mathematics compulsory part paper 2

2021-DSE-MATH-CP 2-1

DR. KOOPA KOO MATHEMATICS ACADEMY

HONG KONG DIPLOMA OF SECONDARY EDUCATION MOCK EXAMINATION 2021

MATHEMATICS Compulsory Part PAPER 2

6.00 pm - 7.15 pm (11/4 hours)

Not to be taken away before the

end of the examination session

INSTRUCTIONS

1. Read carefully the instructions on the Answer Sheet. After the announcement of the start of the examination,

you should first stick a barcode label and insert the information required in the spaces provided. No extra

time will be given for sticking on the barcode label after the ‘Time is up’ announcement.

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. You must mark the answers clearly;

otherwise you will lose marks if the answers cannot be captured.

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.

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.

Section A

1. 7334 ·(−1343

)111

=

A. −7

B. −17

C. 1

D. 7

2. If 3+2bb

=3− x

x, then x =

A. b1+b

B. 2b1+b

C. b2+b

D. 2b2+b

3. Factorize a2 −b2 +1+2a.

A. (a+b+1)(a−b+1)

B. (a+b+1)(a+b−1)

C. (a+b−1)(a−b+1)

D. (a+b−1)(a−b−1)

4. If a,b,c are non-zero constants such that x(x+8a)+a ≡ x2 +2(bx+c), then a : b : c =

A. 1 : 4 : 2

B. 2 : 8 : 1

C. 4 : 1 : 2

D. 4 : 1 : 8

2021-DSE-MATH-CP 2-2 2

5. 0.0865403 =

A. 0.086 (correct to 2 significant figure)B. 0.0865 (correct to 3 decimal places)C. 0.08654 (correct to 4 significant figure)D. 0.086540 (correct to 5 decimal places)

6. If k is a constant such that x3 +4x2 + kx−12 is divisible by x+3, then k =

A. −25

B. −1

C. 1

D. 17

7. If the equation x2 −4x+ k = 1 has no real roots, then the range of values of k is

A. k > 4

B. k ≥ 4

C. k > 5

D. k ≥ 5

8. Solve the equation x(x+2) = 3(x+2).

A. x = 3.B. x =−2.C. x =−2 or x = 3.D. x = 2 or x =−3.

9. It is given that b partly varies directly as a2 and partly varies inversely as a. Whena = 1,b =−4 and when a = 2,b = 5. When a =−2,b =

A. −11

B. −5

C. 5

D. 11


10. Find the range of m such that (m−1)x2−(2m+1)x+m = 0 has two distinct real roots.

A. m <−18

B. m ≤ −18

C. m >−18

D. None of the above.

11. Susan sells two iPhones for $9999 each. She gains 10% on one and loses 10% on theother. After the two transactions, Susan

A. loses $202.B. gains $101.C. gains $202.D. has no gain and no loss.

12. There is a pool of spring water which leaks out an equal amount of water everyminute. 8 vacuum cleaners can dry out all the spring water in 10 hours while 12vacuum cleaners can dry out all the spring water in 6 hours. If 14 vacuum cleanersare used, how long does it take for the entire pool to be dry?

A. 4B. 5C. 6D. 8

13. Let m and n be non-zero numbers. If 2m−nm−2n

= 3, then m : n =

A. 5 : 1

B. 1 : 5

C. 5 : 7

D. 7 : 5


14. If z varies inversely as x−1 and directly as y−2, which of the following may NOT beconstant?

A. xy2z

B. zy2

2x

C. yzx2

D. 888zy2

x

15. Let an be the n th term of a sequence. If a1 = 4, a2 = 5 and an+2 = an +an+1 for anypositive integer n, then a10 =

A. 13B. 157C. 254D. 411

16. In the figure, area of △ABC is 1998 cm2, which is 3 times of the area of the parallelo-gram DEFC. Find the area of the shaded region.

