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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/01
FURTHER MATHEMATICS
October/November 2009
Paper 1
3 hours
*0307464767*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10)
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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
Given that
y = x2 sin x,
(i) show that the mean value of
(ii) find the mean value of
2
dy
with respect to x over the interval 0 ≤ x ≤ 12 π is 12 π ,
dx
d2 y
with respect to x over the interval 0 ≤ x ≤ 12 π .
dx2
j + k,
i + j + θ k,
1
respectively. The shortest distance between the lines AB and OC is √ . Find the value of θ .
2
The curve C has equation
y=
4
[2]
Relative to an origin O, the points A, B, C have position vectors
i,
3
[2]
[6]
x2 − 5x + 4
.
x+1
(i) Obtain the coordinates of the points of intersection of C with the axes.
[2]
(ii) Obtain the equation of each of the asymptotes of C.
[3]
(iii) Draw a sketch of C.
[3]
It is given that
where −π < t < π . Find
x = t + sin t,
dy
in terms of t.
dx
y = t2 + 2 cos t,
[2]
Show that
Show that
5
d2 y
2t sin t
=
.
2
dx
(1 + cos t)3
[4]
dy
increases with x over the given interval of t.
dx
[2]
The equation
x3 + 5x + 3 = 0
has roots α , β , γ . Use the substitution x = −
this equation are βγ , γ α , αβ .
3
to find a cubic equation in y and show that the roots of
y
[4]
Find the exact values of β 2 γ 2 + γ 2 α 2 + α 2 β 2 and β 3 γ 3 + γ 3 α 3 + α 3 β 3 .
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3
6
Show that
d n−1 √
4(n − 1)xn−2
nxn
(4 − x2 ) = √
x
.
−
√
dx
(4 − x2 )
(4 − x2 )
[3]
Let
In = ä
where n ≥ 0. Prove that
1
0
√
xn
dx,
(4 − x2 )
√
nIn = 4(n − 1)In−2 − 3,
for n ≥ 2.
Given that I0 = 16 π , find I4 , leaving your answer in an exact form.
7
[2]
[4]
Use de Moivre’s theorem to express sin6 θ in the form
a + b cos 2θ + c cos 4θ + d cos 6θ ,
where a, b, c, d are constants to be found.
[5]
Hence evaluate
ã
1
π
4
sin6 2x dx,
0
leaving your answer in terms of π .
8
[4]
(a) The curve C1 has equation y = − ln(cos x). Show that the length of the arc of C1 from the point
√
where x = 0 to the point where x = 13 π is ln(2 + 3).
[5]
√
(b) The curve C2 has equation y = 2 (x + 3). The arc of C2 joining the point where x = 0 to the
point where x = 1 is rotated through one complete revolution about the x-axis. Show that the area
of the surface generated is
√
8
π (5 5 − 8).
[5]
3
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4
9
Show that if y depends on x and x = eu then
x2
d2 y d2 y dy
=
−
.
dx2 du2 du
[4]
Given that y satisfies the differential equation
x2
d2 y
dy
+ 5x + 3y = 30x2 ,
2
dx
dx
use the substitution x = eu to show that
d2 y
dy
+4
+ 3y = 30e2u .
2
du
du
Hence find the general solution for y in terms of x.
10
The curve C has polar equation
where 0 ≤ θ ≤ 13 π .
[2]
[5]
r = a sin 3θ ,
(i) Show that the area of the region enclosed by C is
1
π a2 .
12
[3]
(ii) Show that, at the point of C at maximum distance from the initial line,
tan 3θ + 3 tan θ = 0.
(iii) Use the formula
tan 3θ =
to find this maximum distance.
3 tan θ − tan3 θ
1 − 3 tan2 θ
(iv) Draw a sketch of C.
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[4]
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5
11
Answer only one of the following two alternatives.
EITHER
Prove by induction that
N
∑ n3 = 14 N 2 (N + 1)2.
n=1
[5]
N
Use this result, together with the formula for
∑ n2 , to show that
n=1
N
∑ (20n3 + 36n2 ) = N (N + 1)(N + 3)(5N + 2).
n=1
[3]
Let
SN =
N
∑ (20n3 + 36n2 + µ n).
n=1
Find the value of the constant µ such that SN is of the form N 2 (N + 1)(aN + b), where the constants
a and b are to be determined.
[3]
Show that, for this value of µ ,
for all N ≥ 18.
5+
22
23
< N −4 SN < 5 +
,
N
N
[3]
OR
One of the eigenvalues of the matrix
A=
is −2. Find the value of a.
1
2
−3
−4
−4
4
6
2!
a
[3]
Another eigenvalue of A is −5. Find eigenvectors e1 and e2 corresponding to the eigenvalues −2 and
−5 respectively.
[3]
The linear space spanned by e1 and e2 is denoted by V .
(i) Prove that, for any vector x belonging to V , the vector Ax also belongs to V .
[4]
(ii) Find a non-zero vector which is perpendicular to every vector in V , and determine whether it is
an eigenvector of A.
[4]
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