Mathematics 1993 Paper 2

Mathematics HK CE Paper

Form 5

HKCEE 1993 Mathematics II

93 If f(x) = 102x, then f(4y) = 1. A. 104y . B. 102 + 4y . C. 108y . D. 40y . E. 402y .

93 2. If s = n2

[2a + (n 1)d], then d =

A. 2(s an)n(n 1) . B. 2(s an)(n 1) .

C. sn(n 1) . D. as na(n 1) . E.

4(s an)

n(n 1)

.

93 Simplify (x2 3x + 1)(x2 +3x + 1). 3. A. x4 + 1 B. x4 x2 + 1 C. x4 + x2 + 1

D. x4 3x2 23x 1

E. x4 + 3x3 23x2 +3x 1

93 a4. a

b

+

aa

b

.

A. 1a

b

B.

a 2ab b

a b

93-CE-MATHS II C. b a

2a

D. b 2ab aa b

E.

a ba b

93 If 3x2 + ax 5 (bx 1)(2 x) 3, 5. then

A. a = 5, b = 3 . B. a = 5, b = 3 . C. a = 3, b = 5 . D. a = 5, b = 3 . E. a = 3, b = 5 .

93 6.

x

Find the greatest value of 3x + 2y if

(x, y) is a point lying in the region OABCD (including the boundary).

A. 15 B. 13 C. 12 D. 9 E. 8

1

Mathematics HK CE Paper

7.

2 + bx

10. a + b + c = x

93

The diagram shows the graphs of

11. y = ax2 + bx and y = cx + d. The solutions of the equation ax2 + bx = cx + d are

A. 1, 1 B. 1, 2 C. 0, 1 D. 0, 3 E. 1, 3 93

12. 93 If log(p + q) = log p + log q, then 8. A. p = q = 1 . B. p = qq 1 . C. p = qq 1 . D. p = q 1 q . E.

p =

q 193 q

.

13.

93 The expression x2 2x + k is pisible by 9. (x + 1). Find the remainder when it is pided by (x + 3).

A. 1 B. 4 C. 12 D. 16 E. 18

93-CE-MATHS II 2

A. 13 . B. 26 . C. 33 . D. 39 . E. 65 .

Find the H.C.F. and L.C.M. of ab2c and abc3 H.C.F. L.C.M.

A. a a2b3c4

B. abc ab2c3 C. abc a2b3c4 D. ab2c3 abc E. a2b3c4 abc

If and are the roots of the quadratic equation x2 3x 1 = 0, find the value of

1

+

1

.

A. 3 B. 1

C. 13

D.

23

E. 3

If the simultaneous equations

y x2 k

have only

y xone solution, find k. A. 1

B. 14

C. 4

D. 14

E. 1

Mathematics HK CE Paper

14.

16.

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. $25

B. $28.80

C. $31.50 D. $36 E. $54

93

15.

2 rad

93 17.

1.5 cm

Find the perimeter of the sector in the figure.

A. 2.25 cm

B. 3 cm

C.

60

3

cm

D. 4.5 cm E. 6 cm

93-CE-MATHS II 3

h

r

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.

A. 24 : B. 3 : 1 C. 6 : D. 3 : E. 8 : 3

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. 1 : 3 B. 2 : 3 C. 3 : 4 D. 3 : 2 E. 3 : 1

Mathematics HK CE Paper

93 A merchant marks his goods 25% above 18. the cost. He allows 10 % discount on the

marked price for a cash sale. Find the percentage profit the merchant makes for B. C. D.

89

41 a cash sale. A. 12.5% B. 15% C. 22.5% D. 35% E. 37.5%

93 cos 1 cos2

19. 1 sin2 sin A. sin B. cos C. tan

D. 1sin

E.

1cos

93 cos4 sin4 + 2 sin2 = 20. A. 0 B. 1

C. (1 sin2 )2 D. (1 cos2 )2 E. (cos2 sin2 )2

93 B

21.

x

8

C

5

A

In the figure, cosA = 45

. Find a.

A.

93-CE-MATHS II E.

25

93 The largest value of 3sin2 + 2cos2 1 is 22. A. 1 .

B. 32

.

C. 2 . D. 3 . E. 4 .

93 A

23.

B

C

P

In the figure, AB = BC, BP = CP and BP CP. Find tan .

A. 1

4 B. 13

C. 1

2 D. 1

3

E. 32

4

Mathematics HK CE Paper

A. 100o B. 110o o- 1 xC. 120 2

D

D. 135o

4xoE. 140o

o

2x - 10If the points (1, 1), (3, 2) and (7, k) are AB

on the same straight line, then k =

93

24.

C

93 27. In the figure, points A, B, C and D are

concyclic. Find x.

A. 20o

B. 22.5o

C. 25o

D. 27.5o

E. 30o

93

28.

93 A25.

o72o B38CD

In the figure, BA // DE and AC = AD. Find . A. 34o

B. 54o

C. 70o D. 72o

93 E. 76o

29. 93 DC 26. 20o

AB

In the figure, AB is a diameter. Find ADC.

93-CE-MATHS II 5

A. 3 . B. 4 . C. 6 . D. 7 . E. 10 . 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. ABP

II. ABQ III. ABR A. I only

B. II only C. III only

D. I and II only

E. 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. Its centre is in the first quadrant.

II. Its centre lies on the line x y = 0. III. Its centre lies on the line x + y = 1.

A. I only B. II only C. III only

D. I and II only E. I and III only

Mathematics HK CE Paper

30. 93 31.

x2 + y2 10x + 6y 2 = 0?

A. 32 B. 34 C. 36 D. 134 E. 138

Two fair dice are thrown. What is the probability of getting a total of 5 or 10?

A. 1

B. C. D. E.

95

I.

