MATHEMATICS Compulsory Part PAPER 1

Challenging Mathematics for HKDSE Mock Exam Papers

HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION

MATHEMATICS Compulsory Part

PAPER 1

Mock Paper Set 5

Question-Answer Book

Time allowed: 2 hours 15 minutes This paper must be answered in English.

INSTRUCTIONS

1. Write your Candidate Number in the space

provided on .

2. Stick barcode label in the space provided on

Page 1.

3. This paper consists of THREE sections, A(1),

A(2) and B.

4. 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.

5. 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.

6. Unless otherwise specified, all working must

be clearly shown.

7. Unless otherwise specified, numerical answers

should be either exact or correct to 3 significant

figures.

8. The diagrams in this paper are not necessarily

drawn to scale.

Kendy Publishing Company Limited

Candidate Number

Please stick the barcode label here.

Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ2 Page total

SECTION A (1) (35 marks)

1. Simplify

57

293

nm

nm and express your answer with positive indices. (3 marks)

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2. Make A the subject of the formula zAyAx )3( . (3 marks)

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3. Simplify xx 81

4

56

3

. (3 marks)

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4. Factorize

(a) yx 513

(b) 22 103 yxyx

(c) yxyxyx 153103 22 (4 marks)

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5. In a college, there are 420 students and the number of male students is 25% less than the

number of female students. Find the difference of the number of male students and the

number of female students. (4 marks)

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6. Consider the compound inequality

525

311 x

x or 0157 x ………… (*).

(a) Solve (*).

(b) Write down the greatest negative integer satisfying (*). (4 marks)

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7. In a polar coordinate system, O is the pole. The polar coordinates of the point A are (12, 105). If A is rotated anticlockwise about O through 60 to a point B,

(a) find the polar coordinates of B,

(b) find the distance between A and B,

(c) what is the number of folds of rotational symmetry of AOB? (4 marks)

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8. It is given that f(x) is the sum of two parts, one part varies as x2 and the other part varies as x.

Suppose that f(4) = 48 and f(8) = 128.

(a) Find f(x).

(b) Solve the equation f(x) = 105. (5 marks)

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9. The frequency distribution table and the cumulative frequency distribution table below shows the distribution of the heights of the 100 students in Form 6 at a school.

Height (m) Frequency Height less than (m) Cumulative frequency

1.51 – 1.55 a 1.555 5

1.56 – 1.60 20 1.605 x

1.61 – 1.65 b 1.655 54

1.66 – 1.70 c 1.705 y

1.71 – 1.75 18 1.755 96

1.76 – 1.80 d 1.805 z

(a) Find x, y and z.

(b) If a student is randomly selected from the Form 6 students at the school, find the probability that the height of the selected student is less than 1.705 m but not less than 1.605 m. (5 marks)

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SECTION A (2) (35 marks)

10. The coordinates of the points A and B are (6, 8) and (12, 0) respectively. L1 and L2 are two straight lines intersect at A and they cut the x-axis at the origin O and at B respectively. Let P be a moving point in the rectangular coordinate plane such that P is equidistant from L1 and L2. Denote the locus of P by .

(a) Someone claims that consists of two straight lines. Is the claim correct? Explain your answer. (2 marks)

(b) intersects the x-axis and the y-axis at H and K respectively. Let C be the circle which passes through O, H and K. Find the area of C. Give your answer in terms of . (3 marks)

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11. Fig. (a) below shows an inverted right

circular conical vessel made from soft

plastic sheet. The height of the vessel is

27 cm. The dotted circle XY is parallel to

the base and one-third of the height from

the base. Paul pushes up the lower portion

VXY of the cone along circle XY to form

the new vessel in Fig. (b). He then pours

64 cm3 of milk into the new vessel until

overflows.

(a) Find the original volume of the vessel VAB in Fig. (a). Give your answer in terms of . (3 marks)

(b) Paul claims that the final area of the wet curved surface of the new vessel in Fig. (b) is at least 300 cm2. Do you agree? Explain you answer. (3 marks)

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12. The bar chart shows the distribution of the ages of

26 students in a class, where a = b and 4 < a, b < c.

The median of the ages of the students in the class

is 19.5.

(a) Find a, b and c. (3 marks)

(b) Four more students now join the class. It is found that the ages of these four students are all different and the range of the ages of the students in the class remains unchanged. Find

(i) the greatest possible median of the ages of the students in the class, (ii) the least possible mean of the ages of the students in the class. (4 marks)

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A

B C

D

P

Q

13. In the figure, ABCD is a rectangle, P and Q are

points lying on AC such that AP = CQ.

(a) Prove that CDQABP . (3 marks)

(b) Suppose that AB = 10 cm, AD = 24 cm and AP = 6 cm.

(i) Find PQ.

(ii) Is ABP a right-angled triangle? Explain your answer. (5 marks)

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14. Let )52)(53()f( 22 kxxhxxx and 1)g( xx where h and k are constants.

