TWGHs. Mrs. Wu York Yu Memorial College
Second Term Test (2013 - 2014)
Form: 4A
Time Allowed:
Date:
Subject: Mathematics
1.5 hours
08:30 to 10:00
ccccc3-3-2011
No. of pages:
Exam No.: _________
8 (the front page included)
Instructions
1.
Write your examination number on this cover and on the answer sheets.
2.
This paper consists of TWO sections, A (30 marks), and B (60 marks).
3.
Attempt ALL questions in this paper. Write your answers on the space provided.
4.
Unless otherwise specified, all working must be clearly shown.
5.
Unless otherwise specified, numerical answers should either be exact or correct to 3 significant
figures.
6.
The diagrams in this paper are not necessarily drawn to scale.
This material was reproduced under the terms of a licence granted by THE HONG KONG
REPROGRAPHIC RIGHTS LICENSING SOCIETY LIMITED. No further copying permitted.
1
Section A: Multiple choice (@1.5)
Question-1.
Which of the following straight lines is
x y
perpendicular to L:   1 ?
3 5
A.
L1: 3x  y  12  0
B.
L2: 3x  y  15  0
C.
L3: 3x  5 y  15  0
D.
L4: 3x  5 y  12  0
Question-5.
If the two straight lines L1 : 2 x  y  4  0 and
L2 : ax  3 y  8  0 intersect at the x-axis, find the
value of a.
A.
–16
B.
–4
C.
4
D.
16
Question-2.
A straight line L cuts the y-axis at (0, –4) and it
has slope –2, find the equation of L.
2x  y  4  0
A.
2x  y  4  0
B.
x  2y  4  0
C.
x  2y  4  0
D.
Question-6.
When x2468 + 1 is divided by x + 1, the remainder
is
A.
1.
B.
0.
C.
1.
D.
2.
Question-7.
Find the quotient when 2 x 3  x 2  x  6 is divided
by x – 3.
A.
 54
B.
60
2 x 2  7 x  22
C.
2 x 2  7 x  22
D.
Question-3.
In the figure, BC is perpendicular to AB. Find the
equation of BC.
A.
B.
C.
D.
Question-8.
When 2x3 + 5x2 + ax  16 is divided by x2 + bx + 2,
the quotient and the remainder are 2x  3 and 15x
 10 respectively. Find the values of a and b.
A.
a = 1, b = 4
B.
a = 4, b = 5
C.
a = 7, b = 4
D.
a = 10, b = 1
4x  y  1  0
4x  y  1  0
x y40
x y40
Question-4.
Find the equation of the straight line L1 passing
through P(2, 5) and parallel to L2: x + 4y – 7 = 0.
4x  y  3  0
A.
4x  y  6  0
B.
x  4 y  22  0
C.
x  4 y  13  0
D.
Question-9.
Let f(x) = x4 + 2x3 + ax + 8. If f(x) is divisible by x
+ 1, find the remainder when f(x) is divided by x +
2.
A.
6
B.
1
C.
–1
D.
–6
2
Question-14.
Find the H.C.F. and L.C.M. of 5(7 x  2) 2 ,
6(7 x  2)( x  6) and 10( x  6) 2 .
H.C.F.
L.C.M.
30(7 x  2)( x  6)
A.
1
B.
1
30(7 x  2) 2 ( x  6) 2
(7 x  2)( x  6)
C.
30
D.
30
(7 x  2) 2 ( x  6) 2
Question-10.
Let f(x) = 2x3 – 5x2 – 22x – 15. If f(–1) = 0, f(x)
can be factorized as
A.
(x + 1)(x + 5)(2x + 3).
B.
(x + 1)(x + 5)(2x – 3).
C.
(x + 1)(x – 5)(2x + 3).
D.
(x – 1)(x – 5)(2x – 3).
Question-11.
Which of the following is/are factor(s) of (x +
1)(4x2 – 6x) + 2 + 2x?
I.
x–2
II.
x+2
III.
2x – 1
IV.
2x + 1
A.
I only
B.
III only
C.
I and II only
D.
III and IV only
Question-15.
8 x  12
4x 2  6x


