m2 mock exam - munsang college

EP (M2) MOCK 13-1 1

Please stick the barcode label here.

Candidate Number

HKDSE

MATH EP

M2

© 香港教育圖書公司保留版權

Hong Kong Educational Publishing Company

All Rights Reserved 2019

HONG KONG EDUCATIONAL PUBLISHING COMPANY

HONG KONG DIPLOMA OF

SECONDARY EDUCATION EXAMINATION

MATHEMATICS Extended Part

Module 2 (Algebra and Calculus)

Mock Exam 13 (2019)

Question-Answer Book

Time allowed: 2½ hours

This paper must be answered in English

INSTRUCTIONS

1. After the announcement of the start of the examination,

you should first write your Candidate Number in the space

provided on and stick barcode labels in the spaces

provided on Pages 1, 3, 5, 7, 9 and 11.

2. This paper consists of TWO sections, A and B.

3. Attempt ALL questions in this paper. Write your answers

in the spaces provided in this Question-Answer Book. Do

not write in the margins. Answers written in the margins

will not be marked.

4. Graph paper and supplementary answer sheets will be

supplied on request. Write your Candidate Number, mark

the question number box and stick a barcode label on each

sheet, and fasten them with string INSIDE this book.

5. Unless otherwise specified, all working must be clearly

shown.

6. Unless otherwise specified, numerical answers should be

exact.

7. No extra time will be given to candidates for sticking on

the barcode labels or filling in the question number boxes

after the ‘Time is up’ announcement.

EP (M2) MOCK 13-2 2 © Hong Kong Educational Publishing Company

Answers written in the margins will not be marked.

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FORMULAS FOR REFERENCE

BABABA sincoscossin)sin( 2

cos2

sin2sinsinBABA

BA

BABABA sinsincoscos)cos( 2

sin2

cos2sinsinBABA

BA

BA

BABA

tantan1

tantan)tan(

2cos

2cos2coscos

BABABA

)sin()sin(cossin2 BABABA 2

sin2

sin2coscosBABA

BA

)cos()cos(coscos2 BABABA

)cos()cos(sinsin2 BABABA

********************************************************************************

Section A (50 marks)

1. Let 2( ) ( 4) lnf x x x . Find f '(2) from first principles.

(4 marks)

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EP (M2) MOCK 13-3 3



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2. Expand 5(2 3 )x . Hence, find the constant term in the expansion of

2

5

2

1(2 3 ) 3x

x

.

(5 marks)

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3. (a) Prove the identity sin sin

tan2 cos cos

x y x y

x y

.

(b) Using (a), express tan37.5 in surd form. Rationalize the denominator of the answer if

necessary.

(5 marks)

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EP (M2) MOCK 13-5 5




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4. (a) Using integration by parts, find 2 (2 )uu du .

(b) Define 2 3( ) (2 )xf x x for all real numbers x. Find the area of the region bounded by the

graph of ( )y f x , the x-axis, the straight lines 1x and 2x .

(6 marks)

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5. (a) Using integration by substitution, find 5 3 9x x dx , where 3 9x .

(b) At any point (x, y) on the curve , the slope of the tangent to is 5 310 9x x . The

y-intercept of is 100. Find the equation of .

(7 marks)

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EP (M2) MOCK 13-7 7




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6. (a) Using mathematical induction, prove that 2

1

( 1) (2 1)(2 1) 1( 1) (2 1)

2

nnk

k

n nk

for

all positive integers n.

(b) Using (a), evaluate 200

1 2

99

( 1) (2 1)k

k

k

.

(7 marks)

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EP (M2) MOCK 13-9 9




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7. Let n be a positive integer.

(a) Define 1

0 1

cM

, where c is a real number. Evaluate

(i) 2M ,

(ii) nM ,

(iii) 1( )nM .

(b) Evaluate

(i) 1

0

1

3

n

kk

,

(ii)

1 2

10

3

n

.

(8 marks)

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EP (M2) MOCK 13-11 11




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8. Define 2

( )8 12

kf x

x x

, where k is a constant.

It is given that the extreme value of f (x) is 2.

(a) Find ( )f x .

(b) Find the asymptote(s) of the graph of ( )y f x .

(c) Someone claims that there is at least one point of inflexion of the graph of ( )y f x . Do

you agree? Explain your answer.

(8 marks)

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EP (M2) MOCK 13-13 13



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Section B (50 marks)

9.

Figure 1

Peter wants to move a heavy box upstairs by using a wooden board as an inclined plane.

Initially, the wooden board touches the floor and the edge of the highest step of the stairs at the

same time as in Figure 1. Let PQ be the side-view of the wooden board, and E is the foot of

perpendicular from P to AB, so that EB = 128 cm and BC = 54 cm. Let PQC = .

(a) Find the length of PQ in terms of .

(1 mark)

(b) Find the shortest length of the wooden board.

(5 marks)

(c)

Figure 2

Suppose the length of the wooden board is 260 cm. To adjust the inclination of the

wooden board, Peter moves the board slowly so that the end of the board, Q, moves

towards D (see Figure 2). The board touches the floor and the edge of the highest step of

the stairs at the same time. Let x cm be the perpendicular distance from P to BC.

(i) When BQ = 162 cm, the rate of change of is 0.03 rad s1

. Find the rate of change

of x at this moment.

(ii) Peter claims that Q is moving towards D at a speed higher than the horizontal speed

of P moving towards BC. Do you agree? Explain your answer.

(6 marks)

A B

C D

E

P

Q

A B

C D

x cm P

Q

EP (M2) MOCK 13-15 15



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EP (M2) MOCK 13-17 17



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10. (a) (i) Prove that 4 3 21

tan tan tan3

xdx x xdx .

(ii) Evaluate

5π

443π

4

tan xdx .

(5 marks)

(b) (i) Let f (x) be a continuous function for a x b , where a and b are constants, such

that ( ) ( )f x f a b x for all a x b . Prove that ( ) ( )2

b b

a a

a bxf x dx f x dx

.

(ii) Evaluate

5π

443π

4

tanx xdx .

(5 marks)

(c) Consider the curve :2tany x x , where

7π 9π

4 4x . Let R be the region bounded by

, the x-axis, the two lines 7π

4x and

9π

4x . Find the volume of the solid of

revolution generated by revolving R about the x-axis.

(3 marks)

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EP (M2) MOCK 13-19 19



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11. (a) Consider the system of linear equations in real variables x, y, z

2 (2 1) 2

( ) : 3 5 2

3 (3 ) 1

hx y h z

E x y z k

x y h z k

, where h and k are real numbers.

(i) Assume that (E) has a unique solution.

(1) Prove that 2h and 8h .

(2) Solve (E).

(ii) Assume that 2h and (E) is consistent.

(1) Find k.

(2) Solve (E).

(9 marks)

(b) Is there a real solution of the system of linear equations

3 6

2 2 3 2

3 5 10

x y z

x y z

x y z

satisfying 22 1x y z ? Explain your answer.

(3 marks)

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EP (M2) MOCK 13-21 21



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12. OAB is a triangle. It is given that 4 3OA i j k and 2 3 7OB i j k . P is a point on OA

such that OA PB.

(a) (i) Find OA OB .

(ii) Find :OP OA .

(4 marks)

(b) Q is a point on BP such that PQ : QB = 1 : t, where t is positive.

(i) Express OQ in terms of t.

(ii) Suppose that Q is the orthocentre of OAB.

(1) Find the value of t.

(2) Find the area of OPQ : the area of AQB.

(9 marks)

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EP (M2) MOCK 13-23 23



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m2 mock exam - munsang college

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