When ))f(g(x is divided by 2x and when ))f(g(x is divided by 2x , the two

remainders are equal. It is given that nmxxxxx 234 60256)f( , where m and n are

constants.

(a) Find h and k. (5 marks)

(b) How many real roots does the equation 0)f( x have? Explain your answer. (5 marks)

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Section B (35 marks)

15. An examination paper consists of two sections. Section A has 3 questions and Section B has 5

questions. Candidates are required to answer 4 questions. Clara randomly chooses 4

questions to answer. Find the probability that she chooses 2 questions from each section.

(3 marks)

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16. A test consists of two papers, Paper 1 and Paper 2, and the mean of the distribution of the

marks in Paper 1 and Paper 2 are 61 and 46 respectively. The total mark is the sum of the

marks in Paper 1 and Paper 2. The following table shows the marks and the standard scores of

Amy in the test.

Mark Standard score

Paper 1 64 1.5

Paper 2 36 2.5

Billy knows that his standard scores in Paper 1 and Paper 2 are 1.7 and 2.6 respectively. He

claims that his total mark is greater than the total mark of Amy. Is the claim correct? Explain

your answer. (4 marks)

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17. The first term and the 4th term of a geometric sequence are 4374 and 162 respectively. Find

(a) the common ratio of the sequence, (2 marks)

(b) the least value of n such that the sum of the first n terms of the sequence exceeds 6500.

(3 marks)

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18. The figure shows a geometric model ABCD in the

form of tetrahedron. It is given that ACB = 50, AC = AD = 18 cm, BC = BD = 24 cm and

CD = 20 cm.

(a) Find AB. (2 marks)

(b) Find the angle between the faces ACD and BCD. Give your answer correct to the nearest degree. (3 marks)

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19. Let )53(22

1)f( 2 ccxxx where c is a constant.

(a) Using the method of completing the square, find the coordinates of the vertex of the graph y = f(x). Give your answers in terms of c. (2 marks)

(b) If the graph of y = f(x) touches the x-axis, find the possible values of c. (2 marks)

(c) Under a transformation, f(x) is changed to )53(22

1 2 ccxx . Describe the

geometric meaning of the transformation. (2 marks)

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20. PQR is an acute-angled triangle. S is a point such that PQSP and QRSR . Denote

the orthocentre of PQR by H.

(a) Prove that

(i) PQRS is a cyclic quadrilateral,

(ii) PHRS is a parallelogram. (5 marks)

(b) A rectangular coordinate system, with O as the origin, is introduced so that the coordinates of P, Q and R are (0, 12), (8, 0) and (6, 0) respectively.

(i) Find the equation of the circle which passes through P, Q and R.

(ii) Denote the circumcentre of PQR by G. Someone claims that Q, O, H and G are concyclic. Is the claim correct? Explain your answer.

(7 marks)

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END OF PAPER

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Challenging Mathematics for HKDSE ─ Mock Exam Papers

Hong Kong Diploma of Secondary Education Examination Mathematics Compulsory Part

Paper 2 Mock Paper Set 5

Time allowed: 1 hour 15 minutes

1. Read carefully the instructions on the Answer Sheet. Stick a barcode label and insert the information

required in the spaces provided.

2. When told to open this book, you should check that all the questions are there. Look for the words ‘END OF PAPER’ after the last question.

3. All questions carry equal marks.

4. ANSWER ALL QUESTIONS. You are advised to use an HB pencil to mark all the answers on the

Answer Sheet, so that wrong marks can be completely erased with a clean rubber.

5. You should mark only ONE answer for each question. If you mark more than one answer, you will receive

NO MARKS for that question.

6. No marks will be deducted for wrong answers.

Kendy Publishing Co. Ltd.

MP5-MC2

There are 30 questions in Section A and 15 questions in Section B. The diagrams in this paper are not necessarily drawn to scale. Choose the best answer for each question. Section A

1.

)8(

4

1 6722

504

A. 1.

B. 1.

C. 4

1.

D. 4

1 .

2. If b

yx

a

yx , then y =

A. xba

ba

.

B. xba

ab

.

C. xba

ba

.

D. xab

ba

.

3. yxyx 844 22

A. )42)(2( yxyx .

B. )42)(2( yxyx .

C. )42)(2( yxyx .

D. )42)(2( yxyx .

4. 0.0389567 =

A. 0.040 (correct to 2 decimal places).

B. 0.040 (correct to 3 significant figures).

C. 0.0390 (correct to 4 decimal places).

D. 0.03896 (correct to 5 significant figures).

MP5-MC3

5. If 7262 yxyx , then x =

A. 1.

B. 3.

C. 5.

D. 7.

6. Let kxxxx 215)f( , where k is a constant. If )f(x is divisible by 1x , find the remainder when f(x)

is divided by 1x .

A. 0

B. 1

C. 2

D. 2

7. The solution of 625 xx or 522 xx is

A. x > 2.

B. x < 2.

C. x > 1.

D. x > 2 or x > 1.

8. Let a be a constant. If the quadratic equation 1222 aaxx has equal roots, then a

A. 1.

B. 1.

C. 1 or2

1 .

