Kwok Tak Seng Catholic Secondary School 2012-2013 S6 Mock Examination

A Name
Kwok Tak Seng Catholic Secondary Class ( )

School
2012-2013 S6 Mock Examination
Marker’s Examiner’s
Use Only Use Only
Marker No. Examiner No.

MATHEMATICS Compulsory Part
PAPER 1 Question No. Marks Marks

12
Question-Answer Book 34
56
Date: 31-1-2013 78
Time allowed: 2 hours 15 minutes 9
This paper must be answered in English. 10
11
12
INSTRUCTIONS 13
1. Write your Name, Class and Class Number in the 14

spaces provided on Page 1. 15
2. This paper consists of THREE sections, A(1), A(2) 16

and B. Each section carries 35 marks. 17
3. Attempt ALL questions in this paper. Write your 18

answers in the spaces provided in this 19
Question-Answer Book. Do not write in the
margins. Answers written in the margins will not Total
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.
 1
SECTION A(1) (35 marks)
m 2 n 3
1. Simplify and express your answer with positive indices. (3 marks)
( m 2 n 3 ) 2
Answers written in the margins will not be marked.

3mn  2m
2. Make n the subject of the formula k  . (3 marks)
n 1
2 Page total

3. Factorize
(a) 9x2 – 12xy + 4y2,

(b) 9x2 – 12xy + 4y2 + 6x – 4y.
(3 marks)

5x  1
 3
4. (a) Solve  3 and represent the solutions in Figure 1. (3 marks)
 x  4
Figure 1
(b) If x is a positive integer that satisfies the inequalities in (a), write down all the possible
value(s) of x. (1 mark)
3 Go on to the next page Page total

5. In an online game, Nancy has 200 houses. The number of houses owned by Karen is 20%
more than that owned by Nancy.
(a) Find the number of houses owned by Karen.
(b) If Karen gives a certain number of her houses to Nancy, will the number of houses
owned by Nancy be twice that owned by Karen? Explain your answer.
(5 marks)

6. In Figure 2, ABCDE is a circle. AC intersects BE and BD at F and G respectively.
If AF = BF, BE // CD and ∠ABF = 35, find ∠BGC. (4 marks)
B
C
G
35
F
A
D
Figure 2
4 Page total

7. Some marbles are divided between Tom and Fiona in the ratio 8 : 5. If one gets 12 more
than the other, find the total number of marbles.
(4 marks)

8. The weights of 8 students are listed in the following table (in kg):
A B C D E F G H
45.5 50.3 48.7 38 52.3 49.2 55 40.9
(a) By rounding up the weights of the students to the nearest kg, estimate their total weights.
(b) If the maximum loading of the school lift is 400 kg, can they take the school lift at the same
time? Use the result in part (a) to explain your answer. (4 marks)

9. In Figure 3, the coordinates of the point A are (2, 8). A is rotated clockwise about the origin
O through 90o to B. C is the reflection image of A with respect to the y-axis.
(a) Write down the coordinates of B and C.
(b) Is  ABC an isosceles triangle ? Explain your answer. (5 marks).
y
× A ( 2, 8 )
x
O

Figure 3
6 Page total

SECTION A(2) (35 marks)
10. John conducted a survey to study the time of S6 students spent for browsing internet in a
week. He interviewed 10 boys and the result was shown in Figure 4.
Figure 4
The set below shows the data of the survey in ascending order:
{2, 4, a, 5, 6, b, 10, 10, 10, 12}
(a) Write down the values of a and b.
Hence find the mean and the standard deviation of the data.
(b) During examination period, students will spend 1 hour less than half of the
original time spent for browsing internet in a week. Write down the mean and
the standard deviation of the time for browsing internet in a week during the

examination period.
(5 marks)

2
11. It is given that f(x) is the sum of two parts, one part varies as (x + 2) and the other part is a
constant. Suppose that f(1) = 7 and f(2) = 0.
(a) Find f(x).
(b) The graph of y = g(x) is obtained by reflecting the graph of y = f(x) with respect to the
y-axis and then translating upwards by 6 units.
(i) Find g(x).
(ii) The graph of y = h(x) is obtained by translating the graph of y = f(x) upwards by
m units and to the right by n units. If the graphs of y = h(x) and y = g(x) are the
same, find m and n.
(7 marks)
8 Page total

9
Go on to the next page
Page total
3
12. In Figure 5, L is a straight line passing through A(4, 6) with slope and it cuts the y-axis at B.
4
C(4, 1) is a point not lying on L.
y
 A(4, 6)
B
 C(4, 1)
x
O
(a) Find Figure 5
(i) the equation of L,
(ii) the coordinates of B.
(b) D is a point on the y-axis such that A, B, D and C are the vertices of a parallelogram.

