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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certi?cate of Education Advanced Level FURTHER MATHEMATICS Paper 1 May/June 2006 3 hours

Additional Materials: Answer Booklet/Paper Graph paper List of Formulae (MF10)

9231/01

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 on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction ?uid. Answer all the questions. Give non-exact numerical answers correct to 3 signi?cant ?gures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is speci?ed 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. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together.

This document consists of 4 printed pages.

? UCLES 2006

[Turn over

2 1

Express

un =

N

4n 2

1 ?1 [4] [2]

in partial fractions, and hence ?nd

n=1

∑ un in terms of N .

Deduce that the in?nite series u1 + u2 + u3 + . . . is convergent and state the sum to in?nity.

2

Draw a diagram to illustrate the region R which is bounded by the curve whose polar equation is r = cos 2θ and the lines θ = 0 and θ = 1 π. [2] 6 Determine the exact area of R. [4]

3

Prove by induction, or otherwise, that 232n + 312n + 46 is divisible by 48, for all integers n ≥ 0.

4

[6]

4

The linear transformation T :

→

4

is represented by the matrix ?1 ?3 ?6 ?5 2 4 10 8 3 5? ?. ? 14 ? 11 [3]

1 ?2 A=? ?5 ? 4

Show that the dimension of the range space of T is 2.

Let M be a given 4 × 4 matrix and let S be the vector space consisting of vectors of the form MAx, where x ∈ 4 . Show that if M is non-singular then the dimension of S is 2. [4] The curve C has equation

5

y = 2x +

3(x ? 1) . x+1 [2]

(i) Write down the equations of the asymptotes of C.

(ii) Find the set of values of x for which C is above its oblique asymptote and the set of values of x for which C is below its oblique asymptote. [3] (iii) Draw a sketch of C, stating the coordinates of the points of intersection of C with the coordinate axes. [4]

? UCLES 2006

9231/01/M/J/06

3 6 (a) The equation of a curve is

√ 2 3 3 y= x2 . 3 [4]

Find the length of the arc of the curve from the origin to the point where x = 1.

(b) The variables x and y are such that

y3 +

dy dx

3

= x4 + 6. d2 y when x = 1. dx2 [5]

Given that y = ?1 when x = 1, ?nd the value of

7

Given that

In =

where n ≥ 0, prove that

1 π 2

sinn x dx,

0

In+2 =

n+1 I . n+2 n

[4]

The region bounded by the x-axis and the arc of the curve y = sin4 x from x = 0 to x = π is denoted by R. Determine the y-coordinate of the centroid of R. [5]

8

Obtain the general solution of the differential equation d2 y dy + 6 + 25y = 80e?3t . 2 dt dt Given that y = 8 and dy = ?8 when t = 0, show that 0 ≤ ye3t ≤ 10 for all t. dt [5]

[5]

9

Given that

= eiθ and n is a positive integer, show that

n

+

1

n

= 2 cos nθ

and

n

?

1

n

= 2i sin nθ .

[2]

Hence express cos7 θ in the form

p cos 7θ + q cos 5θ + r cos 3θ + s cos θ ,

where the constants p, q, r , s are to be determined. [4]

Find the mean value of cos7 2θ with respect to θ over the interval 0 ≤ θ ≤ 1 π , leaving your answer in 4 terms of π . [5]

? UCLES 2006

9231/01/M/J/06

[Turn over

4 10

The equation of the plane Π is 2x + 3y + 4 = 48. Obtain a vector equation of Π in the form

r = a + λ b + ? c,

where a, b and c are of the form pi, qi + rj and si + tk respectively, and where p, q, r , s, t are integers. [6] The line l has vector equation r = 29i ? 2j ? k + θ (5i ? 6j + 2k). Show that l lies in Π . [3]

Find, in the form ax + by + c = d , the equation of the plane which contains l and is perpendicular to Π . [4] Answer only one of the following two alternatives.

EITHER

11

Obtain the sum of the squares of the roots of the equation

x4 + 3x3 + 5x2 + 12x + 4 = 0.

Deduce that this equation does not have more than 2 real roots. Show that, in fact, the equation has exactly 2 real roots in the interval ?3 < x < 0.

[2] [3] [5]

2 Denoting these roots by α and β , and the other 2 roots by γ and δ , show that | γ | = | δ | = √ . [4] (αβ )

OR

The square matrix A has λ as an eigenvalue with corresponding eigenvector x. The non-singular matrix M is of the same order as A. Show that Mx is an eigenvector of the matrix B, where B = MAM?1 , and that λ is the corresponding eigenvalue. [3] It is now given that

A=

1 a b

0 ?3 c

0 0 ?5

and

M=

1 0 0

0 1 0

1 0 1

.

(i) Write down the eigenvalues of A and obtain corresponding eigenvectors in terms of a, b, c. [4] (ii) Find the eigenvalues and corresponding eigenvectors of B. (iii) Hence ?nd a matrix Q and a diagonal matrix D such that Bn = QDQ?1 .

[4] [3]

[You are not required to ?nd Q?1 .]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable 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 University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. ? UCLES 2006 9231/01/M/J/06

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