Form 5

Form 5
HKCEE 1992
Mathematics II
92
1.
A.
B.
C.
D.
E.
92
2.
B.
C.
D.
E.
C.
D.
E.
92
5.
1
, then b =
1 b
E.
1
1
.
1 a
1
1
.
1 a
1
1+
.
1 a
1
1+
.
1 a
1
1 +
.
1 a
92
6.
1 only
2 only
Any value
Any value except 1
Any value except 2
92-CE-MATHS II
1
log10a, then b =
2
10 a .
10 + a .
5a .
a
.
2
a
1+ .
2
5b  a
5b + a
ab
13b  5a
13a  5b
a c
  k and a, b, c, d are positive,
b d
then which of the following must be
true?
B.
C.
D.
E.
1
1
 5
2
If
A.
5 1
5 1


5 1
5 1
5
Which of the following is a factor of
4(a + b)2  9(a  b)2 ?
A.
B.
C.
D.
E.
92
7.
0
1
2
3
If log10 b = 1 +
A.
B.
C.
D.
For what value(s) of x does the equality
( x  1)( x  2)
= x + 1 hold?
x2
A.
B.
C.
D.
E.
92
4.
ab
ab
ab
ab
1
ab
2
ab
1
ab
If a = 1 
A.
92
3.
A.
B.
1 1
 
a b
ac
k
bd
ab = cd = k
ac = bd = k
a=c=k
ac
k
bd
C.
D.
E.
A.
B.
C.
D.
E.
92
10.
A.
B.
C.
D.
E.
nn  2
n
n2
n2
n
2
1
$16
$400
$416
$800
$816
92
13.
1m
1.2
m
If a and b are greater than 1, which of
the following statements is/are true?
I.
II.
III.
A sum of $ 10 000 is deposited at 4%
p.a., compounded yearly. Find the
interest earned in the second year.
0.3 m
1.8 m
The figure shows a solid platform with
steps on one side and a slope on the
other. Find its volume.
ab  a  b
1
(a + b1)1 = a + b
a2b3 = (ab)6
A.
B.
C.
D.
E.
I only
II only
III only
I and II only
None of them
0.75 m3
0.84 m3
0.858 m3
1.008 m3
1.608 m3
92
14.
If a : b = 2 : 3, a : c = 3 : 4 and b : d =
5 : 2, find c : d.
P
A.
B.
C.
D.
E.
92
11.
1:5
16 : 45
10 : 3
20 : 9
5:1
O
Q
T
In the figure, TP and TQ are tangent to
the circle of radius 3 cm. Find the
length of the minor arc PQ.
15.2%
20%
50%
72.8%
80%
A.
B.
C.
D.
E.
92-CE-MATHS II
3 cm
60o
Suppose x varies directly as y2 and
inversely as z. Find the percentage
increase of x when y is increased by
20% and z is decreased by 20%.
A.
B.
C.
D.
E.
0.2 m
A.
B.
92
9.
92
12.
ntimes



n n   n
Simplify
.
n
n

n



n terms
0.3 m
92
8.
2
3 cm
2 cm
3
cm
2
 cm

cm
2
92
15.
A
92
17.
E
h
F
B
3h
2h
D
H
Find the ratio of the volume of the
tetrahedron ACHD to the volume of the
cube ABCDEFGH in the figure.
In the figure, a cone of height 3h is cut
by a plane parallel to its base into a
smaller cone of height h and a frustum.
Find the ratio of the volume of the
smaller cone to the volume of the
frustum.
A.
B.
C.
D.
E.
A.
B.
C.
D.
E.
C
G
1:8
1:6
1:4
1:3
1:2
92
16.
92
18.
A
E
4 cm
4c
m
C
O
E
2
3

In the figure, the equilateral triangle
ACE of side 4 cm is inscribed in the
circle. Find the area of the inscribed
regular hexagon ABCDEF.
4
In the figure, find cos  .
8 3 cm2
8 2 cm2
4 3 cm2
4 2 cm2
16 cm2
A.
B.
C.
D.
92-CE-MATHS II
5.
3.
1.
0.
1 .
92
19.
D
A.
B.
C.
D.
E.
The greatest value of 1  2sin  is
A.
B.
C.
D.
E.
4c
m
B
1 : 27
1 : 26
1:9
1:8
1:7
3
1
4
11
16
3
4
7
8

E.
92
20.
77
9
In which two quadrants will the
solution(s) of sin  cos  < 0 lie?
A.
B.
C.
D.
E.
In quadrants I and II only
In quadrants I and III only
In quadrants II and III only
In quadrants II and IV only
In quadrants III and IV only
I only
II only
III only
I and II only
II and III only
92
24.