A

B C

D E

F

A. 333 cm2

B. 444 cm2

C. 555 cm2

D. 666 cm2


17. In the figure, ABCD is a rectangle, where AB = 8, AE = 6 and ED = 3. F is themidpoint of the line segment BE and G is the midpoint of the line segment FC. Findthe area of the shaded region.

A B

CD

E

F

G

A. 6B. 12C. 18D. 24

18. In the figure, ABCD is a rectangle. AB = 30 cm, BC = 40 cm. P, Q, and R are pointson BC, AC and BD respectively such that PQ ⊥ AC and PR ⊥ BD. Find PQ+PR.

A

B C

D

A. 24 cmB. 25 cmC. 28 cmD. 30 cm


19. In the figure, the side length of the square ABCD is 96 cm. BE = EF = FG = GC andAM = MN = NP = PC. Find the area of the shaded region.

A

B C

D

E F G

M

N

P

A. 144 cm2

B. 288 cm2

C. 432 cm2

D. 576 cm2

20. In the figure, there is a parallelogram. Two pairs of parallel lines, which are parallelto the sides, divide the parallelogram into 9 smaller parallelograms. If the area of theoriginal parallelogram and the area of the shaded region are 99 cm2 and 19 cm2, findthe area of the parallelogram ABCD.

A

B

C

D

A. 31 cm2

B. 40 cm2

C. 49 cm2

D. 59 cm2


21. In the figure, ABCD is a right-angled trapezium, where AD = 12 cm, AB = 8 cm,BC = 15 cm. The area of △ADE, area of quadrilateral DEBF and area of △CDF areequal. Find the area of the shaded region.

A

BC

D

E

F

A. 18 cm2

B. 24 cm2

C. 30 cm2

D. 36 cm2

22. If P is a moving point such that P is equidistant from the point (−5,−5) and thex-axis, then the equation of the locus of P is

A. y =− 110

x2 − x−5

B. y =− 110

x2 + x−5

C. y =1

10x2 − x−5

D. y =1

10x2 + x−5


23. In the figure, the equations of the straight lines L1 and L2 are px+y = q and rx+y = srespectively. Which of the following are true?

x

y

O

L2

L1

I. p < 0

II. p < r

III. q > s

IV. ps > qr

A. I, II and III only.B. I, II and IV only.C. I, III and IV only.D. II, III and IV only.

24. In the figure, AD is a diameter of the circle ABCD. It is given that XBCY is a straightline. If AD = 40 cm and BC = 24 cm, then AX +DY =

A. 24 cmB. 32 cmC. 64 cmD. 72 cm


25. Which of the following statements about a dodecahedron is/are true?

I. It has 12 facesII. It has 20 verticesIII. It has 30 edges

A. I and II only.B. I and III only.C. II and III only.D. I, II and III.

26. The equation of the circle C is 3x2 +3y2 −12y−10x+12 = 0. Denote the origin, thecentre of C and the point (4,0) by O, P and Q respectively. Which of the followingis/are true?

I. The coordinates of P is(

2,53

)while the radius of C is 5

3.

II. OPQ is an isosceles triangle.III. ∠OPQ is not an acute angle.

A. I and II only.B. I and III only.C. I, II and III.D. None of the above.

27. Find the probability that −1 ≤ sinθ <12

, where θ is chosen randomly in the range0◦ ≤ θ < 360◦.

A. 13

.

B. 12

.

C. 23

.

D. 45

.

2021-DSE-MATH-CP 2-10 10

28. Find the total number of ways to distribute four books A, B, C and D into 2 groups,2 books for each group.

A. 3B. 6C. 12D. 24

29. Let n be a 4 digit number that is a multiple of 5 with distinct digits. Find the totalnumber of such n, where 3000 ≤ n ≤ 7000.

