LIU PO SHAN MEMORIAL COLLEGE

(2007 - 2008) FINAL EXAMINATION

SECONDARY FOUR MATHEMATICS PAPER I

Class: ( ) Date : 16 – 6 – 2008

Name: Time allowed: 2 hours

1. This paper must be answered in English.

2. Write your name, class and class number in the spaces provided on this cover. 3. This paper consists of THREE sections, A(1), A(2) and B. Section A(1) carries 34 marks. Sections

A(2) and B carry 33 marks each. 4. Attempt ALL questions in Sections A(1) and A(2), and any THREE questions in Section B. Write

your answers in the spaces provided in this Question-Answer Book. Supplementary answer sheets will be supplied on request. Write your name, class and class number on this sheet.

5. Unless otherwise specified, all working must be clearly shown.

6. Unless otherwise specified, numerical answers should either be exact or correct to 3 significant

figures.

7. The diagrams in this paper are not necessarily drawn to scale.

FORMULAS FOR REFERENCE

SPHERE Surface area = 4π 2r

Volume = 43

π 3r

CYLINDER Area of curved surface = 2π rh

Volume = π hr 2

CONE Area of curved surface = π lr

Volume = 13

π hr 2

PRISM Volume = base area × height

PYRAMID Volume = 13

× base area × height

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SECTION A(1) (34 marks) Answer ALL questions in this section and write your answer in the spaces provided.

1. Simplify 3

43 )(a

ab and express your answer with positive indices. (3 marks)

2. Solve 1614 1 =+x . (3 marks)

3. Let f(x) = –x3 + 2x2 + x – 4. Find the remainder when f(x) is divided by x – 2 .

(3 marks)

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4. If f(x) = sin(xo + 10o), find the value of f(20)‧f(50). (Leave your answer in surd form.) (3 marks)

5. Solve the equation θθθθ

cossincossin

−+ = 2 , where . (3 marks) oo 3600 <≤ θ

6. Simplify )90tan(

)cos()90sin(A

AAo

o

−−− . (3 marks)

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7. In Figure 1, O is the centre of circle ABC. OBD is a straight line. If ABD = 110∠ o, find x. (4 marks)

8. In Figure 2, O is centre of circle. BF is diameter and BOFE is a straight line. CA and CE are the

tangents to the circle at B and D respectively. If ∠OBD = 22o , find ∠BED. (4 marks)

A

B

O

C

x

110o

Figure 1

D

Figure 2

E C D

A

22o

O

B

F

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9. If log 2 = a and log 3 = b, express the following in terms of a and b. (a) log 6

(b) log 15 (4 marks)

10. In Figure 3, find the values of x and y. (4 marks)

12 cm 9 cm

x cm

yo

Figure 3

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Section A(2) (33 marks)

Answer ALL questions in this section and write your answers in the spaces provided.

11. (a) Factorize x3 – 4x. (2 marks)

(b) (i) Factorize u2 – 6u + 8. (ii) Using (i) or otherwise, factorize x4 – 6x2 + 8. (2 marks)

(c) Using the result of (a) and (b) or otherwise, solve the equation x4 + x3 – 6x2 – 4x + 8 = 0. (3 marks)

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12. Given that z varies directly as x and inversely as y , and z = 144 when x = 12 and y = 4.

(a) Express z in terms of x and y. (3 marks)

(b) Find the value of y when x = 4 and z = 18. (2 marks)

(c) Find the percentage change in z if x is increased by 20% and y is decreased by 10%.

(3 marks)

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13. Figure 4 shows the graph of y = 8x.

(a) Write down the value of a. (1 mark) (b) Sketch the curve y = x on Figure 4. (2 marks) 8log (c) Find the range of values of y such that 0 < x < 1. (2 marks)

(d) John draws the function y = 9

864 +x

on Figure 4.

(i) What equation can be solved from the point(s) of intersection of the graphs of y = 8x

and y = 9

864 +x

?