36o
92
22.
o
If A + B + C = 180o, then
1 + cos A cos (B + C) =
A.
B.
C.
D.
E.
0.
sin2 A .
1 + cos2 A .
1 + sinA cos A .
1  sinA cos A .
In the figure, O is the centre of the
circle. Find .
92
25.



2
42o
36o
24o
21o
18o
A.
B.
C.
D.
E.
y
O


6
A
x
6
E
2
5
The figure shows the graph of the
function
92
23.
tan(x + ) .
tan(x  ) .
 tan x .
 + tan x .
  tan x .
2 cos2  sin2 = 1
2 cos2  sin2 = 2
2 cos2  sin2 = 1
92-CE-MATHS II
C
In the figure, ABCD is a square with
side 6. If BE = CE = 5, find AE.
Which of the following equations
has/have solutions?
I.
II.
III.
B
5
D
A.
B.
C.
D.
E.
O
42
92
21.
A.
B.
C.
D.
E.
4
A.
61
B.
C.
D.
E.
9
10
6 3
109
92
26.
C.
D.
E.
R

92
29.
Q
P
In the figure, the circle is inscribed in a
regular pentagon. P, Q and R are
points of contact. Find .
30o
32o
35o
36o
45o
A.
B.
C.
D.
E.
92
27.
92
30.
S
If 0 < k < h, which of the following
circles intersect(s) the y-axis?
I.
II.
III.
(x  h)2 + (y  k)2 = k2
(x  h)2 + (y  k)2 = h2
(x  h)2 + (y  k)2 = h2 + k2
A.
B.
C.
D.
E.
I only
II only
III only
I and II only
II and III only
If the line y = mx + 3 divides the circle
x2 + y2  4x  2y  5 = 0 into two equal
parts, find m.
A.
C
P
3x
B.
C.
D.
B
A
E.
2x
D
92
31.
T
In the figure, ST is a tangent to the
smaller circle. ABC is a straight line.
If TAD = 2x and DPC = 3x, find x.
A.
B.
C.
D.
E.
92
28.
92
32.
If the two lines 2x  y + 1 = 0 and
ax + 3y  1 = 0 do not intersect, then a
=
A.
B.
(3.5, 3)
(3, 2)
(3, 1)
(1.5, 2)
(1, 2)
The table shows the mean marks of two
classes of students in a
Class A
Class B
6 .
2 .
92-CE-MATHS II
1
4
1
0
5
4
2

The mid-points of the sides of a
triangle are (3, 4), (2, 0) and (4, 2).
Which of the following points is a
vertex of the triangle?
A.
B.
C.
D.
E.
30o
36o
40o
42o
45o
2.
3.
6.
5
Number of students
38
42
Mean mark
72
54
A student in Class A has scored 91
marks. It is found that his score was
wrongly recorded as 19 in the
calculation of the mean mark for Class
A in the above table. Find the correct
mean mark of the 80 students in the
two classes.
III.
c.f.
100
O
A.
B.
C.
D.
E.
92
33.
B.
C.
D.
E.
A.
B.
C.
D.
E.
3
25
3
20
4
25
6
25
3
10
92
35.
II.
III.
c.f.
A.
B.
C.
D.
E.
100
O
II.
1
92
36.
100
92-CE-MATHS II
1
a, b, c form an arithmetic
progression.
a, b, c form a geometric
progression.
b
Both roots are .
a
I only
II only
III only
I and II only
II and III only
x
c.f.
O
I, II, III
I, III, II
II, I, III
II, III, I
III, I, II
If the quadratic equation
ax2  2bx + c = 0 has two equal roots,
which of the following is/are true?
I.
I.
x
The figure shows the cumulative
frequency curves of three distributions.
Arrange the three distributions in the
order of their standard deviations, from
the smallest to the largest.
Two cards are drawn randomly from
five cards A, B, C, D and E. Find the
probability that card A is drawn while
card C is not.
A.
92
34.
61.65
62.55
63
63.45
63.9
1
x
6
Which of the following intervals must
contain a root of 2x3  x2  x  3 = 0?
I.
II.
III.
1 < x < 1
0<x<2
1<x<3
A.
B.
C.
I only
II only
III only
D.
E.
92
37.
I and II only
II and III only
C.
D.
E.
How many integers x satisfy the
inequality 6x2  7x  20  0?
A.
B.
C.
D.
E.
92
38.
92
41.
0
1
2
3
4
x2.
x+2.
x1.
x+2.
x.
y

O
92
42.

