2021-DSE-MATH-MOCK-CP 1-1

Please stick the barcode label here.

DR. KOOPA KOO MATHEMATICS ACADEMY

HONG KONG DIPLOMA OF SECONDARY EDUCATION MOCK EXAMINATION 2021

MATHEMATICS Compulsory Part

PAPER 1

Question-Answer Book

3.30 pm - 5.45 pm (21/4 hours)

This paper must be answered in English

Candidate Number

Candidate Name

Examination Centre

Maximum

Score Marker’s Use Only

Question No.

Marks Marks

1-2 3 + 3 +

3-4 3 + 4 +

5-6 4 + 4 +

7-8 4 + 5 +

9 5

10 8

11 9

12 9

13 9

14 5

15 5

16 9

17 5

18 5

19 6

Total 105

INSTRUCTIONS

(1) After the announcement of the start of the

examination, you should first write your

Candidate Number, Candidate Name and

Examination Centre in the space provided on

Page 1 and stick a barcode label in the space

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

(2) This paper consists of THREE sections, A(1),

A(2) 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 Name and

mark the question number box 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 either exact or correct to 3 significant

figures. (7) The diagrams in this paper are not necessarily

drawn to scale.

(8) 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.

Answers written in the margins will not be marked.

2017-DSE-MATH-CP 1–

Section A(1) (35 marks)

1. Simplify (xy2)2

x5 and express your answer with positive indices. (3 marks)

2. Simplify 1x−2

+1

x+2− 1

x+6. (3 marks)

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3. It is given that c is the sum of three parts. One part is a constant. The secondpart varies directly as a. The third part varies directly as b. When a = 1, b = 1and c = 5; when a = 2, b = 2 and c = 6; when a = 3, b = 5 and c = 3. Express bin terms of a and c. (3 marks)

4. The cost of a bottle of orange juice is the same as the cost of 2 bottles of milk.The total cost of 3 bottles of orange juice and 5 bottles of milk is $132. Findthe cost of a bottle of milk. (4 marks)

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2017-DSE-MATH-CP 1–

5. The coordinates of the point A are (−8,20). A is rotated clockwise about theorigin O through 90◦ to A′. A′′ is the reflection image of A with respect to they-axis.

(a) Write down the coordinates of A′ and A′′.(b) Is OA′′ perpendicular to AA′? Explain your answer.

(4 marks)

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6. The box-and-whisker diagram below shows the distribution of the scores of 40students in a test. It is given that the test consists of 20 multiple-choice questionsand each of them carries one mark. The passing score of the test is 10 marks.

Score (marks)9 10 15 2020

(a) Find the median, the range and the inter-quartile range of the scores of thesestudents.

(b) The teacher later finds that one of the multiple-choice questions is marked wronglyfor all the students. After rechecking, 11 students have their scores changed. Isit possible that for all the students to pass the test after the rechecking? Explainyour answer.

(4 marks)

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2017-DSE-MATH-CP 1–

7. (a) Find the range of values of x which satisfy both 4x+67

> 2(x−3) and 4x−20 ≤ 0

(b) How many positive integers satisfy both the inequalities in (a)? (4 marks)

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2017-DSE-MATH-CP 1–

8. ABFE is a quadrilateral. C is the mid-point of AE and D is a point on BF .AB//CD//EF . AF meets CD at G. Find the ratio of the area of △DFG to thatof △BDA. (5 marks)

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2017-DSE-MATH-CP 1–

9. The stem-and-leaf diagram below shows the distribution of the weights (in kg)of a group of students in a primary school.

Stem (tens) Leaf (units)3 0 1 3 4 4 5 64 1 1 2 2 3 7 95 3 6 8 96 7 9

(a) Find the mean, the inter-quartile range and the range of the above distribution.(b) Two more students now join the group. It is found that both the mean and the

range of the distribution of the weights are increased by 1 kg. Find the weight ofeach of these two students. (5 marks)

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2017-DSE-MATH-CP 1–

Section A(2) (35 marks)

10. The equation of the straight line L is 3x−y= 0. A and C lie on L and the positivex-axis respectively such that AC is perpendicular to L. B is a point on the linesegment AC such that area of △AOB = 35 and area of △BOC = 100. Find thecoordinates of B. (8 marks)

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2017-DSE-MATH-CP 1–

11. The base radius and curved surface area of a solid metal right circular cone is 9cm and 210.6π cm2 respectively.

(a) Find the height of the circular cone.(b) A hemispherical vessel of radius 15 cm is held vertically on a horizontal surface.