D. 2

1or 1.

9. The figure shows the graph of baxxy 22 , where a and b are constants.

The equation of the axis of symmetry of the graph is

A. x = 3.

B. x = 4.

C. x = 6.

D. x = 8.

10. If A is smaller than B by 25% and B is greater than C by %20 , then

A. A is greater than C by10%.

B. A is less than C by10%.

C. C is greater than A by10%.

D. C is less than A by 10%.

MP5-MC4

11. If a and b are positive numbers such that 1123

57

ab

ba, then a : b =

A. 15 : 28.

B. 28 : 15.

C. 28 : 29.

D. 29 : 28.

12. It is given that z varies directly as x and inversely as y. If x is decreased by 36% and z is increased by 28%,

then y

A. is increased by 12.5%.

B. is increased by 22%.

C. is decreased by 37.5%.

D. is decreased by 62.5%.

13. The costs of flour A and flour B are $8/kg and $12/kg respectively. If x kg of flour A and y kg of flour B are

mixed together, the cost of the mixture is $10.5/kg. Find x : y.

A. 1 : 1

B. 2 : 3

C. 3 : 5

D. 4 : 7

14. In the figure, the 1st pattern consists of 5 dots. For any positive integer n, the (n + 1)th pattern is formed by

adding (n + 4) dots to the nth pattern. Find the number of dots in the 8th pattern.

A. 41

B. 50

C. 61

D. 72

15. According to the figure, which of the following must be true?

A. 180zxy

B. 180zyx

C. 360xzy

D. 720zyx

MP5-MC5

P M S

RQ

T

16. In the figure, AB = AC = 10 cm, BC = 12 cm and CD = 5 cm. If AB // CD,

then the area of quadrilateral ABCD is

A. 60 cm2.

B. 72 cm2.

C. 84 cm2.

D. 120 cm2.

17. In the figure, ABCD is a parallelogram and AC = AD. If E is a

point on BC such that AB = AE and BAE = 30, then CAE =

A. 30. B. 35. C. 40. D. 45.

18. The figure shows a right trapezoidal prism. If AB = DC = 5 cm,

AD = 8 cm, BC = 14 cm and CX = 20 cm, then the total surface area of

the prism is

A. 648 cm2.

B. 688 cm2.

C. 728 cm2.

D. 768 cm2.

19. In the figure, OAB and OCD are sectors with centre O. It is

given that the area of the shaded region is 123π cm2. If AC = 9 cm and ∠AOB = 120°, then OA =

A. 14 cm.

B. 16 cm.

C. 18 cm.

D. 20 cm.

20. In the figure, PQRS is a square and M is the mid-point of PS. If

RM and QS meet at T, then area of PQTM : area of QRT =

A. 2 : 1.

B. 3 : 2.

C. 4 : 3.

D. 5 : 4.

21. In the figure, ABCD is a rectangle. If aAD and bDC , then DE =

A. (a + b) cos.

B. (a + b) sin.

C. sincos ba .

D. cossin ba .

A

B C

D

A

B C

D

E

B

X Y A

C

D

WZ

A C

D

B

a b

E

MP5-MC6

P

Q

R

S

22. In the figure, O is the centre of the circle ABCD. If BAC = 13 and

ADC = 58, then AOB =

A. 60 B. 71 C. 84 D. 90

23. In the figure, the rhombus PQRS is divided into nine identical small

rhombuses and five of them are shaded. The number of axes of

reflectional symmetry and the number of folds of rotational

symmetry of the rhombus PQRS is

axes of reflectional symmetry folds of rotational symmetry

A. 2 2

B. 2 4

C. 4 2

D. 4 4

24. If an interior angle of a regular n-sided polygon is 5 times an exterior angle of the polygon, which of the

following are true?

I. The value of n is 12.

II. The number of axes of reflectional symmetry of the polygon is 6.

III. The number of folds of rotational symmetry of the polygon is 12.

A. I and II only

B. I and III only

C. II and III only

D. I, II and III

25. The equations of L1 and L2 are 04154 yx and 0384 yax respectively. If L1 is perpendicular to

L2, find the intersection of L1 and L2.

A. (4, 1)

B. (0, 5)

C. (3, 3)

D. (6, 2)

26. In the figure, ABCD is a square. The coordinates of B are

A. (1, 5)

B. (0, 6)

C. (1, 7)

D. (2, 8)

A B

C

D

O

D

B

MP5-MC7

27. Which of the following about the circle 015201222 22 yxyx are true?

I. The coordinates of the centre of the circle are (3, 5).

II. The radius of the circle is 11.

III. The point (4, 6) lies outside the circle.

A. I and II only

B. I and III only

C. II and III only

D. I , II and III

28. Wilson has two $20 banknotes, two $50 banknotes and one $100 banknot in the pocket. Wilson takes out two

banknotes randomly from his pocket. Find the probabililty that he will get enough money to buy a T-shirt of

price $110.

