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MUNSANG COLLEGE
2017-2018 First Term Examination
F. 5 Mathematics Compulsory Part
Paper 1
Class : _______ Name : _____________________ Class Number : _____
Subject teacher: CHF / CYL / HYC / MKW / WFL (Please circle as appropriate)
Time allowed : 2 hours
Full mark : 93
This question-answer book consists of 20 printed pages.
Instructions to candidates:
1. This paper must be answered in English with a blue / black ball pen, unless otherwise specified.
2. Write your name, class and class number in the space provided on this cover and circle the initial of your subject teacher.
3. This paper consists of THREE sections, A(1), A(2) and B. Section A(1) carries 31 marks, Section A(2) carries 31 marks and Section B carries 31 marks.
4. Answer 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. All diagrams / graphs / charts as part of the answers must be clearly drawn with an HB pencil.
6. Graph paper and supplementary answer sheets will be supplied on request. Write your name, class and class number on each sheet, and fasten them INSIDE this book.
7. Unless otherwise specified, all working must be clearly shown.
8. The diagrams in this paper are not necessarily drawn to scale.
9. Unless otherwise specified, numerical answers must be exact or correct to 3 significant figures.
10. Calculator pad printed with the “HKEA Approved” / “HKEAA Approved” label is allowed. Remove the calculator cover / jacket.
Marking Scheme
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Section A(1) (31 marks)
1. Simplify 2 3 2
3
( )a b
a
and express your answer with positive indices.
(3 marks)
2. Factorize
(a) 3 9x y ,
(b) 2 22 5 3x xy y ,
(c) 2 22 5 3 3 9x xy y x y .
(4 marks)
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3. Solve 2 2
2 1
2
x y
x y
. (4 marks)
4. (a) Find the range of the values of x which satisfy both 7 6
5( 1)4
xx
and 5 13 0x .
(b) Write down all possible integral value(s) of x satisfying both inequalities in (a).
(4 marks)
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5. A chocolate bar is termed regular if its weight is measured as 60 g correct to the nearest 10 g.
(a) Find the least possible weight of a regular chocolate bar.
(b) Someone claims that the total weight of 300 regular chocolate bars can be measured as
15 kg correct to the nearest kg. Do you agree? Explain your answer.
(4 marks)
6. It is given that the quadratic equation 2 6 3 0kx x has two distinct real roots.
(a) Find the range of values of k.
(b) Write down the greatest integral value of k.
(3 marks)
(and 0)k
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7. The graph of f ( )y x cuts the x-axis at ( 2, 0) and (4, 0) respectively. It is given that
2f ( )x x bx c , where b and c are integers.
(a) Find the values of b and c.
(b) (i) Write down the coordinates of the vertex of the graph of f ( )y x .
(ii) Hence, or otherwise, solve the inequality f ( ) 9x .
(5 marks)
Since the vertex of f ( )y x is (1, 9), the minimum value of f ( )x is 9.
The solutions of f ( ) 9x are all real values of x.
1A
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8. In Figure 1, O is the centre of the circle ABCD. AC and BD intersect at E. It is given that
55DBC , BDC x and ACB y . If AB BC , find x and y.
(4 marks)
Figure 1
A
B
C D
O E
x y
55
no reason 1
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Section A(2) (31 marks)
9. (a) Express 5
2 i in the form a + bi, where a and b are real numbers. (2 marks)
(b) It is given that 5
2 i is a root of the quadratic equation 22 0x px q , where p and q
are real numbers. Find the values of p and q.
(4 marks)
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10. (a) Find the remainder when 3 2f ( ) 3 18 43x x x x is divided by 2 2 8x x . (1 mark)
(b) Let 3 2g( ) 3 14 43x x ax x . When g( )x is divided by 2 2 8x x , the quotient and
the remainder are 3x b and 3 respectively.
(i) Find the values of a and b.
(ii) Solve f ( ) g( )
04
x x
x
, where 0x .
