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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/01
FURTHER MATHEMATICS
October/November 2010
Paper 1
3 hours
*5014820134*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10)
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credit.
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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.
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2
1
The curve C has equation y =
x = 0 to the point where x =
2
1
2
e2x + e−2x . Show that the length of the arc of C from the point where
e2 − 1
.
[4]
is
4e
1
4
Use the method of differences to find SN , where
SN =
N
1
[4]
∑ n(n + 2) .
n=1
Deduce the value of lim SN .
[1]
N →∞
3
√
1
x − √ , the x-axis
x
between x = 1 and x = 4, and the line x = 4. Find the exact value of the y-coordinate of the centroid
of R.
[5]
A finite region R in the x-y plane is bounded by the curve with equation y =
4
Prove by mathematical induction that, for all non-negative integers n, 72n+1 + 5n+3 is divisible by 44.
[5]
5
Let In = ã (1 − x)n sin x dx for n ≥ 0. Show that
1
0
In+2 = 1 − (n + 1)(n + 2)In .
Hence find the value of I6 , correct to 4 decimal places.
6
[4]
[4]
The linear transformation T : >4 → >4 is represented by the matrix A, where
1
2

A=
2

0
2
3
1
1
−1
−1
2
−3
α
0

.

−2 
−2
Given that the dimension of the range space of T is 4, show that α ≠ 1.
It is now given that α = 1. Show that the vectors
1
2
 
 ,
2


0
2
3
 
 
1


1
and
[3]
−1
 −1 
 
 
 2


−3
form a basis for the range space of T.
[2]
p
1
 
Given also that the vector   is in the range space of T, find a condition satisfied by p and q.
1


q
[3]
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3
7
1
The roots of the equation x3 + 4x − 1 = 0 are α , β and γ . Use the substitution y =
to show that
1+x
1
1
1
,
and
.
[2]
the equation 6y3 − 7y2 + 3y − 1 = 0 has roots
α +1 β +1
γ +1
For the cases n = 1 and n = 2, find the value of
1
1
1
+
+
.
(α + 1)n (β + 1)n (γ + 1)n
Deduce the value of
Hence show that
8
1
1
1
.
+
+
3
3
(α + 1)
(β + 1)
(γ + 1)3
[2]
(β + 1)(γ + 1) (γ + 1)(α + 1) (α + 1)(β + 1) 73
.
+
+
=
36
(α + 1)2
(β + 1)2
(γ + 1)2
[3]
The curves C1 and C2 have polar equations given by
C1 :
C2 :
9
[2]
r = 3 sin θ ,
r = 1 + sin θ ,
0 ≤ θ < π,
−π < θ ≤ π .
(i) Find the polar coordinates of the points, other than the pole, where C1 and C2 meet.
[2]
(ii) In a single diagram, draw sketch graphs of C1 and C2 .
[3]
(iii) Show that the area of the region which is inside C1 but outside C2 is π .
[5]
Find the eigenvalues and corresponding eigenvectors of the matrix
3

A=
 −1
0
−1
2
−1
0

−1 
.
3
[7]
Find a non-singular matrix M and a diagonal matrix D such that (A − 2I)3 = MDM−1 , where I is the
3 × 3 identity matrix.
[3]
10
By using de Moivre’s theorem to express sin 5θ and cos 5θ in terms of sin θ and cos θ , show that
tan 5θ =
where t = tan θ .
5t − 10t3 + t5
,
1 − 10t2 + 5t4
[5]
Show that the roots of the equation x4 − 10x2 + 5 = 0 are tan 15 nπ for n = 1, 2, 3, 4.
[2]
By considering the product of the roots of this equation, find the exact value of tan 15 π tan 25 π .
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4
11
It is given that x ≠ 0 and
x
Show that if ß = xy then
d2 y
dy
+ 2 + 4xy = 8x2 + 16.
2
dx
dx
d2 ß
+ 4ß = 8x2 + 16.
dx2
Find y in terms of x, given that y = 0 and
12
dy
= −2 when x = 12 π .
dx
[3]
[9]
Answer only one of the following two alternatives.
EITHER
The curve C has equation
y=
where λ is a constant and λ ≠ −1.
x2 + 2λ x
,
x2 − 2x + λ
(i) Show that C has at most two stationary points.
[3]
(ii) Show that if C has exactly two stationary points then λ > − 54 .
[2]
(iii) Find the set of values of λ such that C has two vertical asymptotes.
[2]
(iv) Find the x-coordinates of the points of intersection of C with
(a) the x-axis,
(b) the horizontal asymptote.
[3]
(v) Sketch C in each of the cases
(a) λ < −2,
(b) λ > 2.
[4]
OR
The plane Π1 has equation r = 2i + j + 4k + λ (2i + 3j + 4k) + µ (−i + k). Obtain a cartesian equation
of Π1 in the form px + qy + rß = d.
[4]
The plane Π2 has equation r. (i − 4j + 5k) = 12. Find a vector equation of the line of intersection of
Π1 and Π2 .
[3]
The line l passes through the point A with position vector ai + (2a + 1)j − 3k and is parallel to
3ci − 3j + ck, where a and c are positive constants. Given that the perpendicular distance from A to
15
2
Π1 is √ and that the acute angle between l and Π1 is sin−1 √ , find the values of a and c.
[7]
6
6
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9231/01/O/N/10
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