1. No category

HKCEE 1993
Mathematics II
93
1.
If f(x) = 102x, then f(4y) =
A.
B.
C.
D.
E.
93
2.
If s =
A.
B.
C.
D.
E.
C.
D.
104y .
102 + 4y .
108y .
40y .
402y .
E.
93
5.
n
[2a + (n  1)d], then d =
2
If 3x2 + ax  5  (bx  1)(2  x)  3,
then
A.
B.
C.
D.
E.
2( s  an)
.
n(n  1)
2( s  an)
.
(n  1)
s
.
n(n  1)
as  n
.
a(n  1)
4( s  an)
.
n(n  1)
b a
2 a
b  2 ab  a
ab
ab
ab
93
6.
a = 5, b = 3 .
a = 5, b = 3 .
a = 3, b = 5 .
a = 5, b = 3 .
a = 3, b = 5 .
y
5
4
D
C
3
B
2
93
3.
Simplify (x2  3 x + 1)(x2 + 3 x + 1).
1
A
A.
B.
C.
D.
E.
93
4.
a
+
a b
A.
B.
O
x4 + 1
x4  x2 + 1
x4 + x2 + 1
x4  3x2  2 3 x  1
x4 + 3x3  2 3 x2 + 3 x  1
2
3
4
5
x
3x + 2y = 0
Find the greatest value of 3x + 2y if
(x, y) is a point lying in the region
OABCD (including the boundary).
a
.
a b
A.
B.
C.
D.
E.
1
a b
a  2 ab  b
ab
93-CE-MATHS II
1
1
15
13
12
9
8
93
7.
y
A.
B.
C.
D.
E.
y = ax2 + bx
-1 0
1 2 3 4
x
93
11.
13 .
26 .
33 .
39 .
65 .
Find the H.C.F. and L.C.M. of ab2c and
abc3
y = cx + d
A.
B.
C.
D.
E.
The diagram shows the graphs of
y = ax2 + bx and y = cx + d. The
solutions of the equation
ax2 + bx = cx + d are
A.
B.
C.
D.
E.
93
8.
C.
D.
E.
p=q=1.
q
p=
.
q 1
q
p=
.
q 1
q 1
p=
.
q
q 1
p=
.
q
D.
E.
93
13.
C.
D.
E.
If 3, a, b, c, 23 are in A.P., then
a+b+c=
93-CE-MATHS II
2
3
1
1

3
2
3
3
If the simultaneous equations
 y  x2  k
have only one solution,

 yx
find k.
A.
B.
1
4
12
16
18
L.C.M.
a2b3c4
ab2c3
a2b3c4
abc
abc
If  and  are the roots of the quadratic
equation x2  3x  1 = 0, find the value
1 1
of + .
 
A.
B.
C.
The expression x2 2x + k is divisible
by (x + 1). Find the remainder when it
is divided by (x + 3).
A.
B.
C.
D.
E.
93
10.
93
12.
If log(p + q) = log p + log q, then
A.
B.
93
9.
1, 1
1, 2
0, 1
0, 3
1, 3
H.C.F.
a
abc
abc
ab2c3
a2b3c4
1
1

4
4
1
4
1
93
14.
93
16.
r
h
h
The price of a cylindrical cake of radius
r and height h varies directly as the
volume. If r = 5 cm and h = 4 cm, the
price is $30. Find the price when r = 4
cm and h = 6 cm.
A.
B.
C.
D.
E.
r
$25
$28.80
$31.50
$36
$54
In the figure, the base of the conical
vessel is inscribed in the bottom of the
cubical box. If the box and the conical
vessel have the same capacity, find h :
r.
93
15.
A.
B.
C.
D.
E.
24 : 
3:1
6:
3:
8 : 3
2 rad
93
17.
1.5 cm
h
Find the perimeter of the sector in the
figure.
A.
B.
C.
D.
E.
r
2.25 cm
3 cm


  3  cm
 60

4.5 cm
6 cm
The figure shows a solid consisting of a
cylinder of height h and a hemisphere
of radius r. The area of the curved
surface of the cylinder is twice that of
the hemisphere. Find the ratio
volume of cylinder : volume of hemisphere
A.
B.
C.
D.
E.
93-CE-MATHS II
3
1:3
2:3
3:4
3:2
3:1
93
18.
A merchant marks his goods 25%
above the cost. He allows 10 %
discount on the marked price for a cash
sale. Find the percentage profit the
merchant makes for a cash sale.
A.
B.
C.
D.
E.
93
19.
12.5%
15%
22.5%
35%
37.5%
D.
E.
41
25
The largest value of 3sin2 + 2cos2 
1 is
C.
D.
E.
cos
1  cos 