(ii) Use algebraic method to solve the equation obtained in (d) (i). (4 marks)

a a Figure 4

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14. Figure 5 shows the graph of the function y = a sin x + b cos x for 0º ≤ x ≤ 360º .

(a) Use the graph to find the values of y when x = 0º and x = 90º . (1 mark) (b) Using (a) or otherwise, find the values of a and b. (2 marks) (c) Find the maximum and the minimum values of – 2 sin x + 3 cos x + 1. (3 marks) (d) By adding a suitable straight line in Figure 5, solve the equation 10 sin x – 15 cos x + 10 = 0 ,

where 0º ≤ x ≤ 360º。 (3 marks)

Figure 5

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SECTION B (33 marks)

Answer any THREE questions in this section and write your answers in the spaces provided.

Each question carries 11 marks.

15. Let . 82)2(94)( 23 ++−+−= kxkkxkxxf(a) Show that when is divided by )(xf 2−x , the remainder is 4. (2 marks) (b) When is divided by , the remainder is –50. )(xf 1+x

(i) Find the value of k, hence find the quotient when is divided by . (5 marks) )(xf 2−x(ii) Using the results of part (a) and (b)(i), solve the equation . (2 marks) 4)( =xf

(c) Find the remainder when is divided by [ 2)1( −xf ] 2−x . (2 marks)

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16. When other conditions remain the same, the quality of a cup of a particular kind of Chinese tea depends on the amount of time, t seconds, that tea leaves are soaked in water and the temperature, x℃, of the water. It is proposed that the quality of a cup of this kind of tea can be measured by the indicator Q as follows:

Q = 25 000 + F

where F consists of two parts with one part varying jointly as x and t, and the other part varying as the square of t. The greater the value of Q, the better is the quality of the tea. It is known that Q = 35600 when t = 40, x = 85; and Q = 32475 when t = 65, x = 75.

(a) Show that . (5marks) 24525000 txtQ −+=(b) (i) Find the value of Q when the tea leaves are soaked in water for 45 seconds at a

temperature of 82℃. (1mark) (ii) When the temperature of water is 78℃, is it possible to achieve the same value of Q in

(b)(i) by changing the amount of time that the tea leaves are soaked in water? Explain your answer briefly. (2 marks)

(c) Suppose the temperature of water is 80℃. Using the method of completing the square, find the amount of time the tea leaves need to be soaked in the water in order to achieve the best quality of the tea. (3 marks)

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17.

In figure 6(a), ABC is a triangle right-angled at B. AB = 4 cm and BC = 16 cm. P, Q and R are the points on AB, BC, CA respectively such that AP = x cm, PR//BC and ∠PRQ = 90°. Let the area of the shaded region be y cm2.

Figure 6(a)

(a) Show that . (4 marks) 3282 2 +−= xxy(b) Figure 6(b) shows the graph of for 3282 2 +−= xxy 40 ≤≤ x . Using the graph, find

(i) the minimum area of the shaded region and the corresponding value of x; (2 marks)

(ii) the range of values of x for which the area of the shaded region is less than 1613 of the

area of ΔABC. (3 marks) (c) The function represents the area of ΔPQR and Figure 6(b) shows that graph

of .

228)( xxxf −=)(xfy =

State the two transformations needed for the graph of to become the graph of . (2 marks)

3282 2 +−= xxy)(xfy =

Figure 6(b)

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18. In Figure 7(a), AP and AQ are tangents to the circle at P and Q. A line through A cuts the circle at B and C and a line through Q parallel to AC cuts the circle at R. PR cuts BC at M.

Figure 7(a)

(a) Prove that (i) M, P, A and Q are concyclic; (3 marks)

(ii) MR = MQ. (3 marks) (b) If ∠PAC = 20° and ∠QAC = 50°, find ∠QPR and ∠PQR. (4 marks) (c) In Figure 7(b), the perpendicular line from M to RQ meets RQ at H. Explain briefly why MH

bisects RQ. (1 mark)

Figure 7(b)

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~~ End of paper~~

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