Post on 25-Apr-2020
SECTION A(1) (35 marks)
1. Simplify 54
321 )(nmnm
and express your answer with positive indices. (3 marks)
2. Factorize
(a) 22 32 baba ,
(b) bababa 22 32 . (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.
3. Make y the subject of the formula zy
x32
. (3 marks)
4. (a) Solve the compound inequality
10)4(3or 3 x xx ……(*)
(b) Write down the least positive 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.
5. The total price of an apple and 3 oranges is $20 while the total price of an orange and
2 apples is $18. Find the total price of 2 apples and 3 oranges.
(4 marks)
6. Bank A offers a simple interest rate of r% per annum. If Mary deposits $25 000 into
bank A for 3 years, she will get back an amount of $29 875 after 3 years.
(a) Find the value of r.
(b) Bank B offers an interest rate of (r – 0.5)% per annum compounded monthly.
If Mary wants to deposit $25 000 into a bank such that she can receive
more interest after 3 years, which bank should she choose? 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.
7. The radius and the area of a sector are 10 cm and 2π cm2.
(a) Find the angle of the sector.
(b) Express the perimeter of the sector in terms of π.
(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.
8. In a polar coordinate system, O is the pole. The polar coordinates of the points A and
B are ( 3 , 138°) and (1, 228°) respectively.
(a) Find the length of AB and ∠OAB.
(b) Write down the polar coordinates of all possible point(s) of P such that
△OAB △AOP. (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.
9. In Figure 1, EC is a diameter of the circle and 30ABE .
Figure 1
(a) Find ∠ADC.
(b) Suppose CDAD . Mary claims that CE is the angle bisector of ∠ACD. Do
you agree? Explain your answer. (5 marks)
30°
A
B C
D E
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.
SECTION A(2) (35 marks)
10. In Figure 2, D is a point on AB such that 3:5: BDAD . E is a point on AC such that
DE // BC. It is given that 90ABC and DCBDCE .
Figure 2
(a) Prove that CEDE . (2 marks)
(b) Find AE : CE. Hence, find ∠BAC. (3 marks)
A
D
B
E
C
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.
11. The painting cost (in $) of a solid is a sum of two parts, one part is constant and the
other part varies as the surface area (in cm2) of the solid. The painting cost for a solid
with surface area 200 cm2 is $300 while the painting cost for a solid with surface area
250 cm2 is $360. Suppose the surface area of solid A is 605 cm2.
(a) Find the painting cost of solid A. (4 marks)
(b) Solids A and B are similar solids. The volume of A is 33.1% more than that of
solid B. Mary claims that the painting cost of solid A is at least 20% higher than
that of solid B. Do you agree? 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.
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.
12. The x-intercept and the y-intercept of straight line L1 are –9 and 4.5 respectively. The
equation of straight line L2 is 032 yx . P is a moving point in the rectangular
coordinate plane such that the perpendicular distance from P to L1 is equal to that
from P to L2. Denote the locus of P by .
(a) (i) Describe the geometric relationship between L1 and L2. Explain your answer.
(ii) Find the equation of .
(5 marks)
(b) The equation of the circle C is 9)( 22 nyx , where n is a constant. If L1
cuts C at A and B, while L2 cuts C at H and K and HKAB , find n.
(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.
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.
13. When )(xf is divided by 12 x , the quotient is hx and the remainder is kx 4 . It is given that )(xf is divisible by 1x .
(a) Find the possible value(s) of k. (3 marks)
(b) It is known that there are two distinct real roots for the equation 0)( xf .
Find the possible value(s) of h. (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.
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.
14. A restaurant conducted a survey to find out the customer satisfaction on food. The
stem-and-leaf diagram below shows the distribution of the scores given by a group of
customers.
Stem (tens) Leaf (units)
5 1 2 3 3 6 6 7 9 9
6 1 1 2 2 5 5 a
7 b 0 1 2
It is given that the mode of the scores of the restaurant is 65.
(a) Write down the values of a and b. (2 marks)
(b) Find the mean and the standard deviation of the scores of the restaurant.
(2 marks)
(c) Two more customers completed the survey. Their scores are p61 and
p61 , where 0p .
(i) Find the mean of the scores of all 22 customers completing the survey.
(ii) Peter claims that if p > 7, the standard deviation of the scores is
increased. Do you agree? Explain your 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.
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.
Section B (35 marks)
15. A mathematics test is conducted in a class. If the scores of the top two of the students
and the bottom two of the students in the class are excluded, the mean of the scores
does not change.
(a) If the standard score of the four students are 3.1, 0.1 – k, 0.1 + k and k, find
k. (2 marks)
(b) It is known that the highest score among the four students is 74 marks, and the
mean of the scores of the students in the class is 40 marks. A student in the
class, Peter, claims that he gets 72 marks in the test. Do you agree? Explain
your answer. (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.
16. In a group of 10 students, 4 of them come from school A and the rest come from
school B. 4 students are randomly selected from the group.
(a) Find the probability that at least 3 students from school A are selected.
(2 marks)
(b) It is known that there are only two girls from each school in the group.
(i) Find the probability that exactly 3 students from school A are selected and exactly 3 girls are selected.
(ii) If at least 3 students from school A are selected, find the probability that exactly 3 girls are selected.
(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.
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.
17. (a) In Figure 3, the straight line L1 passes through the origin and intersects with
another straight line L2 at (6, 6). The equation of the straight line L2 is
3032 yx . The shaded region (including the boundary) represents the
solution of a system of inequalities. Find the system of inequalities. (3 marks)
Figure 3
(b) Mary spends $30 to buy some apples and oranges. Each apple costs $2 and
each orange costs $3. The number of apples bought is not more than than that
of oranges bought. She claims that the total number of fruit bought can exceed
12. Do you agree? Explain your answer. (3 marks)
y
x O
L2
L1
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.
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.
18. For any positive integer n, let )2(log 12
nn aA , where a > 0. It is given that
75... 1021 AAA .
(a) Find a. (3 marks)
(b) Find the smallest value of n such that 100...21 nAAA . (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.
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.
19. In Figure 4(a), ABC is a triangular paper card, where G is its circumcentre. AG is
produced to meet BC at D. E is a point on AC such that CEAE . It is given that
cm 8 ACAB and cm 5AG . The paper card in Figure 4(a) is folded along AD
such that AB and AC lie on the horizontal ground as shown in Figure 4(b).
Figure 4(a) Figure 4(b)
(a) Find AD and CD. (4 marks)
(b) Let H be the foot of perpendicular from G to the horizontal ground. AH is
produced to meet BC at F. Suppose H is the incentre of △ABC.
(i) Is ∠AEH = 90°? Explain your answer.
(ii) Prove that △CEH △CFH.
(iii) Find the volume of the pyramid ABCD.
(7 marks)
(c) Peter claims that when ∠BDC = 90°, the volume of the pyramid ABCD in
Figure 4(b) is the greatest. Do you agree? Explain your answer.
(2 marks)
AB
C
D
E
G
A
B C D
E
G
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.
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.
END OF PAPER