2
1  sin 
sin 
A
B

cos4  sin4 + 2 sin2 =
A.
B.
C.
D.
E.
C
A.
B.
93
21.
B
C.
x
D.
8
E.
5
In the figure, cosA = 
A.
B.
A
4
. Find a.
5
153
137
93-CE-MATHS II
P
In the figure, AB = BC, BP = CP and
BP  CP. Find tan .
0
1
(1  sin2)2
(1  cos2)2
(cos2  sin2)2
C
1.
3
.
2
2.
3.
4.
93
23.
sin 
cos 
tan 
1
sin 
1
cos
E.
89
A.
B.
2
A.
B.
C.
D.
93
20.
93
22.
C.
4
1
4
1
3
1
2
1
3
3
2
93
24.
B.
C.
D.
E.
C
o
15
2x -
D
110o
120o
135o
140o
4x + 5o
93
27.
o
2x - 10
A
B
A.
B.
C.
D.
E.
In the figure, points A, B, C and D are
concyclic. Find x.
20o
22.5o
25o
27.5o
30o
A.
B.
C.
D.
E.
93
28.
93
25.
A

o
38
B
C
E
72o
34o
54o
70o
72o
76o
93
26.
93
29.
D
A
20o
C
B
93
30.
In the figure, AB is a diameter. Find
ADC.
A.
A(0, 0), B(5, 0) and C(2, 6) are the
vertices of a triangle. P(9, 5), Q(6, 6)
and R(2, 9) are three points. Which of
the following triangles has/have area(s)
greater than the area of ABC?
I.
II.
III.
ABP
ABQ
ABR
A.
B.
C.
D.
E.
I only
II only
III only
I and II only
II and III only
A circle of radius 1 touches both the
positive x-axis and the positive y-axis.
Which of the following is/are true?
I.
II.
III.
Its centre is in the first quadrant.
Its centre lies on the line x  y = 0.
Its centre lies on the line x + y = 1.
A.
B.
C.
D.
E.
I only
II only
III only
I and II only
I and III only
What is the area of the circle
x2 + y2  10x + 6y  2 = 0?
A.
100o
93-CE-MATHS II
3.
4.
6.
7.
10 .
D
In the figure, BA // DE and AC = AD.
Find .
A.
B.
C.
D.
E.
If the points (1, 1), (3, 2) and (7, k) are
on the same straight line, then k =
5
32
B.
C.
D.
E.
93
31.
Two fair dice are thrown. What is the
probability of getting a total of 5 or 10?
A.
B.
C.
D.
E.
93
32.
34
36
134
138
1
9
5
36
1
6
7
36
2
9
93
34.
A.
B.
C.
D.
E.
I only
II only
III only
I and II only
II and III only
If 9x + 2 = 36, then 3x =
A.
B.
C.
D.
E.
A group of n numbers has mean m. If
the numbers 1, 2 and 6 are removed
from the group, the mean of the
remaining n  3 numbers remains
unchanged. Find m.
A.
B.
C.
D.
E.
III.
Standard deviation of B
Mode of A > Mode of B
93
35.
If a : b = 2 : 3 and b : c = 5 : 3, then
abc
=
abc
A.
B.
1
2
3
6
n3
C.
D.
E.
93
33.
A
93
36.
B
The figure shows the frequency
polygons
of
two
symmetric
distributions A and B with the same
mean. Which of the following is/are
true?
I.
II.
Interquartile range of
Interquartile range of B
Standard deviation of
93-CE-MATHS II
A
<
A
>
2
.
3
4
.
3
2.
6 .
9.
2 .
5
.
2
4.
17
.
2
31 .
x
Sign of f(x)
3.56
3.58
3.57
3.575
+

