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DR. KOOPA KOO MATHEMATICS ACADEMY
HONG KONG DIPLOMA OF SECONDARY EDUCATION MOCK EXAMINATION 2021
MATHEMATICS Compulsory Part
PAPER 1
Question-Answer Book
Candidate Number
Candidate Name
Examination Centre
3.30 pm - 5.45 pm (21/4 hours)
This paper must be answered in English
INSTRUCTIONS
(1)
Maximum
Score
Marker’s
Use Only
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.
Question
No.
Marks
Marks
1-2
3+3
+
3-4
3+4
+
(2)
This paper consists of THREE sections, A(1),
A(2) and B.
5-6
4+4
+
7-8
4+5
+
(3)
Attempt ALL questions in this paper. Write your
answers in the spaces provided in this QuestionAnswer Book. Do not write in the margins.
Answers written in the margins will not be
marked.
9
5
10
8
11
9
12
9
13
9
14
5
15
5
Unless otherwise specified, numerical answers
should be either exact or correct to 3 significant
figures.
16
9
17
5
(7)
The diagrams in this paper are not necessarily
drawn to scale.
18
5
(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.
19
6
Total
105
(4)
(5)
(6)
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.
Unless otherwise specified, all working must be
clearly shown.
2021-DSE-MATH-MOCK-CP 1-1
Section A(1) (35 marks)
1. Simplify
(xy2 )2
and express your answer with positive indices.
x5
(3 marks)
2. Simplify
1
1
1
+
−
.
x−2 x+2 x+6
(3 marks)
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2017-DSE-MATH-CP 1– 2
2
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3. It is given that c is the sum of three parts. One part is a constant. The second
part varies directly as a. The third part varies directly as b. When a = 1, b = 1
and c = 5; when a = 2, b = 2 and c = 6; when a = 3, b = 5 and c = 3. Express b
in 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. Find
the cost of a bottle of milk.
(4 marks)
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2017-DSE-MATH-CP 1– 3
3
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5. The coordinates of the point A are (−8, 20). A is rotated clockwise about the
origin O through 90◦ to A′ . A′′ is the reflection image of A with respect to the
y-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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2017-DSE-MATH-CP 1– 4
4
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6. The box-and-whisker diagram below shows the distribution of the scores of 40
students in a test. It is given that the test consists of 20 multiple-choice questions
and each of them carries one mark. The passing score of the test is 10 marks.
9 10
15
20
Score (marks)
(a) Find the median, the range and the inter-quartile range of the scores of these
students.
(b) The teacher later finds that one of the multiple-choice questions is marked wrongly
for all the students. After rechecking, 11 students have their scores changed. Is
it possible that for all the students to pass the test after the rechecking? Explain
your answer.
(4 marks)
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2017-DSE-MATH-CP 1– 5
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4x + 6
> 2(x − 3) and 4x − 20 ≤ 0
7
(b) How many positive integers satisfy both the inequalities in (a)?
(4 marks)
7. (a) Find the range of values of x which satisfy both
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2017-DSE-MATH-CP 1– 6
6
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2017-DSE-MATH-CP 1– 7
7
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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 that
of △BDA.
(5 marks)
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2017-DSE-MATH-CP 1– 8
8
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2017-DSE-MATH-CP 1– 9
9
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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)
3
4
5
6
Leaf (units)
0 1 3 4 4
1 1 2 2 3
3 6 8 9
7 9
5
7
6
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 of
each of these two students.
(5 marks)
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2017-DSE-MATH-CP 1– 10
10
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2017-DSE-MATH-CP 1– 11
11
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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 positive
x-axis respectively such that AC is perpendicular to L. B is a point on the line
segment AC such that area of △AOB = 35 and area of △BOC = 100. Find the
coordinates of B.
(8 marks)
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2017-DSE-MATH-CP 1– 12
12
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2017-DSE-MATH-CP 1– 13
13
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11. The base radius and curved surface area of a solid metal right circular cone is 9
cm 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 in
the vessel. Mario claims that the volume of the water remaining in the vessel is
less than 0.006 m3 . Do you agree? Explain your answer.
(9 marks)
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2017-DSE-MATH-CP 1– 14
14
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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. Assume
that all angles are folded at right angles and the thickness of the cardboard is
negligible. 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– 16
16
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2017-DSE-MATH-CP 1– 17
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13. The mean score of a class of 37 students in a test is 48 marks. The scores of Alvin
and Koopa in the test are 36 marks and 72 marks respectively. The standard
score 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 due
to the deletion of the test score of Luigi.
(9 marks)
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2017-DSE-MATH-CP 1– 18
18
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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.
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2017-DSE-MATH-CP 1– 20
20
(5 marks)
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2017-DSE-MATH-CP 1– 21
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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.
marks)
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2017-DSE-MATH-CP 1– 22
22
(5
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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 α 2 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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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– 26
26
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18. The figure shows two rectangular display boards ABCD and ADEF, both perpendicular to the ground. FXB is a straight line and AX ⊥ FB. ACX and AEX √
are
two wooden boards supporting the displays boards. It is given that CD = 3 2
m, DE = 1 m and ∠CDE = 135◦ .
(a) Find XB.
7
(b) If EF = m, find the angle between the plane ACX and the plane AEX.
5
marks)
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2017-DSE-MATH-CP 1– 28
28
(5
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2017-DSE-MATH-CP 1– 29
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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).
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(6 marks)
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END OF PAPER
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