The vessel is fully filled with water. The circular cone is now held vertically inthe vessel. Mario claims that the volume of the water remaining in the vessel isless than 0.006 m3. Do you agree? Explain your answer. (9 marks)

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2017-DSE-MATH-CP 1–

12. (a) Let f (x) = x3 −38x2 +312x−735.(i) Compute f (5).(ii) Factorize f (x).

(b) A rectangular piece of cardboard is cut to the shape shown in the figure.

A box of depth x cm with a lid is formed by folding along the dotted lines. Assumethat all angles are folded at right angles and the thickness of the cardboard isnegligible. Let V cm3 be the volume of the pizza box.(i) Express V in terms of x.(ii) Susan claims that one can find three distinct real values of x such that V =

4410, do you agree? Please explain your answer. (9 marks)

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2017-DSE-MATH-CP 1–

13. The mean score of a class of 37 students in a test is 48 marks. The scores of Alvinand Koopa in the test are 36 marks and 72 marks respectively. The standardscore of Alvin in the test is −2.

(a) Find the standard score of Koopa in the test.(b) A student, Luigi, withdraws from the class and his test score is deleted. It is

given that his test score is 48 marks. Find the new standard score of Koopa dueto the deletion of the test score of Luigi. (9 marks)

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2017-DSE-MATH-CP 1–

Section B (35 marks)

14. (a) A coin is flipped four times. Find the probability that it lands ”heads” twice.(b) Four coins are flipped simultaneously. Find the probability that exactly two of

them land ”heads.”(c) 3 cards are selected at random from a standard deck of 52 cards. Find the

probability that all of them are from different suits.(d) Find the number of rearrangements of the word ABCDEFG that contain the

sequences AB, CD, and EF .(e) Find the number of divisors of 3000. (5 marks)

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2017-DSE-MATH-CP 1–

15. For any positive integer n, let A(n) = 3n−1 and B(n) = 2A(n).

(a) Express A(1)+A(2)+A(3)+ ...+A(n) in terms of n.(b) Find the greatest value of n such that B(1)B(2)B(3)...B(n)≤ 1000000000. (5

marks)

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2017-DSE-MATH-CP 1–

16. Let a ̸= 0, b, c and k be integers such that b2 −4ac = k2.

(a) Show that ax2 +bx+ c = 0 has roots that are rational numbers.(b) Let α and β be roots of the equation 3x2 +3x+4 = 0 such that Im(α)> Im(β ).

(i) Express α in the form s+ ti, where s and t are real numbers.(ii) Find the quadratic equation with roots α2 and β 2.(iii) Let f (x) = 3x2 +3x+4, and g(x) = px2 +2p+1. Find the range of values of

p such that f (x)> g(x) for all value of x.(iv) Let q < 100 be a prime number and h(x) = 2x2 +(3− q)x+ 5− q. Find the

total number of q’s such that f (x) = h(x) has roots that are rational numbers.(9 marks)

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2017-DSE-MATH-CP 1–

17. (a) Solve the inequality x(1− x)> 0.

(b) Given that y > 0 and x =y2

y2 +1.

(i) Find the range of x.(ii) Express y in terms of x. (5 marks)

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2017-DSE-MATH-CP 1–

18. The figure shows two rectangular display boards ABCD and ADEF , both perpen-dicular to the ground. FXB is a straight line and AX ⊥ FB. ACX and AEX aretwo wooden boards supporting the displays boards. It is given that CD = 3

√2

m, DE = 1 m and ∠CDE = 135◦.

(a) Find XB.

(b) If EF =75

m, find the angle between the plane ACX and the plane AEX . (5marks)

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2017-DSE-MATH-CP 1–

19. Let C : (x−8)2 +(y−6)2 = 20 be a circle.

(a) Find the tangents to the circle with slope equals 2.(b) Find the tangents to the circle that passes through the point (2,4).(c) Find the acute angle between two tangents in (b). (6 marks)

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2017-DSE-MATH-CP 1–

END OF PAPER

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