QUEEN’S COLLEGE
Half-yearly Examination, 2009 – 2010
Class
Class Number
MATHEMATICS PAPER 1
Question-Answer Book
Secondary 4
1.
2.
3.
Date : 12-1-2010
Time: 8:30 – 10:00
Write your class, class number in the
spaces provided on this cover.
This paper consists of TWO sections, A
and B. Section A and B carry 80 and 40
marks respectively.
Attempt ALL questions in this paper.
Write your answer in the spaces provided
in
this
Question-Answer
Book.
Supplementary answer sheets will be
supplied on request. Write your class and
class number on each sheet and put them
inside this book.
Section A
Question No.
Max Marks
1
5
2
5
3
4
4
6
5
6
6
8
7
10
8
10
9
13
10
13
Section A
Total
80
Section B
Question No.
Max Marks
4.
Unless otherwise specified, all working
must be clearly shown.
11
20
5.
Unless otherwise specified, numerical
answers should either be exact or correct
to 3 significant figures.
12
20
Section B
Total
40
Teacher’s
Use Only
Paper I Total
6.
09-10
Teacher’s Use Only
The diagrams in this paper are not
necessarily drawn to scale.
S4 HY -MATH 1 - 1
- 1 -
Marks
Marks
Page total
SECTION A (80 marks)
Answer ALL questions in this section and write your answers in the spaces provided.
1.
Convert the recurring decimal 0.1 3 into a rational number.
(5 marks)
2.
09-10
Express
 36  49   25 in terms of i, where i 2  1
S4 HY -MATH 1 - 2
- 2 -
(5 marks)
Page total
3.
Give the domain of the following functions,
(i) f ( x ) 
4.
09-10
x 1
x 1
(2 marks)
(ii) f ( x )  x  x  3
(2 marks)
If ( x  2)( x  a)  x 2  3x  b , find the values of a and b.
(6 marks)
S4 HY -MATH 1 - 3
- 3 -
Page total
5.
When a polynomial f(x) is divided by (3x  2) , the quotient is (4 x 2  1) and the
remainder is 7.
6.
09-10
Find f(x).
(6 marks)
If the equation 3x 2  6 x  k  1  0 has equal roots, find the value of k and solve the
equation for x.
(8 marks)
S4 HY -MATH 1 - 4
- 4 -
Page total
7.
Given f ( x )  3x 2  6 x  2
(a) Find f ( x  2) and f(2x)
(b) Solve the equation f ( x  2)  f (2 x )  0
(leave your answer in surd form if necessary)
09-10
S4 HY -MATH 1 - 5
- 5 -
(10 marks)
Page total
8.
(a)
Express
2  4i
in the form of a  ib .
2i
2
3
 2  4i   2  4i   2  4i   2  4i 
(b) Hence evaluate 

 
 

 2i   2i   2i   2i 
09-10
S4 HY -MATH 1 - 6
- 6 -
4
(10 marks)
Page total
9.
The roots of the equation 2 x 2  5 x  1  0 are  and  .
(a) Without solving the equation, find the values of
(i)    and 
(ii)  2   2
(b) Form a quadratic equation with roots
09-10
S4 HY -MATH 1 - 7


and
.


- 7 -
(13 marks)
Page total
10.
When the polynomial f ( x )  ax 3  3x 2  11x  b is divided by (x-1), the remainder is -12.
f(x) is divisible by (x + 3).
(a)
Find the values of a and b.
(b)
Solve the equation f(x) = 0.
09-10
S4 HY -MATH 1 - 8
(13 marks)
- 8 -
Page total
SECTION B (40 marks)
Answer both questions in this section and write your answers in the spaces provided.
11
Given a line L : 2 x  y  6  0 .
(a)
(b)
(c)
If L cuts the x-axis and y-axis at P and Q respectively, find the coordinates of
P and Q.
(3 marks)
Find the coordinates of R which divides PQ in the ratio of 2:1.
(2 marks)
Find the equation of the line L1 which passes through R and is
(d)
(e)
perpendicular to L.
If S(3, k) is a point on L1 , find k.
Find the equation of the line L 2 which passes through S and with
x-intercept = 
09-10
1
.
2
(3 marks)
(2 marks)
(2 marks)
(f)
Find the intersection point T of L and L 2 .
(3 marks)
(g)
Find the area of RST .
(5 marks)
S4 HY -MATH 1 - 9
- 9 -
Page total
09-10
S4 HY -MATH 1 - 10
- 10 -
Page total
12(a).
Fig(a) below shows the graph of y  ax 2  bx  c . It passes through the points (1, 25)
and (4, 19) with y-intercept equal to 35.
y
y  ax 2  bx  c
A
x cm
P
B
S
35
Q
(1, 25)
(4, 19)
D
C
R
7 cm
Fig (b)
Fig (a)
(i)
(ii)
(iii)
Find the values of a, b and c.
Find the vertex of the graph.
Insert the x-axis in a possible position in fig(a)
(12 marks)
(b) Fig (b) shows a rectangle ABCD of dimension 5cm x 7cm. Points P, Q, R and S are
points on AB, BC, CD and DC respectively such that AP = BQ = CR = DS = x cm.
(i)
Find the area of APS and BPQ in terms of x.
(ii)
(iii)
09-10
Deduce the area of the parallelogram PQRS in terms of x.
By using the result of (a), find the value of x such that the area of the
parallelogram is a minimum. What is the minimum area.
(8 marks)
S4 HY -MATH 1 - 11
- 11 -
Page total
09-10
S4 HY -MATH 1 - 12
- 12 -