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```HKDSE
MATH CP
PAPER 2
Set 23
HONG KONG DIPLOMA OF
SECONDARY EDUCATION EXAMINATION
MATHEMATICS Compulsory Part
S6 MOCK EXAM
PAPER 2
Time allowed: 1¼ hours
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.
Not to be taken away before the
end of the examination session
Set 23 PAPER 2-1
1
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.
2.
25456  27345
=
3123
A. 51824 .
B. 15912 .
C. 151356 .
D. 151824 .
If 2 x − 7 = y + 6 = 2 x − 4 y + 37 , then x + y =
A.
B.
C.
D.
3.
11.
16.
20.
23.
1
=
7
A.
B.
C.
D.
0.380 (correct to 3 decimal places).
0.3779 (correct to 4 decimal places).
0.37796 (correct to 5 significant figures).
0.377965 (correct to 6 significant figures).
Set 23 PAPER 2-2
2
4.
If
A.
B.
C.
D.
5.
u
v
, then u =
=
4 − 3u 4 + 3v
v
.
1− 6v
v
.
1+ 6v
2v
.
2 − 3v
2v
.
2 + 3v
2
2
If y − ky + 9  ( y − l ) + m , where k, l and m are constants, then m =
A. 9.
2
B.
C.
D.
6.
Which of the following must have a + b as a factor?
I. a 4 − b 4
II. a 4 + b 4
2
III. a (a + b) − a + b
A.
B.
C.
D.
7.
k
.
4
9− k2.
9+ k2 .
9−
I only
II only
I and III only
II and III only
2
Let f ( x) = − x + bx + 11 , where b is a constant. If f (−2) = −1 , then b =
A. −3.
B. 4.
C. 7.
D. 8.
Set 23 PAPER 2-3
3
8.
3
2
Let p and q be constants. If px + x − 2qx + 8 is divisible by x + 4, then 8p − q =
A.
B.
C.
D.
9.
−24.
−3.
3.
24.
 8 x + 3  x − 11
The solution of 
is
− 2( x − 5)  4
A. x  3.
B. x  −2.
C. x  −2.
D. −2  x  3.
2
10. If 0  c  1, which of the following may represent the graph of y = (c − x) − c ?
A.
B.
C.
D.
Set 23 PAPER 2-4
4
11. If
A.
B.
C.
D.
3a − 2b 2
2a − 3b
= , then
=
a + 4b 3
4a + b
3
− .
2
2
− .
3
1
.
9
23
.
67
12. In the figure, the 1st pattern consists of 5 dots. For any positive integer n, the (n + 1)th pattern
is formed by adding (3n + 4) dots to the nth pattern. Find the number of dots in the 7th pattern.

A.
B.
C.
D.
70
77
87
92
13. The cost of a shirt is \$200. The shirt is sold for \$24 less than its marked price. If the percentage
profit is 25% before the discount, find the percentage profit after the discount.
A. 10.4%
B. 12%
C. 13%
D. 22%
Set 23 PAPER 2-5
5
14. It is given that h partly varies directly as k2 and partly varies inversely as
k . When k = 1,
h = 41 and when k = 4, h = 5. When k = 9, h =
A. −306.
B. −67.
283
C.
.
3
D. 552.
15. In the figure, ABD and BCD are triangles with BC = BD and AD // BC. If BAD = 84 and
ABD : CBD = 2 : 1, then BCD =
A. 66.
B. 69.
C. 74.
D. 84.
16. In the figure, ABC is an isosceles triangle with AC = BC. D and E are points lying on AB and
BC respectively. AE and CD intersect at F. If AF = 64 cm, FE = 36 cm and CFE = 2ABC,
then AC =
A.
B.
C.
D.
75 cm.
80 cm.
81 cm.
82 cm.
Set 23 PAPER 2-6
6
17. OAB is a sector of circle with centre O. OAB is folded such that AO and BO join together to
form a right circular cone. If AO = BO = 20 cm and the angle of the sector OAB is 216, then
the volume of the circular cone is
A. 240 cm3.
B. 768 cm3.
C. 960 cm3.
D. 1024 cm3.
18. In the figure, ABCD is a parallelogram. B and D are points on AS and PC respectively such that
PD : DC = 2 : 3 and AB : BS = 4 : 1. If PS intersects AD and BC at Q and R respectively, then
the area of DQP : the area of BRS =
A. 2 : 1.
B. 4 : 1.
C. 8 : 3.
D. 64 : 9.
19. In the figure, AC =
A.
2x .
B. 2x.
C. 2 3 x .
D.
2 x 2 + 18 .
Set 23 PAPER 2-7
7
20. In the figure, ABCD is a parallelogram. O is the centre of the circle ABD and ODC is a straight
line. AD and OB intersect at E. If BCD = 38, then BED =
A.
B.
C.
D.
104.
109.
114.
142.
O
A
D
E
B
C


