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# A-level 进阶数学真题 03年 纯数学

CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certi?cate of Education Advanced Level FURTHER MATHEMATICS Paper 1 May/June 2003 3 hours

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. 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. 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.

This document consists of 5 printed pages and 3 blank pages.
? CIE 2003

[Turn over

2 1

The diagram shows one loop of the curve whose polar equation is r = a sin 2θ , where a is a positive constant. Find the area of the loop, giving your answer in terms of a and π . [4] Prove by induction that, for all N ≥ 1,

2

∑ n(n + 1)2
n=1

N

n+2

n

=1?

1 . (N + 1)2N

[5]

3

Let v1 , v2 , v3 , . . . be a sequence and let

un = nvn ? (n + 1)vn+1 ,
for n = 1, 2, 3, . . . . Find

n=1

∑ un .

N

[2]

In each of the following cases determine whether the series u1 + u2 + u3 + . . . is convergent, and justify your conclusion. Give the sum to in?nity where this exists.
(i) vn = n 2 . (ii) vn = n 2 .
?3 ?1

[2] [2]

4

The curve C has equation y =

x2 ? 4 . x?3
[3]

(i) Find the equations of the asymptotes of C.

(ii) Draw a sketch of C and its asymptotes. Give the coordinates of the points of intersection of C with the coordinate axes. [4]

[You are not required to ?nd the coordinates of any turning points.]

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3 5

The equation 8x3 + 12x2 + 4x ? 1 = 0 has roots α , β , γ . Show that the equation with roots 2α + 1, 2β + 1, 2γ + 1 is

y3 ? y ? 1 = 0.
The sum (2α + 1)n + (2β + 1)n + (2γ + 1)n is denoted by Sn . Find the values of S3 and S?2 .
6

[3] [5]

Use de Moivre’s theorem to show that cos 6θ = 32 cos6 θ ? 48 cos4 θ + 18 cos2 θ ? 1. Hence solve the equation 64x6 ? 96x4 + 36x2 ? 1 = 0, giving each root in the form cos kπ . The variables x and y are related by the equation x4 + y4 = 1, where 0 < x < 1 and 0 < y < 1.
(i) Obtain an equation which relates x, y,

[5]

[4]

7

dy d2 y , , and deduce that dx dx2 [6]

d2 y 3x2 = ? . y7 dx2

(ii) Given that y = b1 when x = a1 and that y = b2 when x = a2 , where a1 < a2 , prove that the mean d3 y value of 3 with respect to x over the interval a1 ≤ x ≤ a2 is dx

3(a2 b7 ? a2 b7 ) 1 2 2 1

b7 b7 (a ? a1 ) 1 2 2

.

[4]

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4 8

The linear transformation T :

4

4

is represented by the matrix A, where ?1 ?1 ?2 ?3 ?2 ?1 ?3 ?5 3 11 ? ?. ? 14 ? 17 [7]

1 ?2 A=? ?3 ? 4

Find the rank of A and a basis for the null space of T.

1 ? ?2 ? ? is denoted by e. Show that there is a solution of the equation Ax = Ae of the form The vector ? ? ?1 ? ? ? ?1 p ?q? ?, where p and q are to be found. x=? [4] ?1? ? ? 1 The variables x and t, where x > 0 and 0 ≤ t ≤ 1 π , are related by 2 d2 x dx x 2 + dt dt
2

9

+ 5x

dx + 3x2 = 3 sin 2t + 15 cos 2t, dt

and the variables x and y are related by y = x2 . Show that d2 y dy + 5 + 6y = 6 sin 2t + 30 cos 2t. 2 dt dt Hence ?nd x in terms of t, given that x = 2 and 3 dx = ? when t = 0. dt 2 [3]

[10]

10

Find the acute angle between the planes with equations

x ? 2y + ? 9 = 0

and

x + y ? + 2 = 0.

[3]

The planes meet in the line l, and A is the point on l whose position vector is pi + qj + k.
(i) Find p and q. (ii) Find a vector equation for l.

[2] [3]

The non-coincident planes Π1 and Π2 are both perpendicular to l. The perpendicular distance from A √ √ to Π1 is 14 and the perpendicular distance from A to Π2 is also 14. Find equations for Π1 and Π2 in the form ax + by + c = d. [5]

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5 11

Answer only one of the following two alternatives.

EITHER

Given that

In =

1 0

xn e?α x dx,

where α is a positive constant and n is a non-negative integer, show that for n ≥ 1,

α In = nIn?1 ? e?α .

[3]

Hence, or otherwise, ?nd the coordinates of the centroid of the ?nite region bounded by the x-axis, the line x = 1 and the curve y = x e?x , giving your answers in terms of e. [11]

OR

The vector e is an eigenvector of each of the n × n matrices A and B, with corresponding eigenvalues λ and ? respectively. Prove that e is an eigenvector of the matrix AB with eigenvalue λ ? . [3] Find the eigenvalues and corresponding eigenvectors of the matrix C, where
C=

0 1 2

1 2 1

4 ?1 2

.

[8]

Verify that one of the eigenvectors of C is an eigenvector of the matrix D, where
D=

?3 0 0

1 ?2 0

1 4 ?4

.

[2]

Hence ?nd an eigenvalue of the matrix CD.

[1]

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