 = 0 to 1 = 14 0.
[6]
[Question 11 is printed on the next page.]
© UCLES 2013
9231/13/O/N/13
[Turn over
4
11
Answer only one of the following two alternatives.
EITHER
1
Let In = Ó 1 + x2 dx. Show that, for all integers n,
n
0
2n + 1In = 2nIn−1 + 2n .
5
Evaluate I0 and hence find I3 .
[4]
Given that I−1 = 14 0, find I−3 .
[5]
OR
The vector e is an eigenvector of each of the 3 × 3 matrices A and B, with corresponding eigenvalues
, and - respectively. Justifying your answer, state an eigenvalue of A + B.
[3]
The matrix A, where
A=
`6
1
3
−1
0
−1
−6 a
−2 ,
−3
` a ` a ` a
1
1
2
has eigenvectors 1 , −1 , 0 . Find the corresponding eigenvalues.
1
1
1
The matrix B, where
B=
`8
2
4
−2
0
−2
[4]
−8 a
−4 ,
−4
` a ` a ` a
1
1
2
also has eigenvectors 1 , −1 , 0 , for which −2, 2, 4, respectively, are corresponding
1
1
1
eigenvalues. The matrix M is given by M = A + B − 5I, where I is the 3 × 3 identity matrix.
State the eigenvalues of M.
[1]
Find matrices R and S and a diagonal matrix D such that M5 = RDS.
[6]
[You should show clearly all the elements of the matrices R, S and D.]
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© UCLES 2013
9231/13/O/N/13