Mathematics 1992 Paper 2

Form 5

HKCEE 1992 Mathematics II 92 1.

A. B. C. D. E.

92 2.

B. C. D. E.

1 . 1− a 1 1− . 1+ a 1 1+ . 1− a 1 1+ . 1+ a 1 −1 + . 1+ a 1−

−1 only 2 only Any value Any value except −1 Any value except 2

5 +1 5 −1 − = 5 −1 5 +1 A.

C. D. E.

92 5.

0

92-CE-MATHS II

E.

92 6.

5 1 + 5 2 1 log10a, then b = 2

10 a . 10 + a . 5a . a . 2 a 1+ . 2

Which of the following is a factor of 4(a + b)2 − 9(a − b)2 ? A. B. C. D. E.

92 7.

1 2 3

If log10 b = 1 + A. B. C. D.

1 , then b = 1− b

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.

B.

1 1 + = a b

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? If

A. B. C. D. E.

a+c =k b+d ab = cd = k ac = bd = k a=c=k ac =k bd

C. D. E.

nn − 2 n 2

n n−2 n 2 1

92 13. 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. C. D. E.

I. a+b = a + b II. (a−1 + b−1)−1 = a + b III. a2b3 = (ab)6 I only II only III only I and II only None of them

1m

1.2 m

If a and b are greater than 1, which of the following statements is/are true?

A. B. C. D. E.

$16 $400 $416 $800 $816

0.75 m3 0.84 m3 0.858 m3 1.008 m3 1.608 m3

92 14. P

92 If a : b = 2 : 3, a : c = 3 : 4 and b : d = 10. 5 : 2, find c : d. A. B. C. D. E. 92 11.

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.

92 12.

1:5 16 : 45 10 : 3 20 : 9 5:1

15.2% 20% 50% 72.8% 80%

A sum of $ 10 000 is deposited at 4% p.a., compounded yearly. Find the interest earned in the second year.

92-CE-MATHS II

0.2 m

A. B.

92 9.

A. B. C. D. E.

n times   n× n ×  × n Simplify . n+   n +   + n n terms

0.3 m

92 8.

3 cm

O Q

60

o

T In the figure, TP and TQ are tangent to the circle of radius 3 cm. Find the length of the minor arc PQ. A. B. C. D. E.

3π cm 2π cm 3π cm 2 π cm π cm 2

92 15.

A

E

92 17.

h

F

B

3h 2h D H

C

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.

G

Find the ratio of the volume of the tetrahedron ACHD to the volume of the cube ABCDEFGH in the figure. A. B. C. D. E.

1:8 1:6 1:4 1:3 1:2

A. B. C. D. E.

92 16.

A

92 18.

B

E 4 cm

4 cm

C

4 cm

E D

The greatest value of 1 − 2sin θ is A. B. C. D. E.

O

1 : 27 1 : 26 1:9 1:8 1:7

5. 3. 1. 0. −1 .

92 19.

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. A. B. C. D. E.

8 3 cm2 8 2 cm2 4 3 cm2 4 2 cm2 16 cm2

θ 4 In the figure, find cos θ . A. B. C. D.

92-CE-MATHS II

2

3

1 4 11 16 3 4 7 8 −

E.

77 9

A. B. C. D. E.

92 In which two quadrants will the 20. solution(s) of sin θ cos θ < 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.

θ 36 o

92 22.

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 θ.

92 25.

π π 2

π

42o 36o 24o 21o 18o

A. B. C. D. E.

y

O

3π 2

2π

6

A

x

The figure shows the graph of the function tan(x + π) . tan(x − π) . π tan x . π + tan x . π − tan x .

92 Which of the following equations 23. has/have solutions? I. 2 cos2θ − sin2θ = 1 II. 2 cos2θ − sin2θ = 2 III. 2 cos2θ − sin2θ = 1

92-CE-MATHS II

B 5

6

E 5

D

A. B. C. D. E.

O

42

92 If A + B + C = 180o, then 21. 1 + cos A cos (B + C) = A. B. C. D. E.

o

C

In the figure, ABCD is a square with side 6. If BE = CE = 5, find AE. A. B. C. D. E.

61 9 10 6 3 109

92 26.

D. E.

R

θ

92 If 0 < k < h, which of the following 29. circles intersect(s) the y-axis? Q

I. (x − h)2 + (y − k)2 = k2 II. (x − h)2 + (y − k)2 = h2 III. (x − h)2 + (y − k)2 = h2 + k2

P

A. B. C. D. E.

In the figure, the circle is inscribed in a regular pentagon. P, Q and R are points of contact. Find θ. 30o 32o 35o 36o 45o

A. B. C. D. E.

A.

