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)

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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)

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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)

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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)

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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)

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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

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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

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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)

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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)

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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)

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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)

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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)

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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)

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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

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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)

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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

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END OF PAPER