A. 0.2

B. 0.3

C. 0.4

D. 0.5

29. Vincent plays the following game once. A ball falls randomly into the tubes A, B, C or D with equal chances.

The ball which falls into tubes A, B, C and D can get back $4, $1, $1 and $10 respectively. Find the expected

money he can get.

A. $4

B. $4.25

C. $4.75

D. $5

30. If the mean and the mode of the 11 numbers : 19 , 10 , 12 , 12 , 13 , 14 , 15 , 16 , a , b and c are 14 and 15

respectively, then the median of these 11 numbers is

A. 13.

B. 14.

C. 15.

D. 16.

MP5-MC8

Section B

31. The H.C.F. of 18 3 x , 144 2 xx and 14 2 x is

A. 2x 1.

B. (2x 1)2.

C. (2x 1)(2x + 1)(4x2 + 2x + 1).

D. (2x 1) 2 (2x + 1)(4x2 + 2x + 1).

32. The figure shows the linear relation between x2log and y2log .

Which of the following must be true?

A. 409643 yx

B. 409634 yx

C. 409643 yx

D. 409634 yx

33. 2257 223229

A. 100010101002.

B. 100101010002. C. 101001010002. D. 101010010002.

34. If and

425

4252

2

ββ

αα, then

A. 2.

B. 2.

C. 4.

D. 5.

35. The figure shows a shaded region (including the boundary).

If (a, b) is a point lying in the shaded region, which of the

following are true?

I. 1b

II. 12 ab

III. ab 24

A. I only

B. III only

C. I and II only

D. I and III only

MP5-MC9

A

B D

C

36. Let an be the nth term of a geometric sequence. If 4326 a and 8129 a , which of the following must be

true?

I. 14583 a

II. 14

2 a

a

III. 2000321 naaaa

A. I and II only

B. I and III only

C. II and III only

D. I, II and III

37. The figure shows the graph of 2

tan3 x

hy where h is a

constant and x0 . Which of the following are true?

I. h > 0

II. 45

III. h

3tan

A. I and II only

B. I and III only

C. II and III only

D. I, II and III

38. For 3600 x , how many roots does the equation 0cos10sin3 2 xx have?

A. 2

B. 3

C. 4

D. 5

39. The figure shows a regular tetrahedron ABCD. Find the angle

between the line AC and the plane BCD correct to the nearest degree.

A. 45 B. 50 C. 55 D. 60

MP5-MC10

40. In the figure, O is the centre of the semi-circle DABE. The

circle ABC cuts the semi-circle at A and B and touches the

diameter DE of the semi-circle at O. BD cuts the circle at C.

If BDO = 24, then BAC =

A. 72°

B. 84°

C. 96°

D. 108°

41. Let O be the origin. The coordinates of the points A and B are (0, 66) and (99, 33) respectively. The

x-coordinate of the orthocentre of OAB is

A. 9.

B. 11.

C. 22.

D. 33.

42. Box P contains 3 blue coins and 5 yellow coins while box Q contains 2 blue coins and 4 yellow coins. If a box

is randomly chosen and then a coin is randomly drawn from the box, find the probability that a yellow coin is

drawn.

A. 7

2

B. 14

9

C. 24

9

D. 48

31

43. A queue is formed by 2 boys and 5 girls. If no boys are next to each other, how many different queues can be

formed?

A. 720

B. 1440

C. 4320

D. 5040

A B

C

D EO

MP5-MC11

44. The stem-and-leaf diagram below shows the distribution of the scores of 18 participants in a singing contest.

Stem (tens) Leaf (units) 4 4 5 h 7

5 2 6 6 7 h 9

6 0 0 3 3 9 k

8 2 k

Which of the following must be true?

I. The median of the distribution is 58.

II. The range of the distribution is 44.

III. The inter-quartile range of the distribution is 11.

A. I only

B. II only

C. I and III only

D. II and III only

45. If the variance of the seven numbers x1, x2, x3, x4, x5, x6 and x7 is 5, then the variance of the seven numbers 2x1 + 3, 2x2 + 3, 2x3 + 3, 2x4 + 3, 2x5 + 3, 2x6 + 3, 2x7 + 3 is

A. 10.

B. 15.

C. 20.

D. 25.

END OF PAPER

41

41

Section A(1) [35 marks]

Marks

1. 57

186

57

293

nm

nm

nm

nm 1A

231nm 1A

m

n23

1A

3

2. zAyAx )3(

AzyzAx 3 1A

yzAzAx 3

yzzxA )3( 1A

zx

yzA

3 1A

3

3. )81)(56(

)56(4)81(3

81

4

56

3

xx

xx

xx

1M

)81)(56(

2024243

xx

xx

)81)(56(

17

xx 1A

)18)(56(

17

xx 1A

3

4. (a) )5(3513 yxyx 1A

(b) )5)(2(103 22 yxyxyxyx 1A

(c) yxyxyx 153103 22

)5(3)5)(2( yxyxyx 1A

)32)(5( yxyx 1A

4

Mock Paper Set 5 (Paper 1) Answers and Solutions


42

Marks

5. Let the number of male students and the number of female students be x and y

respectively. Then we have

)2(%)251(

)1(420

yx

yx

2M

From (2), x = 0.75y …(3)

