QUEEN’S COLLEGE
Half-yearly Examination, 2011– 2012
Class
Class Number
MATHEMATICS PAPER 1
Question-Answer Book
Secondary 4
Write your class, class number in the
spaces provided on this cover.
2.
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
8
5
8
6
8
7
10
8
11
9
10
10
11
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.
11-12
Date : 9-1-2012
Time: 3:00p.m. – 4:30p.m.
1.
3.
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 0.2 7 into a rational number.
2.
Express
11-12
 16   144   9   49 in terms of i, where i 2  1
S4 HY -MATH 1 - 2
-2-
(5 marks)
(5 marks)
Page total
3.
Find the domain of the following functions,
(a) f ( x) 
x
x5
(b)
f ( x)  2 x  3
Two points A(2, –3) and B(–4, 5) are given.
(a) Find the coordinates of the mid-point of AB.
(b) Find the equation of the perpendicular bisector of AB
4.
`
11-12
S4 HY -MATH 1 - 3
-3-
(4 marks)
(2 marks)
(6 marks)
Page total
5.
The figure shows the graph of y  a( x  2) 2  8
which cuts the y-axis at C(0, 6), and cuts the
x-axis at P and Q.
(a) Find the coordinates of the vertex.
(b) Find the axis of symmetry of the graph.
(c) (i) Find the value of a.
(ii) Find the coordinates of P and Q.
11-12
S4 HY -MATH 1 - 4
-4-
(1 mark)
(1 mark)
(2 marks)
(4 marks)
Page total
6.
Let  and  be the roots of the equation x 2  ( p  2) x  9  0 where p is a constant.
(a) Express + and  in terms of p.
(2 marks)
(b) Find a quadratic equation in x whose roots are ( 2  1) and (2   1) in terms of p.
(6 marks)
11-12
S4 HY -MATH 1 - 5
-5-
Page total
7.
3  9i
in the form of a  bi .
3i
(a)
Express
(b)
If the result of part (a) is a solution of equation
x 4  x 3  x 2  ax  6b  0 , find the values of a and b.
11-12
S4 HY -MATH 1 - 6
-6-
(4 marks)
(6 marks)
Page total
8.
11-12
The equation x 2  10 x  m  0 , where m is a real constant, has unequal real roots  and .
(a) Find the range of the values of m.
(4 marks)
3
3
(b) If     370 , find the value of m.
(7 marks)
S4 HY -MATH 1 - 7
-7-
Page total
9.
Given that f ( x)  2 x  1 and g ( x)  x 2  3x  7
(a)
Find f (2 x 2 ) and g ( x  1) .
(b)
11-12
Find the values of x when f (2 x )  g ( x  1)
S4 HY -MATH 1 - 8
2
-8-
(5 marks)
(5 marks)
y
10.
Page total
In the figure O is the origin and P is (3, 4).
(a) Find the equation of OP
(2 marks)
(b) If AP = OP and A is a point on the y-axis,
find the coordinates of A.
(3 marks)
(c) If APB is a straight line, find
the coordinates of B
(3 marks)
(d) Find the equation of AB in intercept-form. (1 mark)
(e) Find the area of OPB.
11-12
S4 HY -MATH 1 - 9
(2 marks)
-9-
A
P
O
x
B
Page total
SECTION B
(40 marks)
Answer both questions in this section and write your answers in the spaces provided.
A
11
2x cm
P
p
B
x cm
Q
S
D
R
8 cm
C
The figure above shows a rectangle ABCD of dimension 6cm  8cm. Points P, Q, R and S are
on AB, BC, CD and DA respectively such that AP = CR = 2x cm, BQ = DS = x cm.
11-12
(a) Find the areas of APS and BPQ in terms of x.
(b) Deduce the area of PQRS in terms of x.
(c) Express the area of PQRS obtained in (b) as a function of y  a( x  h) 2  k .
(3 marks)
(3 marks)
(5 marks)
(d) If M(5, m) is a point on the graph of function in (c), find the value of m.
(e) Sketch the graph of quadratic function obtained in (c) on page 12.
Indicate clearly on your sketch the vertex, y-intercept, the line of symmetry
(2 marks)
and point M on the graph.
(f) From the graph, find the minimum area of PQRS and
the corresponding value of x.
(5 marks)
S4 HY -MATH 1 - 10
- 10 -
(2 marks)
Page total
11-12
S4 HY -MATH 1 - 11
- 11 -
Page total
11 (e)
y
x
11-12
S4 HY -MATH 1 - 12
- 12 -
Page total
12
When f ( x)  ax 3 13x 2  bx  60 is divided by (3 x  1)( x  5) ,
the remainder is 7 x  35 .
(a) Express f(x) in terms of quotient, divisor and remainder.
1
(b) Use result (a), find f ( ) and f (5) .
3
(2 marks)
(c) Find the values a and b.
(d) Factorize f(x).
(e) If g ( x)  9 x 3 60 x 2  28x  48 and f(x) have a common factor (3 x  4) ,
(i) factorize g(x).
(ii) Solve 3 f ( x)  2 g ( x) .
11-12
S4 HY -MATH 1 - 13
(1 mark)
(6 marks)
(3 marks)
(3 marks)
(5 marks)
- 13 -
Page total
END OF PAPER
11-12
S4 HY -MATH 1 - 14
- 14 -