+
+
From the table, a root of the equation
f(x) = 0 is
A.
B.
C.
D.
E.
6
3.57 (correct to 3 sig. fig.).
3.575 (correct to 4 sig. fig.).
3.5775 (correct to 5 sig. fig.).
3.5725 (correct to 4 sig. fig.).
3.58 (correct to 3 sig. fig.).
93
37.
93
40.
Given that the positive numbers p, q, r,
s are in G.P., which of the following
must be true?
I.
II.
III.
A.
B.
C.
D.
E.
93
38.
If the solution of the inequality
x2  ax + 6  0 is c  x  3, then
A.
B.
C.
D.
E.
kp, kq, kr, ks are in G.P., where k
is a non-zero constant.
ap, aq, ar, as are in G.P., where a
is a positive constant.
log p, log q, log r, log s are in
A.P.
a = 5, c = 2 .
a = 5, c = 2 .
a = 5, c = 2 .
a = 1, c = 2 .
a = 1, c = 2 .
93
41.
D
I only
II only
I and II only
I and III only
I, II and III only
D
A
C
A.
x
1
4
1
3
B
B.
In the figure, the rectangle has
perimeter 16 cm and area 15 cm2. Find
the length of its diagonal AC.
C.
3
8
3
4
3
2
D.
A.
B.
C.
D.
E.
93
39.
B
In the figure, ABCD is a square and
ABE is an equilateral triangle.
Area of ABE
=
Area of ABCD
y
A
C
E
32 cm
E.
34 cm
7 cm
226 cm
93
42.
241 cm
In factorizing the expression
a4 + a2b2 + b4, we find that
Q
P
S
A.
B.
C.
D.
E.
(a  b ) is a factor.
(a2 + b2) is a factor.
(a2  ab  b2) is a factor.
(a2  ab + b2) is a factor.
it cannot be factorized.
2
R
2
93-CE-MATHS II
O
In the figure, the radii of the sectors
OPQ and ORS are 5 cm and 3 cm
Area of shaded region
respectively.
=
Area of sector OPQ
7
A.
B.
C.
D.
E.
93
43.
4
.
25
2
.
5
9
.
25
16
.
25
21
.
25
Solve tan4 + 2tan2  3 = 0 for
0o   < 360o.
45o, 135o only
45o, 225o only
45o, 60o, 225o, 240o
45o, 120o, 225o, 300o
45o, 135o, 225o, 315o
A.
B.
C.
D.
E.
93
46.
y
1
Which of the following gives the
compound interest on $ 10 000 at 6%
p.a. for one year, compounded
monthly?
A.
B.
C.
D.
E.
93
44.
93
45.
0
80o 170o 260o 350o
x
-1
0.06
 12
12
$ 10 000(1.0612  1)
$ 10 000 
The figure shows the graph of the
function
12
 0.06 
$ 10 000 1 

12 

 0.06 12 
$10 000 1 
  1
12 


y = sin(350o  x) .
y = sin(x + 10o) .
y = cos(x + 10o) .
y = sin(x  10o) .
y = cos(x  10o) .
A.
B.
C.
D.
E.
 0.6 12 
$10 000 1 
  1
12 


93
47.
2
of the students in a class
3
failed in an examination. After taking a
re-examination, 40% of the failed
students passed. Find the total pass
percentage of the class.
C
Originally
A.
B.
C.
D.
E.
A
2
%
3
1
33 %
3
40%
60%
1
73 %
3
26
93-CE-MATHS II
B
In the figure, ABC is an equilateral
triangle and the radii of the three circles
are each equal to 1. Find the perimeter
of the triangle.
A.
B.
C.
8
12
3(1 + tan30o)
6(1 + tan30o)
D.
E.
1 

3 1 
o 
 tan 30 
1 

6 1 
o 
 tan 30 
93
48.
E
II.
III.
a : b : c = 1: 2 : 3
sinA : sinB : sinC = 1 : 2 : 3
A.
B.
C.
D.
E.
I only
II only
III only
I and II only
I, II and III only
H
D
93
50.
3
F
P
C
G
A
T
M

12
B
5
B
In the figure, ABCDEFGH is a cuboid.
The diagonal AH makes an angle 
with the base ABCD. Find tan  .
In the figure, TP and TQ are tangent to
the circle at P and Q respectively. if M
is a point on the minor arc PQ and
PMQ = , then PTQ =
A.
A.
B.
C.
D.
E.
3
5
3
12
3
13
3
178
B.
C.
D.
E.

.
2
  90o .
180o .
180o 2 .
2  180o .
93
51.
H
153
5
O
93
49.
M
C
K
b
A
A
a
c
A : B : C = 1: 2 : 3
93-CE-MATHS II
B
In the figure, O is the centre of the
circle. AB touches the circle at N.
Which of the following is/are correct?
B
In the figure, if arc BC : arc CA : arc AB
= 1 : 2 : 3, which of the following is/are
true?
I.
N
9
I.
II.
III.
M, N, K, O are concyclic.
HNB ~ NKB
OAN = NOB
A.
B.
I only
II only
C.
D.
E.
III only
I and II only
I, II and III only
93
52.
93
54.
H
G
X
A
In the figure, the three circles touch one
another. XY is their common tangent.
The two larger circles are equal. If the
radius of the smaller circle is 4 cm, find
the radii of the larger circles.
B
E
F
A.
B.
C.
D.
E.
In the figure ABCD and EFGH are two
squares and ACH is an equilateral
triangle. Find AB : EF.
A.
B.
C.
D.
E.
1:2
1:3
1: 2
1: 3
2: 3
93
53.
D
D
C
F
A
B
E
D
A
C
B
F
E
A
B
F
C
E
In the figure, a rectangular piece of
paper ABCD is folded along EF so that
C and A coincide. If AB = 12 cm, BC =
16 cm, find BE.
A.
B.
C.
D.
E.
3.5 cm
4.5 cm
5 cm
8 cm
12.5 cm
93-CE-MATHS II
Y
C
D
10
8 cm
10 cm
12 cm
14 cm
16 cm