21. In the figure, AB produced and DC produced meet at E. It is given that AB = 3 , CD = 4 and

A.
B.
C.
D.
30.
40.
48.
66.
22. In the figure, AB and CD are two vertical poles standing on the horizontal ground. The angle of
elevation of C from A is 50 and the angle of depression of D from A is 35. If AB = 2 m, then
AC =
2 tan 35
C
A.
m.
sin 50
2 tan 35
B.
m.
cos 50
2
C.
m.
sin 50 tan 35
A
2
D.
m.
cos 50 tan 35
B
Set 23 PAPER 2-8
8
D
23. The rectangular coordinates of points P and Q are (2k, −k + 2) and (−2, 8) respectively, where k
is a constant. If the length of the line segment joining P and Q is 2 5 , then the polar
coordinates of P are
A. (4, 0) .
B.
(4 2 , 135) .
C.
(4 2 , 225) .
D.
(4 2 , 315) .
24. The straight line L and the straight line 2 x + 7 y − 28 = 0 do not intersect each other. If the
x-intercept of L is 5, then the equation of L is
A. 2 x + 7 y − 10 = 0 .
B. 2 x + 7 y − 35 = 0 .
C. 7 x − 2 y + 10 = 0 .
D. 7 x − 2 y − 35 = 0 .
25. In the figure, the straight line L1: y = ax + b and the straight line L2: y = px + q intersect at a
point on the negative x-axis. Which of the following must be true?
I. ab  0
II. pq  0
III. aq = bp
A. I only
B. I and II only
C. II and III only
D. I, II and III
Set 23 PAPER 2-9
9
26. S is a line segment of length 5 cm. Let P be a moving point such that the distance from P to S
is always 2 cm. Find the area of the region bounded by the locus of P correct to the nearest
0.1 cm2.
A. 32.6 cm2
B. 36.0 cm2
C. 45.1 cm2
D. 63.6 cm2
27. The bar chart below shows the numbers of ice-creams of three flavours sold at an ice-cream
shop on a certain day.
Number of ice-creams sold
80
70
60
50
40
30
Vanilla
Chocolate
Strawberry
One of the staff makes the following claims:
I.
On that day, the total number of vanilla ice-creams and strawberry ice-creams sold is
less than the number of chocolate ice-creams sold.
II. On that day, the number of chocolate ice-creams sold is two times the number of
strawberry ice-creams sold.
III. On that day, the number of vanilla ice-creams sold is 50% more than the number of
strawberry ice-creams sold.
Which of the above claim(s) is/are true?
A. I only
B. II only
C. I and III only
D. II and III only
Set 23 PAPER 2-10
10
28. There are 12 black balls and k white balls in a bag. Peter repeats drawing a ball at a time
randomly from the bag with replacement for 90 times. If he draws a white ball for 60 times,
which of the following is the best estimate of the value of k?
A. 8
B. 18
C. 24
D. 60
29. The frequency distribution table below shows the distribution of the heights of 25 ten-year-old
boys.
Height (cm)
Frequency
126 − 130
2
131 − 135
3
136 − 140
8
141 − 145
7
146 − 150
5
Find an estimate of the standard deviation of the heights of the 25 boys correct to the nearest
0.1 cm.
A. 5.8 cm
B. 6.2 cm
C. 34.0 cm
D. 38.0 cm
30. Consider the following positive integers:
3
6
6
7
13
13
17
18
20
x
y
z
If the mean of x and y is 4, and the mean of all the above data is 10, then the median of the
above data is
A. 8.
B. 9.
C. 11.
D. 15.
Set 23 PAPER 2-11
11
Section B
31.
x
2
A.
B.
C.
D.
1
1
− 2
=
+ 2 x − 3 3x − 2 x − 1
2
.
( x + 3)(3 x + 1)
2
.
( x − 3)(3 x − 1)
2x − 4
.
( x + 1)( x − 3)(3 x − 1)
2x + 4
.
( x − 1)( x + 3)(3 x + 1)
32. 3  47 − 815 + 816 =
A. D000000B00016.
B. E000000C00016.
C. D0000000B00016.
D. E0000000C00016.
33. The figure shows the graph of y = log a x and the graph of y = log b x on the same
rectangular coordinate system, where a and b are positive constants. If a vertical line L cuts the
x-axis, the graph of y = log a x and the graph of y = log b x at P, Q and R respectively, which
of the following is/are true?
a−b0
a+b2
QR
= log b a − 1
III
PQ
y
I
II
A.
B.
C.
D.
L
R
y = logb x
y = loga x
Q
II only
I and II only
I and III only
II and III only
O
P
x
2
34. Let f ( x) = ax − 4 x + a , where a is a non-zero constant. If the maximum value of f (x) is 3,
then a =
A. 4.
B. 1.
C. −1.
D. −4.
Set 23 PAPER 2-12
12
40i
is a purely imaginary number, where k is a real number, then k =
3+i
−5.
−4.
4.
5.
35. If k +
A.
B.
C.
D.
36. The sum of all the positive terms in the arithmetic sequence 999, 991, 983,  is
A. 62 372.
B. 62 375.
C. 62 874.
D. 62 875.
2  x + 2 y  8

