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2009-CE-A MATH1 1

HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 2009

Kitzone Website

All Rights Reserved 2009

2009-CE

A MATH

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2009-CE-A MATH2 2

Section A (62 marks)

Answer ALL questions in this section and write your answers in the spaces provided in this Question-Answer

Book.

1. Find 286(7 8) x dx .

(2 marks)

2. Find the range ofy such that2

2 3

2 1

xy

x x

+=

for all real values ofx.

(5 marks)

3. Find the general solution of the equation sin3 3sin 2 sin 0 x x x + = .

(4 marks)

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ed.

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ed.

Answers written in margins will not be marked.

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2009-CE-A MATH3 3


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2009-CE-A MATH4 4

4. (a) Solve 2 6 0y y =

(b) Solve 2( 3) 3 6 0x x =

(5 marks)

5. Find, in the expression of ( )6

4 13 2 1

2x

x

+

,

(a) the constant term, and

(b) the coefficient of x .

(4 marks)


Answ

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Answ

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2009-CE-A MATH5 5


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2009-CE-A MATH6 6

6. Prove, by mathematical induction,

( ) ( ) ( )33 3 3 3 5 5 5 51 2 3 3 1 2 3 4 1 2 3n n n+ + + + + + + + + = + + + +

for all positive integers n.

(6 marks)

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Answerswritteninmar



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2009-CE-A MATH7 7

7. Differentiate1

1y

x=

from first principle.

(4 marks)

8. A straight lineL1 : y mx c= + , where m and c are constants, makes an angle of 45 with the line

L2 : 7 3 0x y+ = .

(a) Find the two values ofm.

(b) If the distance from the point (2,0) toL1 is 4 and 0m > , find the two values ofc.

(6 marks)

9. Given 5OA j=

, 7OB i j= +

. P is a point such that AP tAB=

.

(c) Express OP

in terms oft.

(d) IfOP is perpendicular toAB, find

i. the value oft

ii. OP

(6 marks)


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2009-CE-A MATH8 8




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2009-CE-A MATH9 9

10. Find the value of cos22.5

in surd form.

(4 marks)

11. Let , be the roots of the quadratic equation2

2 4 9 0x x + = . Find the quadratic equation whose

roots are 3+ and 3+ .

(5 marks)


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2009-CE-A MATH10 10


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2009-CE-A MATH11 11

12. The slope at any point ( , )x y of a curve Cis given by 2 3dy

xdx

= . IfCpasses through

the point (2,5) , find the equation of the curve C.

(5 marks)

13. A family of circles which passes through a straight line 2 4y x= is given by the equation2 2 (2 4) (4 ) (8 4 ) 0 x y k x k y k + + + + = , where kis a constant.

(a) Find the equation of the circle in the family which passes through the point (3,1) .

(b) Find the equation of the circle with the centre passing through the line 2 8 0x y + = .

(6 marks)




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2009-CE-A MATH12 12


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2009-CE-A MATH13 13

SECTION B (48 marks)

Answer any FOUR questions in this section. Each question carries 12 marks.

Write your answers in the spaces provided.

14.

In this figure,AB is a pipe 10 m long leaning against the two axes. The units on the axes are in meter.

Mis the mid-point ofAB.A is sliding along the positivey axis whileB is along the positivex axis.

(a) Find the equation of the locus ofM.

(5 marks)

(b) When OB = 6 m,B is moving away from O at 2 m/s. At this moment, find

i. the length ofOA;ii. the rate of change of the position ofA;

iii. the rate of change of OBA.(7 marks)


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2009-CE-A MATH14 14


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2009-CE-A MATH15 15

15. A circle Ccentred at S passes throughA(6, 0) and touches the lineL:x + 3y 16 = 0 atB(4, 4).

(a) Find the equation ofSB.

(3 marks)

(b) Find the coordinates ofS.

Hence find the equation of the circle C.

(4 marks)

(c) Fis the family of circles passing through the points of intersection ofCand the liney 1 = 0.

i. Write down the equation ofF.

ii. IfL also touches another circle C1 in F, find the equation ofC1.

(5 marks)


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2009-CE-A MATH17 17

16.

In the above figure, OAB is an equilateral triangle with OA = 1.Mis the mid. point ofAB and P divides

the line segment OA in the ratio of 2 :1 . Q is a point on OB such that PQ intersects OM at G and

: 4 : 3PG GQ = . Let OA

and OB

be a and b respectively.

(a) Find OM

in terms ofa and b.

(1 marks)

(b) Let : : (1 )OQ QB k k =

i. Find OG

in terms of k, a and b.

ii. Hence find PQ

.

(4 marks)

(c) i. Find a b and hence find PQ

ii. Find QGMand correct the answer to the nearest degree.(7 marks)


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2009-CE-A MATH18 18


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17.

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