2010 pure mathematics paper 1

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2010-AL

PMATH HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY

PAPER 1 HONG KONG ADVANCED LEVEL EXAMINATION 2010

PURE MATHEMATICS A-LEVEL PAPER 1

8.30 am - 11.30 am (3 hours)

This paper must be answered in English

1. This paper consists of Section A and Section B.

2. Answer ALL questions in Section A, using the AL(E) answer book.

3. Answer any FOUR questions in Section B, using the AL(C) answer book.

4. Unless otherwise specified, all working must be clearly shown .

@w ¥ i ~ . : & j f H ~ fiU ~ & ; f . I Hong Kong Examinations and Assessment Authority

All Rights Reserved 2010

201O-AL-P MATIll-l

Not to be taken away before theend of the examination session



FORMULAS FOR REFERENCE

sin(A ± B) = sin A cos B±cos A sin B

cos(A ± B) = cos A cos B+sin A sin B

tanA±tanBtan(A±B)1+ tan A tan B

. A . B 2' A+B A -Bsm +sm = sm--cos-

2 2

A+B . A -Bsin A -sin B 2 cos sm-

2 2

A+B A -Bcos A + cos B = 2 cos--cos-

2 2

. A+B . A-BcosA-cosB -2sm sm

2 2

2 sin A cos B = sin(A + B) + sin(A B)

2 cos A cos B =cos(A + B) + cos(A - B)

2 sin A sin B cos(A - B) - cos(A + B)

201O-AL-P MATH 1-2 2



SECTION A (40 marks)

Answer ALL questions in this section.

Write your answers in the AL(E) answer book.

Let n be a positive integer. Denote the coefficient of Xk in the expansion of (1+ 2X)" by ak .

Prove that

n n

(a) 'L>k =3 ,

k=O

n

(b) L kak =2n3n

-1

,

k=!

n

( c ) L ( 3 k+ l ) a k =(2n+l)3n

•

k=O

(7 marks)

2. (a) Resolvex

(x2-1)(x

2- 4)

into partial fractions.

(b) By differentiatingx

(x 2 _1)(x2 -4), or otherwise, resolve

3x4-5x

2 -4(x

2-li(x

2-4i

into partial

fractions.

(c) Evaluate00 3k 4 - 5 e - 4L 2 2 2 2

k=3 (k -I ) (k - 4)

(6 marks)

3. Let f(x) = x3 + g(x) , where g(x) is a guadratic polynomial with real coefficients. When f(x) is

divided by (x I)(x 4) and when f(x) is divided by (x 4)2, the remainders are -x + k and

kx - 10 respectively, where k is a real number. Find

(a) k,

,(b) g(x) ,

(c) the remainder when (g(x)i is divided by x +1 .

(7 marks)

201O-AL-P MATH 1-3 3 Go on to the next page>



4. (a) Write down the matrix which represents the anticlockwise rotation about the origin by 7r in the2

Cartesian plane.

(b) Let 0 be the origin. It is given that 0 , P(1, 3) and Q are the vertices of an isosceles triangle,

where LPOQ == 7r and Q lies in the second quadrant.2

(i) Find the coordinates of Q .

Oi) Let T be the transformation which transforms the points (1,0) and (0, 1) to the

points (0,-1) and (-1,0) respectively and M be the 2x2 realmatrixwhich

represents the transformation T.

(1) Write down the matrix M.

(2) Describe the geometric meaning of the transformation T.

(3) The transformation T transforms 0 , P and Q to 0 ' , P' and Q'

respectively. Find the area of f..O'P'Q' .

(7 marks)

@ Let S {ZEC: zz=(12+16i)z+(12-16i)z 375}.

(a) Prove that S is represented by a circle on the Argand diagram. Also [rod the centre and the

radius of the circle.

(b) Find Zl E S such that Izl I IZ I for all Z E S .

(7 marks)

6. Let a, /3, rand 0 be positive real numbers. Prove that

I I 1 I )(a) (a+/3+r+o) -+ + + - ~ 1 6 ,( a /3 r 0

__ 3__ 3 3 3 > _ _ _16__b) + + + -/3+r+ o r+o+a o+a+/3 a+/3+r a+/3+r+ o

a /3 r 0 4l (c); - - -+ + + ~ \J {3+r+o r+ o +a o+a+/3 a+/3+r 3

(6 marks)

..


I



SECTION B (60 marks)

Answer any FOUR questions in this section. Each question carries 15 marks.