37. Which of the following shaded regions represents the solution of  x  0
?
0  y  3

A.
B.
C.
D.
Set 23 PAPER 2-13
13
38. Which of the following figures show the graph of y = 1− cos x ?
A.
B.
C.
D.
39. In the figure, AO is a vertical pole standing on the horizontal ground OBC. The bearing of C
from B is
A.
B.
C.
D.
A
N30W.
N60W.
S30E.
S60E.
60
West
O
C
45
B
South
Set 23 PAPER 2-14
14
40. In the figure, EB is the tangent to the circle ABCD at B and DAE is a straight line. If
AEB = 76 and ACB = 21, then ACD =
A.
B.
C.
D.
D
55.
62.
76.
83.
C
21
A
E
76
B
41. The coordinates of the points P, Q and R are (3, 2), (15, 2) and (3, 7) respectively. If C is the
inscribed circle of PQR, then the equation of C is
2
2
A. x + y − 5 x − 4 y + 37 = 0 .
9
B. x 2 + y 2 − 9 x − y + 59 = 0 .
2
2
2
C. x + y − 10 x − 8 y + 37 = 0 .
D.
x 2 + y 2 − 18x − 9 y + 59 = 0 .
42. A queue is formed by 5 boys and 5 girls. If boys and girls stand alternately in the queue, how
many different queues can be formed?
A. 14 400
B. 28 800
C. 30 240
D. 86 400
Set 23 PAPER 2-15
15
43. There are two cities, X and Y. In a morning, the probabilities of raining in X and Y are
2
and
5
1
respectively. Given that at most one of the two cities is raining in the morning, find the
4
probability that it is raining in X in the morning.
3
A.
10
1
B.
3
2
C.
5
4
D.
9
44. A computer salesman designs a questionnare to collect the opinions about a top-selling
computer from its users. The salesman has 10 friends who are users of the computer. He
randomly selects three of his friends and only these three friends are selected as a sample to fill
in the questionaire. Which of the following is/are disadvantage(s) of this sampling method?
I. The sample size is too small.
II. He selects three of his friends by himself.
III. Not all the users of the computer have an equal chance of being selected.
A. I only
B. I and III only
C. II and III only
D. I, II and III
45. Let a1, b1 and c1 be the mean, the variance and the inter-quartile range of a group of numbers
{x1, x2, x3, …, x40} respectively while a2, b2 and c2 be the mean, the variance and the
inter-quartile range of the group of numbers {x1, x2, x3, …, x40, x41, x42, x43, x44} respectively. If
x41 = x42 = x43 = x44 = a2, which of the following must be true?
I. a1 = a2
II. b1  b2
III. c1  c2
A. I and II only
B. I and III only
C. II and III only
D. I, II and III
END OF PAPER
Set 23 PAPER 2-16
16
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