Substituting (3) into (1),

240

42075.1

4200.75

y

y

yy

Substituting y = 420 into (3),

180

)240(75.0

x

The difference = 240 180 = 60

1M+1A

4

6. (a) 52

5

311 x

x

x

x

xx

2

714

2510311

or

7

15

157

0157

x

x

x

2A

2x 1A

(b) 1 1A

4

7. (a) The polar coordinates of B

= (12, 105 + 60)

= (12, 165)

1A

(b) OAB is an isosceles triangle with OA = OB = 12 and AOB = 60.

The base angles are equal and both are 60.

i.e. OAB is an equilateral triangle.

1M

The distance between A and B is 12. 1A

(c) The number of folds of rotational symmetry of the equilateral AOB is 3. 1A

4


43

Marks

8. (a) Let bxaxx 2)f( , where a and b are constants. Then

)2(168

)1(124

128864

48416

ba

ba

ba

ba

1A

(2) (1):

1

44

a

a

Substituting a = 1 into (1),

8

124

b

b

xxx 8)f( 2 1M+1A

(b) 10582 xx

0)15)(7(

010582

xx

xx

x = 7 or 15

2A

5

9. (a) 205 x

25

78

9618

y

y

1A

1A

z = 100 1A

(b) The required probability

=

100

2578

= 0.53

1M+1A

5


44

Section A(2) [35 marks]

Marks

10. (a) Yes, consists of two straight lines.

They are the two straight lines bisecting the angles between L1 and L2.

1A

1A

(b) AOB is an isosceles triangle with AO = AB.

consists of the vertical and horizontal lines passing through A.

The coordinates of H and K are (6, 0) and (0, 8) respectively. 1A

Since HOK = 90, HK is a diameter of the circle C.

Radius of C = 22 86

2

1 = 5

Area of C = 2(5) = 25 square units 1M+1A

5

11. (a) Let the radius of the circular base be r cm.

Volume of cone VAB = )27(

3

1 2r = 29 r

Volume of cone VXY =

273

2

3

2

3

12

r = 2

3

8r

Volume of the part of the cone above AB in the new vessel

=

273

1

3

1

3

12

r = 2

3

1r

4

644

643

1

3

8

3

89

2

2222

r

r

rrrr

1M+1A

The original volume = 2)4(9 = 146 cm3 1A

(b) The final area of the wet curved surface

=

3

274

3

4274(4)

2222

1M+1A

= 725

9

32

> 300 cm2 1A

6


45

Marks

12. (a) Median = 19.5

1343 cba

a = b = 5 and c = 9 3A

(b) The ages of the four students may be {17, 18, 19, 20} or {18, 19, 20, 21}.

(i) When the ages are {18, 19, 20, 21}, the median will be the

greatest.

The greatest possible median = 19.5 1M+1A

(ii) When the ages are {17, 18, 19, 20}, the mean will be the least.

The least possible mean

= 4)2110206196184(1730

1

=30

419

1M+1A

7

13. (a) AB = CD and AB // DC (opp. sides of rectangle)

BAP = DCQ (alt. s, AB // DC)

AP = CQ (given)

CDQABP (SAS) (3)

(b) (i) 262410 22 AC

PQ = 26 6 6 = 14 cm 1M+1A

(ii) Area of ABC = (10)(24)2

1=120 cm2

Suppose BP AC. Then

26

2120

1202

1

BP

ACBP

i.e. 13

39BP cm

On the other hand, by Pythagoras theorem,

BP = 22 610 = 8 cm 1A

This leads to a contradiction.

ABP is not a right-angled triangle 1M+1A

8


46

Marks

14. (a) kxxhxxx )1(5)1(2)1(5)1(3))f(g( 22

)32)(23( 22 kxxhxx 1M

)7)(8()328)(2212())2f(g( khkh

)3)(12()328)(2212())2f(g( khkh

From the question,

5

361235687

)3)(12()7)(8(

hk

khhkkhhk

khkh

1A

Then

)5()52(5)8(5256

)5()5255()225153(256

)552)(53()f(

232

232

22

hhxhxhxx

hhxhhxhhxx

hxxhxxx

1M

By comparing the coefficient of x2,

4

128

60)85(

h

h

h

1A

k = 4 + 5 = 9 1A

(b) )952)(453()f( 22 xxxxx

Considering 0453 2 xx ,

= (5)2 4(3)(4) = 23 < 0 1M

This equation has no real roots. 1A

Considering 0952 2 xx ,

= (5)2 4(2)(9) = 47 < 0 1M

This equation has no real roots. 1A

The equation 0)f( x has no real roots. 1A

10


47

Section B [35 marks]

Marks

15. The required probability

= 84

52

32

C

CC

1M+1A

=7

3 1A

3

16. Let the standard deviation of the marks in Paper 1 and Paper 2 be 1 and 2

respectively.