(5 marks)
6a b
3 23 ( 6) (2 24) 8 3x b x b x b
11
2 2
2
( ) ( )0
4
[( 2 8)( 5) 3] [( 2 8)(3 5) 3]0
4
( 2 8)( 5 3 5)0
4
( 2)( 4)(4 )0
4
( 2)( 4) 0
4 or 2
f x g x
x
x x x x x x
x
x x x x
x
x x x
x
x x
x x
1A
1A
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11. In Figure 2, the curve 2y x mx n cuts the x-axis at ( , 0)P and ( , 0)Q respectively.
The axis of symmetry of the graph of 2y x mx n is 1x .
(a) Find the value of m. (3 marks)
(b) It is given that the distance between P and Q is 8 units. Find the value of n. (3 marks)
Figure 2
y
x ( , 0)P ( , 0)Q
2y x mx n
O
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12. In Figure 3, PQ is the tangent to the circle ABCDE at A. CP and DQ intersect at F and cut the
circle at E and B respectively. It is given that PQ // EB.
(a) Prove that P, Q, C and D are concyclic. (3 marks)
(b) It is given that O is the mid-point of PQ and O is also the centre of the circle PQCD in (a).
If BD is a diameter of the circle ABCDE, prove that the line joining P and D is the tangent
to the circle ABCDE at D. (3 marks)
Figure 3
A
B
C
D
E
F
P Q
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13. The equation of the circle C is 2 2 10 8 16 0x y x y . Denote the centre of C by P.
(a) Write down the coordinates of P and the radius of C. (2 marks)
(b) The circle cuts the x-axis at Q and R. Find the area of PQR. (3 marks)
(c) Determine whether the y-axis is a tangent to C. (2 marks)
= distance between the centre and the y-axis
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Section B (31 marks)
14. Consider the function of f ( ) logkx x a , where a and k are constants.
The intercept of the graph of f ( )y x against logk x on the vertical axis is 3 .
The graph of f ( )y x against x passes through the point (32, 2).
Find the values of a and k.
(4 marks)
3
3
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15. Let 2f ( ) 8x x x k and g( ) 2 6x x , where k is a constant. Denote the graph of f ( )y x
and g( )y x by F and G respectively.
(a) Using the method of completing the square, express the coordinates of the vertex of F in
terms of k. (2 marks)
(b) It is given that F and G intersect at (2, 2)A and B respectively.
(i) Find the value of k.
(ii) Find the coordinates of B.
(iii) Find the equation of the circle with diameter AB.
(iv) Determine whether the vertex of F lies on, inside or outside the circle in (b)(iii).
(8 marks)
F
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F
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F
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16. (a) In Figure 4, the equation of the straight line L1 is 3 2 480x y and the slope of the
straight line L2 is 4
5 . L1 and L2 intersect at the point (80,120) . The shaded region,
including the boundary, represents the solution of a system of inequalities. Find the system
of inequalities. (3 marks)
(b) A coffee shop produces two types of coffee, Cappuccino and Latte. Each cup of
Cappuccino requires 3 g of coffee powder and 4 g of milk powder while each cup of Latte
requires 2 g of coffee powder and 5 g of milk powder. On a certain day, there are 480 g of
coffee powder and 920 g of milk powder in the coffee shop. The profits for producing a cup
of Cappuccino and a cup of Latte are $30 and $28 respectively. The manager of the coffee
shop claims that the total profit will exceed $6000 that day. Do you agree? Explain your
answer. (4 marks)
O x
y
L1
L2
Figure 4
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The greatest possible total profit is $5760 < $6000.
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17. In Figure 5(a), ABCD is a trapezoidal paper card such that AB//DC, 5 cmDA AB BC ,
10 cmCD and 120ABC .
(a) (i) Find the length of AC.
(ii) Find the area of the trapezium ABCD.
(4 marks)
(b) The paper card in Figure 5(a) is folded along AC such that BC and DC lie on the horizontal
ground as shown in Figure 5(b). In Figure 5(b), 90DAB and M is the mid-point of
BD.
(i) Find the length of AM.
(ii) It is given that 5 2 cmMC . Find the angle between AC and MC.
(iii) Someone claims that ACM is the angle between AC and the ground.
Do you agree? Explain your answer.
(6 marks)
A B
C D
Figure 5(a)
Figure 5(b)
A
B
C D
M
or 8.66 cm (cor. to 3 sig. fig.)
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