Write down two possible pairs of coordinates for D.
(c) Are the parallelograms obtained in (b) rhombuses? Explain your answer.
(7 marks)
 10  Page total
 11 
Page total
Figure 6 shows a parabola y  3x  18x  27  k where k is a constant. The parabola cuts
2
13.
the x-axis at two distinct points B and C and has a vertex at A.
y
y  3x 2  18x  27  k
x
O C B
Figure 6
(a) Find the range of values of k. (3 marks)
(b) If the parabola passes through D (4, –9), find

(i) the value of k,
(ii) the area of ΔABC. (5 marks)
(c) By considering the graph, find the range of values of x satisfying the inequality
3x 2  18x  27  k  0 , where k is the value obtained in (b) (i). (1 mark)
 13 
Page total
14. (a) The height and the base radius of a right circular conical solid X are 18 cm and 15 cm
respectively. Find the volume of the solid X in terms of .
(2 marks)
(b) Figure 7 shows a solid Y consisting of two parts. The upper part is a hemisphere of radius
x cm. The lower part is a frustum which is made by cutting off a right circular cone of base
radius x cm from the solid X in (a). The height of the frustum is 7.2 cm.
x cm
7.2 cm
15 cm
Solid Y
Figure 7
(i) Find the value of x.

(ii) Is the volume of the solid Y greater than the volume of the solid X by more than 15%?
Explain your answer.
(iii) If the solid Y is cut into two identical parts, find the increase in the total surface area,
correct to the nearest cm2.
(7 marks)
 15 
Page total
SECTION B (35 marks)
15. Let f (x) = (x + 1)(x + ai)(x  bi) + 2 , where a and b are positive integers. It is given that f (1) = 12 .
(a) (i) Show that (ai + 1)(bi 1) = 5 .
(ii) Find the values of a and b.
(b) Let g(x) = x 4  2x3  8x  20. Solve 2 f (x) = g(x) , expressing your answers in the form
p + qi if necessary.
(7 marks)
16. In a college, there are four F.5 classes and the numbers of students in each class are shown
as follows:
Class 5A 5B 5C 5D
No. of students 36 37 38 39
Now 4 students are selected from all S.5 students randomly for uniform checking. Peter is a
student from 5B. Find the probability that
(a) Peter is selected,
(b) exactly two 5C students are selected,
(c) Peter is selected given that exactly two 5C students are selected.
(6 marks)
 17  Go on to the next page Page total

17. All S.6 students, including Peter and Paul, attended a Mathematics test. The following
table shows their marks and standard scores in the test.
Peter Paul
Mark 37 50
Standard score – 0.6 2
(a) Find the mean and the standard deviation of the test marks of all S.6 students.
(b) A mark of 40 is deleted due to repeated recording. How do the standard scores of
Peter and Paul change respectively? Explain your answer.
(5 marks)
18. In Figure 8(a), ABCD is a rectangular paper card. AB = 12 cm and BC = 24 cm. M is the
mid-point of AD. The paper card is folded along BM and MC. Then the paper card is placed
on a horizontal plane with AB, BC and CD lying on a horizontal plane such that A, M and C
lie on the same vertical plane, and B, M and D lie on another vertical plane as shown in
Figure 8(b). It is known that BAC = 90 in Figure 8(b).
top
A M. D M
12 cm
C
B 24 cm C B
A D
Figure 8 (a)
front

Figure 8(b)
(a) Find the length of MB. (Leave your answer in surd form.) (2 marks)
(b) Find the length of the projection of MC on the horizontal plane. (4 marks)

19. In figure 9, a point X moves such that it keeps a constant distance of 1 unit from a fixed point I(1, 1).
(a) Describe the locus of the point X.
(1 mark)
(b) The locus of the point X touches OA, OB and AB at P, Q and R respectively, where the
coordinates of O, A and B are (0, 0), (4, 0) and (0, 3) respectively.
(i) Is OPIQ a square? Explain your answer.
(ii) A student claims that the point R is the circumcentre of OAB. Do you agree? Explain
your answer. Hence find the coordinates of the circumcentre of OAB.
(iii) Find the equation of the circumcircle of OAB.
(10 marks)
B(0, 3)

 I(1, 1)
x
O A(4, 0)
Figure 9
END OF PAPER

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A Name

Kwok Tak Seng Catholic Secondary Class ( )

Marker No. Examiner No.

PAPER 1 Question No. Marks Marks

1. Write your Name, Class and Class Number in the 14

2. This paper consists of THREE sections, A(1), A(2) 16

3. Attempt ALL questions in this paper. Write your 18

4. Graph paper and supplementary answer sheets will

5. Unless otherwise specified, all working must be

6. Unless otherwise specified, numerical answers

7. The diagrams in this paper are not necessarily

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

2 Page total

(a) 9x2 – 12xy + 4y2,

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

3 Go on to the next page Page total

Answers written in the margins will not be marked.

If AF = BF, BE // CD and ∠ABF = 35, find ∠BGC. (4 marks)

Answers written in the margins will not be marked.

4 Page total

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

5 Go on to the next page Page total

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

6 Page total

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

7 Go on to the next page Page total

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

8 Page total

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

3x 2  18x  27  k  0 , where k is the value obtained in (b) (i). (1 mark)

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

 17  Go on to the next page Page total

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

 19  Go on to the next page Page total

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.

Answers written in the margins will not be marked.