S

C

P

B. C. D.

3x B A

E. 2x D

T

30o 36o 40o 42o 45o

92 If the two lines 2x − y + 1 = 0 and 28. ax + 3y − 1 = 0 do not intersect, then a = −6 . −2 . 2.

92-CE-MATHS II

1 4 −1 0 5 4 2 −

92 The mid-points of the sides of a triangle 31. are (3, 4), (2, 0) and (4, 2). Which of the following points is a vertex of the triangle?

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.

I only II only III only I and II only II and III only

92 If the line y = mx + 3 divides the circle 30. x2 + y2 − 4x − 2y − 5 = 0 into two equal parts, find m.

92 27.

A. B. C. D. E.

3. 6.

A. B. C. D. E. 92 32.

(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

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. A. B. C. D. E.

B. C. D. E.

92 I. 34.

c.f. 100

O

61.65 62.55 63 63.45 63.9

92 Two cards are drawn randomly from 33. five cards A, B, C, D and E. Find the probability that card A is drawn while card C is not. A.

III.

3 25 3 20 4 25 6 25 3 10

1

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. A. B. C. D. E.

I, II, III I, III, II II, I, III II, III, I III, I, II

92 If the quadratic equation 35. ax2 − 2bx + c = 0 has two equal roots, which of the following is/are true? I.

a, b, c form an progression. II. a, b, c form a progression. III. b Both roots are . a

c.f.

A. B. C. D. E.

100

O II.

1

x

c.f. 100

92-CE-MATHS II

1

x

geometric

I only II only III only I and II only II and III only

92 Which of the following intervals must 36. contain a root of 2x3 − x2 − x − 3 = 0? I. −1 < x < 1 II. 0 < x < 2 III. 1 < x < 3

O

arithmetic

A. B. C. D. E.

I only II only III only I and II only II and III only

92 How many integers x satisfy the 37. inequality 6x2 − 7x − 20 ≤ 0? A. B. C. D. E. 92 38.

0 1 2 3 4

A. B. C. D. E.

y y = mx

O

α

β y = ax

+ k

x 2

+ bx + c

From the figure, if α ≤ x ≤ β, then A. B. C. D. E.

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 . ax2 + (b − m)x + (c − k) ≥ 0 .

92 Under which of the following 39. conditions must the mean of n consecutive positive integers also be an integer? A. B. C. D. E.

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

92 The L.C.M. of P and Q is 12ab3c2. The 40. 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. C. D. E.

92 If a polynomial f(x) is divisible by x − 1, 41. then f(x − 1) is divisible by

360a3b6c3 60a2b3c2 60ab3c2 6a2b3c 6ab3c

92-CE-MATHS II

x−2. x+2. x−1. x+2. x.

92 Find the (2n)th term of G.P. 42. 1 − , 1, −2, 4, … 2 A. B. C. D. E.

22n −22n −22n − 3 22n − 2 −22n − 2

92 If the price of an orange rises by $1, 43. then 5 fewer oranges could be bought for $100. Which of the following equations gives the original price $x of an orange? A. B. C. D. E.

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

92 By selling an article at 10% discount off 44. 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. B. C. D. E.

$21 600 . $26 136 . $26 400 . $27 225 . $27 500 .

92 Coffee A and coffee B are mixed in the 45. 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.

2:3 5:6 6:5 3:2 55 : 34

92 46.

A. B. C. D. E.

θ 2 = . 2 3 θ 3 sin = . 2 4 θ 1 sin = . 2 3 2 sin θ = 3 3 sin θ = 4 sin

92 48. o

135

2

O

2 cm

N

θ

3

M 2 cm

A In the figure, find tan θ.

C. D. E.

92 47.

1 3 1 8 3 8 2 7 1 2

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.

H

B. C. D.

G

E

1 cm2 2 1 cm2 2 cm2 3 cm2 2 3 cm2

F E. 4 92 49.

θ B

C

In the figure, if θ is the angle between the diagonals AG and BH of the cuboid, then

92-CE-MATHS II

B

6

4

2

B

a

c=

D A

2 cm

B

6

B.

2

2 cm

30 A

c=

A.

C

o

30 C

A

a

o

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.

03 a>6

30o 35o 40o

C. D. E. 92 52.

B

O

φ

θ

A

92 50. C D

A

24

In the figure, O is the centre of the circle. If the diameter AOB rotates about O, which of the following is/are constant?

B

o

θ

E

C

I. θ+φ II. AC + BD III. AC × BD

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 θ. 24o 38o 45o 52o 66o

A. B. C. D. E.

D

A. B. C. D. E.

I only II only III only I and II only II and III only

92 53.

B 9

D

F

92 51.

A

16

E

8

?

40

o

4

E

2

G

?

C

A B

C

D

In the figure, EB and EC are the angle bisectors of ∠ABC and ∠ACD respectively. If ∠A = 40o, find ∠E. A. B.

20o 25o

92-CE-MATHS II

In the figure, AB = 16, CD = 8, BF = 9, GD = 4, EG = 2. Find GC. A. B. C. D. E.

4.5 5 6 8 10

92 54.

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