x
From the figure, if   x  , then
A.
B.
C.
D.
E.
92
43.
ax + (b  m)x + (c  k)  0 .
ax2 + (b  m)x + (c  k) < 0 .
ax2 + (b  m)x + (c  k) = 0 .
ax2 + (b  m)x + (c  k) > 0 .
ax2 + (b  m)x + (c  k)  0 .
2
B.
C.
n is any positive integer
n is any positive odd integer
n is any positive even integer
n is any multiple of 3
n is the square of any positive
integer
D.
E.
92
44.
The L.C.M. of P and Q is 12ab3c2. The
L.C.M. of X, Y and Z is 30a2b3c. What
is the L.C.M. of P, Q, X, Y and Z?
A.
B.
360a3b6c3
60a2b3c2
92-CE-MATHS II
100
=5
x 1
100
100

x 1
x
100
100

x
x 1
100
100

x 1
x
100
100

x
x 1
=5
=5
=5
=5
By selling an article at 10% discount
off the marked price, a shop still makes
20% profit. If the cost price of the
article is $19 800, then the marked
price is
A.
7
22n
22n
22n  3
22n  2
22n  2
If the price of an orange rises by $1,
then 5 fewer oranges could be bought
for $100. Which of the following
equations gives the original price $x of
an orange?
A.
Under which of the following
conditions must the mean of n
consecutive positive integers also be an
integer?
A.
B.
C.
D.
E.
Find the (2n)th term of G.P.
1
 , 1, 2, 4, …
2
A.
B.
C.
D.
E.
y = ax2 + bx + c
92
40.
If a polynomial f(x) is divisible by x 
1, then f(x  1) is divisible by
A.
B.
C.
D.
E.
y = mx + k
92
39.
60ab3c2
6a2b3c
6ab3c
$21 600 .
B.
C.
D.
E.
92
45.
$26 136 .
$26 400 .
$27 225 .
$27 500 .
92
47.
E
A.
B.
135 o
D.
3
E.
In the figure, find tan .
C.
D.
E.
C
92
48.
1
3
1
8
3
8
2
7
1
2

2
.
2 3
 3
sin  .
2 4
 1
sin  .
2 3
2
sin  =
3
3
sin  =
4
sin

O
N
2 cm
B.
B
2
In the figure, if  is the angle between
the diagonals AG and BH of the cuboid,
then

A.
D
4
A
C.
2
F

2:3
5:6
6:5
3:2
55 : 34
92
46.
G
4
Coffee A and coffee B are mixed in the
ratio x : y by weight. A costs $50/kg
and B costs $40/kg. If the cost of A is
increased by 10% which that of B is
decreased by 15%, the cost of the
mixture per kg remains unchanged.
Find x : y.
A.
B.
C.
D.
E.
H
M
C
2 cm
A
2 2 cm
2c
m
B
In the figure, OA is perpendicular to the
plane ABC. OA = AB = AC = 2 cm and
BC = 2 2 cm. If M and N are the midpoint of OB and OC respectively, find
the area of AMN.
A.
B.
92-CE-MATHS II
8
1
cm2
2
1 cm2
E.
92
49.
92
51.
A
E
?
2 cm2
3
cm2
2
3 cm2
C.
D.
40o
B
B
a
30o
A
a
D
In the figure, EB and EC are the angle
bisectors of ABC and ACD
respectively. If A = 40o, find E.
30o
C
A
C
In ABC, A = 30o, c = 6. If it is
possible to draw two distinct triangles
as shown in the figure, find the range of
values of a.
A.
B.
C.
D.
E.
C
6
c=
c=
6
B
A.
B.
C.
D.
E.
0<a<3
0<a<6
3<a<6
a>3
a>6
20o
25o
30o
35o
40o
92
52.
B
O


A
92
50.
D
A
C
B
24o

In the figure, O is the centre of the
circle. If the diameter AOB rotates
about O, which of the following is/are
constant?
E
C
In the figure, the two circles touch each
other at C. The diameter AB of the
bigger circle is tangent to the smaller
circle at D. If DE bisects ADC, find
.
A.
B.
C.
D.
E.
24o
38o
45o
52o
66o
92-CE-MATHS II
D
9
I.
II.
III.
+
A.
B.
C.
D.
E.
I only
II only
III only
I and II only
II and III only
AC + BD
AC  BD
92
53.
B
9
D
F
4
16
8
E
G
2
?
C
A
In the figure, AB = 16, CD = 8, BF = 9,
GD = 4, EG = 2. Find GC.
A.
B.
C.
D.
E.
92
54.
4.5
5
6
8
10
F
a
A
B
a
E
D
C
In the figure, ABCD is a square of side
a and BDEF is a rhombus. CEF is a
straight line. Find the length of the
perpendicular from B to DE.
A.
B.
C.
1
a
2
2a
3
a
2
D.
E.
3
a
2
a
92-CE-MATHS II
10

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