2

5.16164

1

1

1A

4

5.24636

2

2

1A

Let the marks of Billy in Paper 1 and Paper 2 be b1 and b2 respectively.

4.64

7.12

61

1

1

b

b

6.35

6.24

46

1

2

b

b

Total mark of Billy = 64.4 + 35.6 = 100 1A

His total mark is not greater than that of Amy. 1A

4

17. (a) Let the common ratio be r.

1624374 3 r 1M

3

127

13

r

r 1A


48

Marks

(b) Sum of the first n terms

=

3

11

3

114374

n

=

n

3

116561

= 65003

116561

n

1A

258.43log61

6561log

61

65613

6561

61

3

1

6561

6500

3

11

n

n

n

n

1M

The least value of n is 5. 1A

5

18. (a) In ABC,

50cos)24)(18(22418 222AB 1M

fig.) sig. 6 to(cor. 632.344

fig.) sig. 4 to(cor. cm 56.18AB 1A

(b) Let M be the mid-point of CD.

Since AC = AD and BC = BD,

AM CD and BM CD.

The angle between the faces ACD and BCD is AMB. 1M

fig.) sig. 6 to(cor. cm 9666.14

1018 22

AM

fig.) sig. 6 to(cor. cm 8174.21

1024 22

BM

In AMB,

AMB cos)1.81742)(14.9666(2476224344.632 1M

degree)nearest the to(cor. 57

fig.) sig. 4 to(cor. 5442.0cos

AMB

AMB 1A

5


49

Marks 19.

(a) )53()4(2

1)53(2

2

1 22 ccxxccxx

)532()2(

)53(2)2(2

1

)53()444(2

1

22

22

222

cccx

cccx

ccccxx

1A

The vertex is (2c, 2c2 + 3c 5). 1A

(b) If the graph of y = f(x) touches the x-axis, then

0)1)(52(

0532 2

cc

cc 1M

c = 2

5 or 1 1A

(c) )53()(2)(2

1)f( 2 cxcxx

)53(22

1 2 ccxx

1M

The graph of f(x) is reflected along the y-axis. 1A 6

20. (a) (i) QPS = QRS = 90

QPS + QRS = 180

PQRS is a cyclic quadrilateral. (opp. s sup.) (2)

(ii) G lies on the altitude from P to QR.

PG QR

PG // SR

G lies on the altitude from R to PQ.

RG PQ

RG // SP

PHRS is a parallelogram. (opp. sides //) (3)


50

Marks (b) (i) Let the equation of the circle be 022 FEyDxyx .

Substituting the coordinates of P, Q and R into the equation, we have

0)0()6(06

0)0()8(0)8(

0)12()0(120

22

22

22

FED

FED

FED

)3...(0636

)2...(0864

)1...(012144

FD

FD

FE

1M

(3) (2):

2

01428

D

D

Substituting D = 2 into (3),

48

0)2(636

F

F

Substituting F = 48 into (1),

8

04812144

E

E 1A

The equation of the circle is 0488222 yxyx . 1A

OR 65)4()1( 22 yx

(ii) G is the centre and QS is a diameter of the circle which passes through

P, Q and R.

The coordinates of G are (1, 4) and the coordinates of S

are (6, 8). 1A

From (a)(ii), PHRS is a parallelogram.

PH // RS and PH = RS

The coordinates of H are (0, 4). 1A

HG is horizontal and 90HGQ . 1M

Note that HOQ = 90.

180HOQHGQ

Q, O, H and G are not concyclic. 1A 12

92

92

1. A 16. B 31. A

2. A 17. D 32. A

3. A 18. C 33. B

4. C 19. B 34. B

5. B 20. D 35. A

6. A 21. C 36. A

7. A 22. D 37. B

8. A 23. A 38. A

9. A 24. B 39. C

10. B 25. D 40. A

11. C 26. A 41. B

12. C 27. B 42. D

13. C 28. C 43. C

14. C 29. A 44. C

15. C 30. B 45. C

1. )8(4

1 6722

504

))2((

)2(

1 6723

2

5042

)2(2

1 20162

1008

)2(2

1 20162016

1 (A)

2. b

yx

a

yx

xba

bay

baxbay

bxaxbyay

ayaxbybx

yxayxb

)()(

)()(

(A)

3. yxyx 844 22

)2(4)2)(2( yxyxyx

)42)(2( yxyx (A)

4. (A) : 0.0389567 = 0.04 (cor. to 2 d.p.)

(B) : 0.0389567 = 0.0390 (cor. to 3 sig. fig.)

(C) : 0.0389567 = 0.0390 (cor. to 4 d.p.)

(D) : 0.0389567 = 0.038957 (cor. to 5 sig. fig.)

