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 Page 1.
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
2. Make A the subject of the formula zAyAx )3( . (3 marks)
__________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ3 Page total
3. Simplify xx 81
4
56
3
. (3 marks)
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
4. Factorize
(a) yx 513
(b) 22 103 yxyx
(c) yxyxyx 153103 22 (4 marks)
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ4 Page total
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)
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
6. Consider the compound inequality
525
311 x
x or 0157 x ………… (*).
(a) Solve (*).
(b) Write down the greatest negative integer satisfying (*). (4 marks)
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ5 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ6 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ7 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ8 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ9 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ10 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ11 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ12 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ13 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ14 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ15 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ16 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ17 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ18 Page total
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ19 Page total
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)
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Answers written in the margins will not be marked. Kendy Publishing Company Limited MP5-SQ20 Page total
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
END OF PAPER
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
Ans
wer
s w
ritte
n in
the
mar
gins
will
not
be
mar
ked.
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
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
Mock Paper Set 5 (Paper 1) Answers and Solutions
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
Mock Paper Set 5 (Paper 1) Answers and Solutions
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
Mock Paper Set 5 (Paper 1) Answers and Solutions
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
Mock Paper Set 5 (Paper 1) Answers and Solutions
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
Mock Paper Set 5 (Paper 1) Answers and Solutions
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
Mock Paper Set 5 (Paper 1) Answers and Solutions
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
Mock Paper Set 5 (Paper 1) Answers and Solutions
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)
Mock Paper Set 5 (Paper 1) Answers and Solutions
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)
Mock Paper Set 5 (Paper 2) Answers and Solutions
Mock Paper Set 5 (Paper 2) Answers and Solutions
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
Mock Paper Set 5 (Paper 2) Answers and Solutions
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.
With the notation in the figure,
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.
With the notation in the figure,
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
Mock Paper Set 5 (Paper 2) Answers and Solutions
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)
Mock Paper Set 5 (Paper 2) Answers and Solutions
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)
Mock Paper Set 5 (Paper 2) Answers and Solutions
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
Mock Paper Set 5 (Paper 2) Answers and Solutions
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°