Write your answers in the AL(C) answer book.

7. (a) Consider the system of linear equations in x , y, z

y + z 2

(£ )

+

4z 2 , where a, b E R .

3x + 4y + (a+4)z = b

(i ) Find the range of values of a for which (E) has a unique solution, and solve (E) when

(E) has a unique solution.

Oi) Suppose that a 2 . Find the value(s) of b for which (E) is consistent, and solve (E)

for such value(s) of b.

(S marks)

(b) Consider the system of linear equations in x , y, z

, + y + z 2

+ 2z(F) : /!. ' where /!., 11 E R .

3x + 4y + 2z

1x + 17y 3z

Find the values of /!. and 11 for which (F) is consistent.

(4 marks)

(c) Consider the system of linear equations in x, y , z

y + z 2: +

6z 3(G) :

9x + 12y + 14z 15{

5x 2y lSz 16

Is (G) consistent? Explain your answer.

(3 marks)

S. (a) Let /!. be a real number and p(x) be a polynomial with real coefficients. Prove that /!. is a

repeated root of the equation p(x) =0 if and only if p(/!.) p/(/!.) "= 0 .

(4 marks)

(b) Let f(x) x6 + ax +bx

4 + ex +bx2 + ax + 1 , where a, b and e are real numbers.

Suppose that a real number11

is a repeated root of the equation f(x) = 0 . Prove that

Oi) is a repeated root of the equation f(x) = 0 .11

(6 marks)

(c) Let g(x) 4x6 16x5 +17x4 _7x3 +17x2 -16x+4 .

(i) Find a repeated root of the equation g(x) O.

(ii) Can g(x) be factorized as a product oflinear polynomials with real coefficients?

Explain your answer.

(5 marks)

201O-AL-P MATH 1-5 5 Go on to the nextpage)



9. Let xl and YI be real numbers. For any positive integer n, define

5 1 2 7 xn+l -xn + Yn and Yn+l =-Xn +-Yn .

6 6 9 9

(a) Suppose that xl > Yl .

(i) Prove that Xn > Yn •

(ii) Prove that {xn} is a strictly decreasing sequence and {Yn} is a strictly increasing

sequence.

(iii) Prove that {xn} and {yn} are convergent sequences.

(iv) Prove that lim Xn = lim Ynn ~ o o n ~ o o

(v) Prove that 4xn+1 + 3Yn+1 = 4xn +3Yn

(vi) Express lim xn in terms of Xl and YI .n ~ " "

(12 marks)

(b) Suppose that XI < Yl . Are {xn} and {Yn} convergent sequences? Explain your answer.

(3 marks)

10. Denote the 2 x 2 identity matrix by I . A 2 x 2 real matrix M is said to be orthogonal if and only if

MMT I, where MT is the transpose of M.

(a) Prove that a 2 x 2 real matrix M is orthogonal if and only ifthere exists 0 E R such that

COS0 sin 0] [COS0 M= or M

- sin 0 cos0 sin0(6 marks)

cosO

[ ' : ~ J sin 0] If cos nO sin no]

(b) (i) Suppose that M = r . Prove that Mn

= for all

l-sinO cosO -sinnO cosnO

positive integers n.

f cos 0SinO]

Mn(ii) Suppose that M = . Evaluate for all positive integers n.

sin 0 -cosOI(3 marks)

X 400(c) Find all 2 x 2 real orthogonal matrices X such that '" - I .

(4 marks)

M 401(d) Suppose that M is a 2 x 2 real orthogonal matrix. Is orthogonal? Explain your answer.

(2 marks)

201O-AL-P MATI! l'-{) 6



r

11. (a) (i ) Prove that In x s x 1 for all x > 0 .

(ii) Let ai ' a2 , •• • ,an and bl , b2 , •• . ,bn be positive real numbers satisfYing

al +a2 +"'+an =albl +a2b2 +"'+anbn =1. Provethat bjG'b2Gz · .. bnG• s l

(4 marks)

(b) (i) Prove that xx;:>: x for all x> 0 .

(ii) Let C I' C2' •• • , Cn be positive real numbers satisfying Cj C2 '" cn 1 .

(3 marks)

(c) Let xl ' x2"" 'X n be positive real numbers. Prove that

(i)

x x x ((ii) Xl 'X 2 2 •• • Xn n :2::

(8 marks)

END OF PAPER


2010 pure mathematics paper 1

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