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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/11
FURTHER MATHEMATICS
Paper 1
May/June 2013
3 hours
*2772170948*
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_11/RP
© UCLES 2013
[Turn over
2
1
Find the area of the region enclosed by the curve with polar equation r = 2 1 + cos 1, for 0 ≤ 1 < 20.
[4]
2
Prove by mathematical induction that 52n − 1 is divisible by 8 for every positive integer n.
3
The cubic equation x3 − 2x2 − 3x + 4 = 0 has roots !, ", ' . Given that c = ! + " + ' , state the value
of c.
[1]
Use the substitution y = c − x to find a cubic equation whose roots are ! + ", " + ' , ' + !.
Find a cubic equation whose roots are
Hence evaluate
4
Let In = Ô
1
0
1
1
1
,
,
.
!+" "+' '+!
1
1
1
+
+
.
2
2
! + "
" + '
' + !2
2nIn+1 = 2−n + 2n − 1In .
Use the method of differences to show that
Ð
2N
6
[2]
1
n dx. Prove that, for every positive integer n,
1 + x2 Ð
N
Deduce that
[3]
[2]
Given that I1 = 14 0, find the exact value of I3 .
5
[5]
r=N +1
r=1
5
[3]
1
1
1
= −
.
2r + 1 2r + 3 6 2 2N + 3
1
1
<
.
2r + 1 2r + 3 8N
[5]
[4]
The matrix A is given by
A=
`4
3
1
−5
−4
−1
3a
3 .
2
` a
1
Show that e = 1 is an eigenvector of A and state the corresponding eigenvalue.
1
[2]
Find the other two eigenvalues of A.
[4]
The matrix B is given by
B=
` −1
−1
1
4
3
−1
0a
1 .
3
Show that e is an eigenvector of B and deduce an eigenvector of the matrix AB, stating the
corresponding eigenvalue.
[3]
© UCLES 2013
9231/11/M/J/13
3
@
7
1
By considering the binomial expansion of Ï −
Ï
form
A6
, where Ï = cos 1 + i sin 1, express sin6 1 in the
p + q cos 21 + r cos 41 + s cos 61,
1
32
where p, q, r and s are integers to be determined.
Hence find the exact value of Ó
10
4
0
8
sin6 1 d1.
[6]
[4]
The linear transformations T1 : >4 → >4 and T2 : >4 → >4 are represented by the matrices M1 and M2
respectively, where
€

€

1 −2
3
5
1
−2 0
−3
−1 0
0
3 −4 17 33
2
M1 =
and M2 =
.
5 −9 20 36
4
−7 1
−9
4 −7 16 29
6 −10 0 −14
The null spaces of T1 and T2 are denoted by K1 and K2 respectively. Find a basis for K1 and a basis
for K2 .
[6]
€ 
1
2
It is given that a =
. The vectors x1 and x2 are such that M1 x1 = M1 a and M2 x2 = M2 a. Given
3
€  4
p
5
that x1 − x2 =
, find p and q.
[4]
7
q
9
Find x in terms of t given that
4
dx
d2 x
+4
+ x = 6e−2t ,
2
d
t
dt
dx 7
and that, when t = 0, x = 53 and
= .
dt 6
[9]
State lim x.
[1]
t→∞
[Questions 10 and 11 are printed on the next page.]
© UCLES 2013
9231/11/M/J/13
[Turn over
4
10
The curve C has equation y =
Show that y ≤
25
12
2x2 − 3x − 2
. State the equations of the asymptotes of C.
x2 − 2x + 1
[2]
at all points of C.
[4]
Find the coordinates of any stationary points of C.
[3]
Sketch C, stating the coordinates of any intersections of C with the coordinate axes and the asymptotes.
[4]
11
Answer only one of the following two alternatives.
EITHER
The curve C has equation y = 2 sec x, for 0 ≤ x ≤ 14 0. Show that the arc length s of C is given by
s=Ó
10
4
0
2 sec2 x − 1 dx.
4
Find the exact value of s.
[2]
The surface area generated when C is rotated through 20 radians about the x-axis is denoted by S.
Show that
(i) S = 40 Ó
(ii)
10
4
0
2 sec3 x − sec x dx,
[3]
d
sec x tan x = 2 sec3 x − sec x.
dx
[3]
Hence find the exact value of S.
[2]
OR
The points A, B, C and D have coordinates as follows:
A 2, 1, −2,
B 4, 1, −1,
C 3, −2, −1
and
D 3, 6, 2.
The plane 1 passes through the points A, B and C. Find a cartesian equation of 1 .
[4]
Find the area of triangle ABC and hence, or otherwise, find the volume of the tetrahedron ABCD.
[The volume of a tetrahedron is
1
3
× area of base × perpendicular height.]
The plane 2 passes through the points A, B and D. Find the acute angle between 1 and 2 .
[6]
[4]
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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
9231/11/M/J/13