(C) 5. 7262 yxyx

(2)72

(1)762

yx

yx

(1) + (2) 2:

3

2165

x

x (B)

6. Since f(x) is divisible by 1x , 0)1f(

1

0111

0)1()1()1( 215

k

k

k

When )f(x is divided by 1x ,

the remainder

)1f(

1)1()1()1( 215

0 (A)

7. 625 xx 522 xx

2

63

x

x or

1

33

x

x

2x (A) 8. Rewrite 1222 aaxx as

0)12(22 aaxx .

Since 0)12(22 aaxx has equal roots,

0

1

0)1(

012

0484

0)]12([4)2(

2

2

2

2

a

a

aa

aa

aa

(A)



93

9. y-intercept 16 b = 16 Substituting (2 , 0) into 162 2 axxy ,

12

242

16)2()2(20 2

a

a

a

Line of symmetry: 3)2(2

12 x (A)

Alternative Method Since the graph cuts the x-axis at x = 2, the function can be rewritten as ))(2(2 cxxy .

Substituting (0, 16) into ))(2(2 cxxy ,

4

16)0)(2(02

c

c

The another x-intercept is 4. Line of symmetry: x = 3

10. BBA 75.0%)251( CCB 2.1%)201(

CCA 0.9)2.1)(75.0(

A is less than C by 10%. (B)

11. 1123

57

ab

ba

29

28

2829

223357

)23(1157

b

a

ba

abba

abba

29:28: ba (C)

12. z varies directly as x and inversely as y.

Let y

xkz , where k is a non-zero constant.

z

xky

Percentage change of y

%100%)281(

%)361(

z

xk

z

xk

z

xk

%100128.1

8.0

%5.37

∴ y is decreased by 37.5%. (C)

13. 5.10128

yx

yx

5

3

5.25.1

5.105.10128

y

x

xy

yxyx

5:3: yx (C)

14. Number of dots in the 8th pattern 1110987655

61 (C)

15.

With the notation in the figure,

xa 180 (supp. s, // lines)

yx

xy

ayb

180

)180(360

pt.) aat s( 360

180zcb (alt. s, // lines)

360

180180

xzy

zyx (C)

16.

Let X be the mid-point of BC. Then BCAX . (prop. of isos. , AB = AC)

62

12

2 BC

BX

8610 22 AX

Area of ABC = (12)(8)2

1= 48 cm2

A

B C

D

X

10

12

10

5


94

2cm 24

4810

5 of Area

of Area

of Area

ACD

AB

CD

ABC

ACD

Area of ABCD = 48 + 24 = 72 cm2 (B) 17.

Since AB = AE, AEBABE (base s, isos. )

752

30180ABE

Since AC = AD, BC = AD = AC BAC = ABC = 75 (base s, isos. ) CAE = 75 30 = 45 (D)

18.


cm32

814 CKBH

cm435 22 AH

Area of the trapezium =2

)4)(148( = 44 cm2

Total surface area of the prism = 44 2 + (8 + 5 + 5 + 14) 20 = 728 cm2 (C)

19. Let OA = r cm.

16

28818

3698118

123360

120)9(

360

120

22

22

r

r

rrr

rr

OA = 16 cm (B)

20. QRT ~ SMT (AAA) QR = PS = 2SM TR = 2TM (corr. sides, ~s) Let the area of SMT be a.

aSTRTM

TR

SMT

STR

2 of Area of Area

of Area

a

aSTR

SM

QR

SMT

QRT

4

2 of Area

of Area

of Area

2

2

Area of PQTM = area of PQS area of SMT = area of QRS area of SMT = 2a + 4a a = 5a area of PQTM : area of QRT = 5 : 4 (D)

21.


cosax and sinby

sincos bayxDE (C)

A

B C

D

E

3 cm

A

B C

D 8 cm

5 cm 5 cm

3 cm H K

A

P

C

a b

E B

D

x

y


95

22.

Join OC.

26)13(22 BACBOC

( at centre twice at circumference) 116)58(22 ADCAOC

( at centre twice at circumference) AOB

BOCAOC 26116

90 (D)

23. There are 2 axes of reflectional symmetry.

The figure repeats itself when it rotates every 180°. There are 2 folds of rotational symmetry. (A)

24. I. : Exterior angle =n

360

Interior angle = 5360

n

12

252

180360

5360

nnn

II. : The number of axes of reflectional symmetry of a regular 12-sided polygon is 12.

III. : The number of folds of rotational symmetry of a regular 12-sided polygon is 12.

(B)

25. Slope of L1 Slope of L2 = 1

5

145

4

a

a

)2(03845

)1(01454

yx

yx

(1) (4) + (2) 5:

6

024641

x

x

Substituting x = 6 into 01454 yx ,

2

0105

0145)6(4

y

y

y

The intersection of L1 and L2 is (6, 2).

(D)

26.