A. 376B. 392C. 448D. 800

30. The stem-and-leaf diagram below shows the distribution of the ages of 24 members ofa committee.

Stem (tens) Leaf (units)2 x3 2 2 3 7 8 84 3 3 4 5 5 6 7 95 1 1 y 66 0 5 87 0 1

If the range and the inter-quartile range of the distribution are 42 and 18 respectively,then

A. x = 8 and y = 5

B. x = 8 and y = 6

C. x = 9 and y = 5

D. x = 9 and y = 6

2021-DSE-MATH-CP 2-11 11

Section B

31. Simplify√

x2 +2x+1+√

x2 −2x+1.

A. 2B. 2x

C. −2x

D. None of the above.

32. Solve the equation log2xlog(x−1)

= 2.

A. x = 2−√

3

B. x = 2+√

3

C. x = 99

D. x = 2−√

3 or x = 2+√

3

33. FACE16 =

A. 11111010110011102

B. 11111010110111102

C. 11111011110011102

D. 11111011110111102

34. Which of the following is a complex number?

I. 0

II. 3+ i

III. (i+π)4

A. I only.B. II only.C. II and III only.D. I, II and III.

2021-DSE-MATH-CP 2-12 12

35. Consider the following system of inequalities:x ≥ 0

y ≥ 2

5x+4y ≤ 88

4x+3y ≤ 68

If (x,y) is a point lying in the region defined by the above system of inequalities, thenthe greatest value of x+ y+1 is

A. 18B. 20C. 21D. 23

36. The product of the 1st term and 2nd term of a geometric sequence is 72 while theproduct of the 3rd term and the 4th term of the sequence is 1152. The product of the4th term and the 5th term of the sequence is

A. 2304B. 3456C. 4608D. 20736

37. If 270◦ < x < 360◦, which of the following must be true?

I. sinxsin(90◦− x)≤ 0

II. tanx tan(270◦− x) = 1

III. sinx− cosx < 0

A. II only.B. III only.C. I and II only.D. I, II and III.

2021-DSE-MATH-CP 2-13 13

38. Let a and b be constants. If the figure shows the graph of y = acos(2x+ 120◦)+ b,then

A. a = 2 and b = 6

B. a = 4 and b = 4

C. a = 6 and b = 2

D. a = 8 and b = 0

39. Find the minimum value of 808sin2 x+5sin2(90◦+ x)

.

A. 8B. 10C. 16D. 40

40. The angle between two adjacent planes of a regular octahedron is

A. 106◦

B. 107◦

C. 108◦

D. 109◦

2021-DSE-MATH-CP 2-14 14

41. The circle C passes through the points L(0,0),M(8,0) and N(0,6). Which of thefollowing is/are true?

I. The equation of C is x2 + y2 −8y−6x = 0.II. Area of △LMN : Area of C = 12 : 5π.III. If a point lying in C is randomly selected, given that it lies in quadrant

I, the probability that the point lies inside △LMN is 0.379 (correct to 3significant figures).

A. I only.B. III only.C. I and II only.D. I, II and III.

42. Find the range of values of k such that the straight line y = x+ k is tangential to thereal circle x2 + y2 − kx− ky = 0.

A. k = 0

B. All real numbers.C. All real numbers except 0.D. No real solutions.

43. Using 5 points and 4 points from two different planes as vertices, find the maximumnumber of tetrahedrons that can be formed.

A. 60B. 96C. 120D. 128

2021-DSE-MATH-CP 2-15 15

44. Three cards are randomly drawn at the same time from the cards numbered 0, 1, 2,3, 4, 5, 6, 7, 8, 9 and 11 respectively. Find the probability that the product of thenumbers of the cards drawn is odd.

A. 433

B. 111

C. 3133

D. 1011

45. A set of numbers has a mode of 33, an inter-quartile range of 28 and a variance of 26.If 3 is added to each number of the set and each resulting number is then doubled toform a new set of numbers, find the mode, the inter-quartile range and the varianceof the new set of numbers.

Mode Interquartile range VarianceA. 66 62 52B. 72 62 104C. 72 56 52D. 72 56 104

END OF PAPER

2021-DSE-MATH-CP 2-16 16

mathematics compulsory part paper 2

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