36167

93 34. A. 93 35.

93 36.

B. C. D.

Interquartile range of A < Interquartile range of B

II. Standard deviation of A > Standard

deviation of B

III. Mode of A > Mode of B

A. I only B. II only C. III only D. I and II only E. II and III only

If 9x + 2 = 36, then 3x =

2343

. .

6

3629

2 .

.

93 A group of n numbers has mean m. If the 32. numbers 1, 2 and 6 are removed from the

group, the mean of the remaining n 3 numbers remains unchanged. Find m. A. 1 B. 2 C. 3 D. 6 E. n 3 93

A33.

E. 9 .

If a : b = 2 : 3 and b : c = 5 : 3, then

a b ca b c

=

A. B. C. D. E. 2 .

52

.

4 .

172

.

31 .

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?

From the table, a root of the equation f(x) = 0 is

93-CE-MATHS II 6

Mathematics HK CE Paper

93 A. 3.57 (correct to 3 sig. fig.). B. 3.575 (correct to 4 sig. fig.). C. 3.5775 (correct to 5 sig. fig.). D. 3.5725 (correct to 4 sig. fig.). E. 3.58 (correct to 3 sig. fig.).

Given that the positive numbers p, q, r, s 93 40. B. (2 + b2) is a factor.

C. (a2 ab b2) is a factor. D. (a2 ab + b2) is a factor. E. it cannot be factorized.

If the solution of the inequality x2 ax + 6 0 is c x 3, then 37. true?

I. kp, kq, kr, ks are in G.P., where k

is a non-zero constant.

II. ap, aq, ar, as are in G.P., where a is

a positive constant.

III. log p, log q, log r, log s are in A.P. 93

41.

A. I only B. II only

C. I and II only D. I and III only E. I, II and III only

93 D

C

38.

y

A

x

B

In the figure, the rectangle has perimeter 16 cm and area 15 cm2. Find the length of its diagonal AC. A. 32 cm B. 34 cm C. 7 cm D. 226 cm

E. 241 cm

93 In factorizing the expression 39. a4 + a2b2 + b4, we find that

93-CE-MATHS II 7

A. a = 5, c = 2 . B. a = 5, c = 2 . C. a = 5, c = 2 . D. a = 1, c = 2 . E. a = 1, c = 2 .

D

C

AB

In the figure, ABCD is a square and ABE is an equilateral triangle.

Area of ABEArea of ABCD

=

A. 14 B. 13

C. 3

8 D. 34 E. 32

Mathematics HK CE Paper

42.

Originally of the students in a class 44. 3

Q

O

In the figure, the radii of the sectors OPQ and ORS are 5 cm and 3 cm respectively.

Area of shaded regionArea of sector OPQ

=

A. 4 25 .

B. 2 5 . 93 C. 9

45. 25 . D. 16 25 .

E.

21 25

.

93 Which of the following gives the 43. compound interest on $ 10 000 at 6%

93 p.a. for one year, compounded monthly? 46.

A.

$ 10 000 0.0612

12

B. $ 10 000(1.0612

1)

C. 0.12

$ 10 000

1 06 12

D. $10 000 12

1 0.06

1

12 E. $10 000 1 0.6

12

12 1

93-CE-MATHS II 8

failed in an examination. After taking a re-examination, 40% of the failed students passed. Find the total pass percentage of the class.

A.

2623%

B. 331

3

%

C. 40% D. 60% E.

731

3%

Solve tan4 + 2tan2 3 = 0 for 0o < 360o.

A. 45o, 135o only B. 45o, 225o only

C. 45o, 60o, 225o, 240o D. 45o, 120o, 225o, 300o E. 45o, 135o, 225o, 315o

x

The figure shows the graph of the function

A. y = sin(350o x) . B. y = sin(x + 10o) . C. y = cos(x + 10o) . D. y = sin(x 10o) . E. y = cos(x 10o) .

Mathematics HK CE Paper

47.

C

E.

5

C

93 49.

A

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. 12

B. 3(1 + tan30o) C. 6(1 + tan30o) D. 3 1 1

tan30 o

E.

6 1

1

tan30o

93 E48.

D

H F

3 C

93 50.

A

G

12

5

B

In the figure, ABCDEFGH is a cuboid. The diagonal AH makes an angle with the base ABCD. Find tan .

A. 3 5

B.

312

C. 3 13

93-CE-MATHS II 9

b

a

A

c

B

In the figure, if arc BC : arc CA : arc AB = 1 : 2 : 3, which of the following is/are true?

I. A : B : C = 1: 2 : 3 II. a : b : c = 1: 2 : 3

III. sinA : sinB : sinC = 1 : 2 : 3

A. I only B. II only C. III only D. I and II only E. I, II and III only

P

T

M

B

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.

2

.

B. 90o . C. 180o . D. 180o 2 . E.

2 180o .

Mathematics HK CE Paper

93 H

93 53.

51.

O

M

K

A

N

B

In the figure, O is the centre of the circle. AB touches the circle at N. Which of the following is/are correct?

I. M, N, K, O are concyclic. II. HNB ~ NKB III. OAN = NOB

A. I only 93 B. II only 54.

C. III only D. I and II only E. I, II and III only

93 H

G

52.

D

C

A

B

E

F

In the figure ABCD and EFGH are two squares and ACH is an equilateral triangle. Find AB : EF.

A. 1 : 2 B. 1 : 3 C. 1 : 2 D. 1 : 3 E. 2:3

93-CE-MATHS II 10

A

F

D

F

B

E

C

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. 3.5 cm B. 4.5 cm C. 5 cm D. 8 cm E. 12.5 cm

XY

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.

A. 8 cm B. 10 cm C. 12 cm D. 14 cm E. 16 cm

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