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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/13
FURTHER MATHEMATICS
May/June 2012
Paper 1
3 hours
*0973024008*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10)
READ THESE INSTRUCTIONS FIRST
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Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
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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.
JC12 06_9231_13/RP
© UCLES 2012
[Turn over
2
1
Find the sum of the first n terms of the series
1
1
1
+
+
+ ...
1×3 2×4 3×5
and deduce the sum to infinity.
[5]
3ur − 2
for all r. Prove by
4
n
[5]
mathematical induction that un = 4 34 − 2, for all positive integers n.
2
For the sequence u1 , u2 , u3 , . . . , it is given that u1 = 1 and ur+1 =
3
The curve C has equation
xy + (x + y)3 = 1.
Show that
dy
= − 34 at the point A (1, 0) on C .
dx
Find the value of
4
[3]
d2 y
at A.
dx2
[5]
Let
e
In = ã x2 (ln x)n dx,
1
for n ≥ 0. Show that, for all n ≥ 1,
In = 13 e3 − 13 nIn−1 .
Find the exact value of I3 .
5
[4]
[4]
The matrix A has an eigenvalue λ with corresponding eigenvector e. Prove that the matrix (A + kI),
where k is a real constant and I is the identity matrix, has an eigenvalue (λ + k) with corresponding
eigenvector e.
[2]
The matrix B is given by
2

B=
 2
−3
2
2
3
−3

3
.
3
Two of the eigenvalues of B are −3 and 4. Find corresponding eigenvectors.
[3]
1
−1 ! is an eigenvector of B, find the corresponding eigenvalue.
−2
[1]
Given that
Hence find the eigenvalues of C, where
−1

C=
 2
−3
2
−1
3
and state corresponding eigenvectors.
© UCLES 2012
−3

3
,
0
[3]
9231/13/M/J/12
3
6
7
The curve C has equation y =
x2
. Find the equations of the asymptotes of C.
x−2
[3]
Find the coordinates of the turning points on C.
[3]
Draw a sketch of C.
[3]
Expand ß +
1
1
ß − and, by substituting ß = cos θ + i sin θ , find integers p, q, r , s such that
ß
ß
4
2
64 sin2 θ cos4 θ = p + q cos 2θ + r cos 4θ + s cos 6θ .
[6]
Using the substitution x = 2 cos θ , show that
2
√
√
ã x4 (4 − x2 ) dx = 43 π + 3.
[4]
1
8
The cubic equation x3 − x2 − 3x − 10 = 0 has roots α , β , γ .
(i) Let u = −α + β + γ . Show that u + 2α = 1, and hence find a cubic equation having roots −α + β + γ ,
α − β + γ, α + β − γ.
[5]
(ii) State the value of αβγ and hence find a cubic equation having roots
9
1 1
1
,
,
.
βγ γ α αβ
[5]
The plane Π1 has parametric equation
r = 2i − 3j + k + λ (i − 2j − k) + µ (i + 2j − 2k).
10
Find a cartesian equation of Π1 .
[4]
The plane Π2 has cartesian equation 3x − 2y − 3ß = 4. Find the acute angle between Π1 and Π2 .
[3]
Find a vector equation of the line of intersection of Π1 and Π2 .
[4]
Find the set of values of a for which the system of equations
x − 2y − 2ß = −7,
2x + (a − 9)y − 10ß = −11,
3x − 6y + 2aß = −29,
has a unique solution.
[4]
Show that the system has no solution in the case a = −3.
[2]
Given that a = 5,
(i) show that the number of solutions is infinite,
[2]
(ii) find the solution for which x + y + ß = 2.
[3]
© UCLES 2012
9231/13/M/J/12
[Turn over
4
11
Answer only one of the following two alternatives.
EITHER
The curve C has cartesian equation
(x2 + y2 ) = a2 (x2 − y2 ),
2
where a is a positive constant. Show that C has polar equation
r2 = a2 cos 2θ .
[2]
Sketch C for −π < θ ≤ π .
[2]
Find the area of the sector between θ = − 14 π and θ = 14 π .
[3]
Find the polar coordinates of all points of C where the tangent is parallel to the initial line.
[7]
OR
Show that the substitution y = xß reduces the differential equation
1 d2 y
6
2 dy
9
6
2
+ − 2
+ − 2 + 3 y = 169 sin 2x
2
x dx
x x dx
x x
x
to the differential equation
d2 ß
dß
+6
+ 9ß = 169 sin 2x.
2
dx
dx
Find the particular solution for y in terms of x, given that when x = 0, ß = −10 and
[4]
dß
= 5.
dx
[10]
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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 2012
9231/13/M/J/12