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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/13
FURTHER MATHEMATICS
October/November 2012
Paper 1
3 hours
*1588720257*
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.
JC12 11_9231_13/RP
© UCLES 2012
[Turn over
2
2n
∑ r2 = 16 n(2n + 1)(7n + 1).
[4]
1
Show that
2
Find the set of values of a for which the system of equations
r=n+1
ax + y + 2ß = 0,
= 4,
3x − 2y
3x − 4y − 6aß = 14,
has a unique solution.
3
4
[4]
1
2
3
N
+
+
+ ... +
. Prove by mathematical induction that, for all positive
2! 3! 4!
(N + 1)!
integers N ,
1
SN = 1 −
.
[5]
(N + 1)!
Let SN =
The points A, B and C have position vectors i + 2j + 2k, 2i + 4j + 5k and 2i + 3j + 4k respectively.
−−→ −−→
Find AB × AC.
[3]
Deduce, in either order, the exact value of
(i) the area of the triangle ABC,
(ii) the perpendicular distance from C to AB.
[3]
5
6
The curve C has polar equation r = 1 + 2 cos θ . Sketch the curve for − 23 π ≤ θ < 23 π .
[2]
Find the area bounded by C and the half-lines θ = − 13 π , θ = 13 π .
[4]
The curve C has parametric equations
x = t2 ,
y = 14 t4 − ln t,
for 1 ≤ t ≤ 2. Find the area of the surface generated when C is rotated through 2π radians about the
y-axis.
[7]
7
A cubic equation has roots α , β and γ such that
α + β + γ = 4,
Find the value of αβ + βγ + γ α .
α 2 + β 2 + γ 2 = 14,
α 3 + β 3 + γ 3 = 34.
[2]
Show that the cubic equation is
x3 − 4x2 + x + 6 = 0,
and solve this equation.
© UCLES 2012
[6]
9231/13/O/N/12
3
8
Let ß = cos θ + i sin θ . Show that
1 + ß = 2 cos 12 θ (cos 21 θ + i sin 12 θ ).
[3]
By considering (1 + ß)n , where n is a positive integer, deduce the sum of the series
n
n
n
sin θ + sin 2θ + . . . + sin nθ .
1
2
n
9
10
The curve C has equation y =
x2 − 3x + 3
. Find the equations of the asymptotes of C .
x−2
[6]
[3]
Show that there are no points on C for which −1 < y < 3.
[4]
Find the coordinates of the turning points of C.
[3]
Sketch C.
[2]
The curve C has equation x3 + y3 = 3xy, for x > 0 and y > 0. Find a relationship between x and y when
dy
= 0.
[4]
dx
Find the exact coordinates of the turning point of C , and determine the nature of this turning point.
[8]
Show that ã x 1 − x2 2 dx = − 13 1 − x2 2 + c, where c is a constant.
1
11
3
Given that In = ã xn 1 − x2 2 dx, prove that, for n ≥ 2,
1
[1]
1
0
(n + 2)In = (n − 1)In−2 .
[5]
Use the substitution x = sin u to show that
ã
1
0
1 − x2 2 dx = 14 π .
1
Find I4 .
© UCLES 2012
[5]
[2]
9231/13/O/N/12
[Turn over
4
12
Answer only one of the following two alternatives.
EITHER
The vector e is an eigenvector of each of the n × n matrices A and B, with corresponding eigenvalues
λ and µ respectively. Prove that e is an eigenvector of the matrix AB with eigenvalue λ µ .
[2]
It is given that the matrix A, where
3

A=
 −2
1
2

−2 
,
2
1
0 !. Find the corresponding eigenvalues.
−1
[2]
Given that 2 is also an eigenvalue of A, find a corresponding eigenvector.
[2]
has eigenvectors
0
1 ! and
−1
2
−2
2
The matrix B, where
−1

B=
 2
−3
2
2
−6
2

2
,
−6
has the same eigenvectors as A. Given that AB = C, find a non-singular matrix P and a diagonal
matrix D such that
P−1 C2 P = D.
[8]
OR
Obtain the general solution of the differential equation
d2 x
dx
+6
+ 13x = 75 cos 2t.
2
dt
dt
Given that x = 5 and
dx
= 0 when t = 0, find x in terms of t.
dt
[7]
[4]
Show that, for large positive values of t and for any initial conditions,
x ≈ 5 cos(2t − φ ),
where the constant φ is such that tan φ = 43 .
[3]
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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.
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/O/N/12