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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/01
FURTHER MATHEMATICS
May/June 2007
Paper 1
3 hours
*0222815537*
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.
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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 5 printed pages and 3 blank pages.
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2
1
Verify that
n2
1
1
2n + 1
−
= 2
.
2
+ 1 (n + 1) + 1 (n + 1)(n2 + 2n + 2)
[1]
Use the method of differences to show that, for all N ≥ 1,
N
∑ (n
2
n=1
2n + 1
< 1.
+ 1)(n2 + 2n + 2) 2
[3]
2n + 1
.
+ 1)(n2 + 2n + 2)
[1]
Write down the value of
∞
∑ (n
2
n=1
2
The curve C is defined parametrically by
x = t − ln t,
1
y = 4t 2 ,
where t > 0. The arc of C joining the point where t = 1 to the point where t = 4 is rotated through one
complete revolution about the x-axis. Show that the area of the surface generated is 160
π.
[6]
3
3
Find the general solution of the differential equation
dy
d2 y
+ 4 + 29y = 58x + 37.
2
dx
dx
4
[6]
Given that the variables x and y are related by
y = x + e−xy ,
find the value of
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d2 y
when x = 0.
dx2
[6]
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3
5
The diagram shows the curve C with polar equation r = eθ , where 0 ≤ θ ≤ 12 π . Find the maximum
distance of a point of C from the line θ = 12 π , giving the answer in exact form.
[4]
Find the area of the finite region bounded by C and the lines θ = 0 and θ = 12 π , giving the answer in
exact form.
[3]
6
The matrix A, given by
7
A= 2
−3
−4
2
4
6
2,
−2
has eigenvalues 1, 2, 4. Find a set of corresponding eigenvectors.
[4]
Hence find the eigenvalues of B, where
10
B= 2
−3
−4
5
4
6
2,
1
and state a set of corresponding eigenvectors.
7
[3]
The equation
x3 + 3x − 1 = 0
has roots α , β , γ . Use the substitution y = x3 to show that the equation whose roots are α 3 , β 3 , γ 3 is
y3 − 3y2 + 30y − 1 = 0.
Find the value of α 9 + β 9 + γ 9 .
8
[2]
[5]
The sequence x1 , x2 , x3 , . . . is such that x1 = 1 and
xn+1 =
Prove by induction that xn >
1
2
1 + 4xn
5 + 2xn
.
for all n ≥ 1.
Prove also that xn > xn+1 for all n ≥ 1.
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[5]
[3]
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4
9
Let
1
In = 0
1
dx,
(4 − x2 )n
for n = 1, 2, 3, . . . . Verify that
d
x
1 − 2n
8n
=
+
n
n
dx (4 − x2 )
(4 − x2 )
(4 − x2 )n+1
and hence prove that
8nIn+1 = (2n − 1)In +
1
.
3n
[5]
Find the y-coordinate of the centroid of the region bounded by the axes, the line x = 1 and the curve
y=
1
,
4 − x2
giving your answer correct to 3 decimal places.
10
[5]
The line l1 passes through the points P and Q whose position vectors are i − j − 2k and −2i + 5j + 13k
respectively. The line l2 passes through the point S whose position vector is i − 2j + 8k and is parallel
to the vector i − j − 3k. The point whose position vector is −i + 3j + 2k is on the line l3 , the common
perpendicular to l1 and l2 .
(i) Find, in the form r = a + tb, an equation for l3 .
[3]
(ii) Find the perpendicular distance from S to l3 .
[4]
(iii) Find the perpendicular distance from S to the plane which contains l3 and passes through P. [4]
11
(a) Use de Moivre’s theorem to show that sin 8θ can be expressed in the form
sin θ cos θ (a sin6 θ + b sin4 θ + c sin2 θ + d),
where the value of the constant a is to be found and b, c, d are constants whose values need not
be found.
[6]
(b) Use de Moivre’s theorem to show that
N
∑
n=1
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sin nθ 2N +1 sin θ + sin N θ − 2 sin(N + 1)θ
=
.
2n
2N (5 − 4 cos θ )
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5
12
Answer only one of the following two alternatives.
EITHER
The curve C has equation
y = λx +
x
,
x+2
where λ is a non-zero constant.
(i) Find the asymptotes of C.
(ii) Show that if λ > 0 then
[3]
dy
> 0 at all points of C.
dx
[2]
(iii) Show that, for λ < − 12 , C has two distinct stationary points, both to the left of the y-axis.
[3]
(iv) In separate diagrams draw sketches of C for each of the cases λ > 0 and λ < − 12 .
[6]
OR
The linear transformation T : 4 → 4 is represented by the matrix
1
⎛2
M=⎜
⎝3
⎜
5
−2
−4
−6
−10
2
5
8
12
4
9⎞
⎟.
⎠
14 ⎟
22
(i) Find the rank of M.
[3]
(ii) Obtain a basis for the null space, K , of T.
[3]
(iii) Evaluate
−1
⎛ 2⎞
⎟,
M⎜
⎝ −3 ⎟
⎜
⎠
4
and hence show that any solution of
5
⎛ 11 ⎞
⎟
Mx = ⎜
⎠
⎝ 17 ⎟
⎜
27
(∗)
has the form
−1
⎛ 2⎞
⎜ ⎟ + λ e + µe ,
⎝ −3 ⎟
⎜
⎠
1
2
4
where λ and µ are constants and {e1 , e2 } is a basis for K .
[3]
(iv) Find the solution x1 of (∗) such that the first component of x1 is A, and the sum of all the
components of x1 is B.
[5]
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