Let the coordinates of B be (a, b). With the notation in the figure. Note that BCFABE . AE = BF and BE = CF (corr. sides, s)

ba

ab

4)2(

106

)2(6

)1(4

ba

ba

(1) + (2):

1

22

a

a

Substituting a = 1 into (2),

5

61

b

b

The coordinates of B are (1, 5). (A)

A B

C

D

O

13°

58°

D

B(a, b)


96

27. 015201222 22 yxyx

02

1510622 yxyx …(*)

I. : Centre = (3, 5)

II. : Radius2

53

2

15)5(3 22

III. : Substituting (4, 6) into (*),

L.H.S.2

15)6(10)4(664 22

02

191

The point (4, 6) lies outside the circle. (B)

28. The cells shaded grey are the favourable

outcomes. Second banknote drawn

20 20 50 50 100First

banknote drawn

20 / 40 70 70 12020 40 / 70 70 12050 70 70 / 100 15050 70 70 100 / 150100 120 120 150 150 /

P (enough money) 4.020

8 (C)

29. Expected money he can get

= 4

110

4

21

4

14

= $4 (A)

30. Mean = 14

43

154111

1411

1615141312121019

cba

cba

cba

Mode = 15 At least two of the numbers a, b and c are 15. a, b and c are 13, 15 and 15. Arrange the 11 numbers in ascending order: 10 , 12 , 12 , 13 , 13 , 14 , 15 , 15 , 15 , 16 , 19 Median = 14 (B)

31. 333 1)2(18 xx ,

)124)(12(

]12)[(2)12(2

22

xxx

xxx

2

222

)12(

1)2(2)2(144

x

xxxx

)12)(12(

1)2(14 222

xx

xx

H.C.F. = 12 x Note: L.C.M. =

)124)(12()12( 22 xxxx (A)

32. The equation of the straight line is

3log4

3log 22 xy

4096

8

2

2log

2logloglog

43

4

3

4

33

4

33

2

32

4

3

22

yx

yx

xy

x

xy

(A)

33. 25732257 2221)2(2223229

2

35710

01001010100

2222

(B)

34. , are the distinct roots of the quadratic

equation 0452 2 xx .

Product of roots 22

4 (B)

35. (a, b) is on the upper part above the line 1y

1b

(a, b) is on the lower part below the line 12 xy 12 ab

(a, b) is on the lower part below the line xy 24 ab 24

I only (A)


97

36. Let r be the common ratio of the geometric

sequence.

3

227

8432

128

3

3

3

6

9

r

r

r

ra

a

1458

27

8432

36

3

r

aa

I is true.

4

9

3

2

1122

4

2

ra

a

II is true.

.532804

91458

23

1 r

aa

2000321 naaaa when n = 1

III is not true.

Remark: Although the sum to infinity of the sequence

3.1968

3

21

.53280

11

321

r

aaaa

which is less than 2000, the sequence is alternating and the sum of the first few terms may be larger than 2000. (A)

37. When x = 0, y = 1.

12

0tan31

h

h

I is true.

When x , y = 0.

60

302

3

1

2tan

2tan310

II is not true.

h

3

3

60tantan

III is true. (B)

38. 3 sin2x + 8cosx = 0

rejected)(3or 3

1cos

0)3)(cos1cos3(

03cos8cos3

0cos8cos33

0cos8)cos13(

2

2

2

x

xx

xx

xx

xx

The equation has 2 roots for 3600 x . (A)

39.

Let the length of each side of the tetrahedron be 2a. Let M be the mid-point of BD. Then AM BD and CM BD. The angle between the line AC and the plane BCD is ACM.

AM = CM = 22)2( aa = a3

In ACM,

3

134

4cos

cos)2)(3(2)2(33

cos2

2

2

222

222

a

aACM

ACMaaaaa

ACMACCMACCMAM

ACM = 55 (cor. to the nearest degree) (C)

M

A

BD

C

2a


98

40.

Join OA and OB.

ODOB (radii) 24BDODBO (base s, isos. ) 482424BOE (ext. of ) 48BOEOAB ( in alt. segment)

24OBDOAC (∠s in the same

segment)

72

2448

OACOABBAC

(A)

41. Slope of OB =99

33=

3

1

Slope of the orthogonal from A to OB =

3

11

= 3

The equation of the orthogonal from A to OB is y = 3x + 66.

OA is along the y-axis. The equation of the orthogonal from B to OA

is y = 33.

Solving

33

663

y

xy, we get

11

33663

x

x (B)

42.

P(yellow coin)

= P(box P and yellow coin) + P(box Q and yellow coin)

=6

4

2

1

8

5

2

1

=48

31 (D)

43. Number of queues formed by the 7 children = 7!

In counting the number of queues formed with the 2 boys next to each other, the 2 boys are considered as a group. The number = 6! Number of queues formed with no boys are next to each other = 7! 6! = 4320 (C)

44. Since 75 h and 97 h , h = 7 Moreover, k = 9

I. : median =2

5957= 58

II. : range = 89 44 = 45 III. : inter-quartile range = 63 52 = 11 (C)

45. )2Var()32Var( XX

20

52

)Var(22

2

X

(C)

A B

C

D EO

24°

MATHEMATICS Compulsory Part PAPER 1

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