2 x 3  3x 2 4 x 2  9
4
A.
2x  3
4
B.
2x  3
8
C.
x(2 x  3)
8
D.
x(2 x  3)
Question-12.
Find the quotient and the remainder of (16x2 –
12x3 – 37x + 7) ÷ (6x2 – 2x + 7).
A.
Quotient = –2x – 2, remainder = 19x – 7
B.
Quotient = –2x – 2, remainder = –19x – 7
C.
Quotient = –2x + 2, remainder = 19x – 7
D.
Quotient = –2x + 2, remainder = –19x – 7
Question-16.
Find the H.C.F. and L.C.M. of 12(x + 1)2(3x  1)2
and 18(x + 1)3(3x  1).
H.C.F.
L.C.M.
A.
B.
C.
D.
Question-13.
p 4  4q 4 p 2  2q 2

Simplify
.
4
2q 2  p 2
4
A.
2
p  2q 2
B.
C.
D.
(x + 1)2(3x  1)
(x + 1)2(3x  1)
6(x + 1)2(3x  1)
6(x + 1)2(3x  1)
Question-17.
 x
y  x y 
 2  2      1 
x  y x 
y
xy
A.
x y
xy
B.
x y
1 1
C.

y x
1 1
D.

y x
p 2  2q 2
p 2  2q 2
4
–4
3
216(x + 1)5(3x  1)3
36(x + 1)3(3x  1)2
216(x + 1)5(3x  1)3
36(x + 1)3(3x  1)2
Question-18.
x 2  3x  2 x 2  3x  4
Simplify
.
 2
x4
2x  x 1
2x 1
A.
x 1
x 1
B.
2x 1
( x  2)( x  1)
C.
2x 1
2x 1
D.
( x  2)( x  1)
Question-19.
Given that a 
A.
B.
C.
D.
x( x  1)
x 1
x(1  x)
x 1
x( x  1)
1 x
x( x  1)
x 1
Question-20
Factorize 4(6r – 4s) – (2s – 3r)2.
A.
(2s – 3r)(2s – 3r + 8)
B.
(2s – 3r)(2s – 3r  8)
C.
(3r – 2s)(8 – 3r + 2s)
D.
(3r – 2s)(8 – 3r – 2s)
x 1
x 1
a 1
and b 
, express
x 1
x 1
b 1
in terms of x.
Section B (Questions)
Question-1. (4 marks)
3
In the figure, the straight line L : y   x and L1 are parallel. The y-intercept of L1 is –4. Find the equation
2
of L1.
4
Question-2. (4 marks)
When 8x3 – 2x2 + kx + 1 is divided by 2x – 3, the remainder is 31. Find the value of k.
Question-3. (4 marks)
Use the factor theorem to determine whether each of the following is a factor of
f ( x)  7 x 3  5 x 2  11x  23 .
(a)
x+1
(b)
x–3
Question-4. (4 marks)
Find the H.C.F. and L.C.M. of 12p2q2r2, 18p3q6r9 and 30p2q5r8.
Question-5. (4 marks)
2
3
 2
Simplify 2
.
x  3x x  5 x
5
Question-6. (6 marks)
In the figure, the vertices of an isosceles triangle OAB are O, A and B(8, 6) respectively. A is a point on the
x-axis, and P is a point on AB such that OP⊥AB.
Find
(a) the coordinates of A and P,
(b) the equations of OP and AB.
Question-7. (6 marks)
When a polynomial f (x) is divided by 2x2 + 4x  6, the quotient and the remainder are 3x2 + 2 and 4x + 8
respectively.
(a)
Find the polynomial f(x).
(b)
Find the quotient and the remainder of f ( x)  (4  2 x) .
Question-8. (4 marks)
(a)
Find the remainder when x9999  1 is divided by x  1.
(b)
If today is Monday, what is the day after 999999 days?
6
Question-9. (6 marks)
It is given that when a polynomial f(x) is divided by x + 1 and x – 3, the remainders are –9 and –1
respectively. Find the remainder when f(x) is divided by (x + 1)(x – 3).
Question-10. (4 marks)
b2  b b2 1
a
2ab  2b
Simplify
.

 2 
a 1
a
a 1
b 1
Question-11. (4 marks)
x 1
x 1
Given that a 
and b 
, express
x 1
x 1
1
b in terms of x.
1
1
a
1
7
Question-12. (10 marks)
In the figure, the straight line L1 cuts the x-axis and the y-axis at A(8, 0) and B(0, 8). The straight line L2 cuts
the x-axis at P. Q and R are points on L1, and S is a point on L2 such that PQRS is a rectangle, where
PQ  3 2 units.
(a) Find the equation of L1.
(b) (i)
Find the coordinates of P.
(ii) Hence, find the equation of L2.
(c) (i)
Find the equation of PQ.
(ii) Hence, find the coordinates of Q.
(End)
8