Mathematics 1993 Paper 2

Form 5

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.

93 3.

E.

104y . 102 + 4y . 108y . 40y . 402y .

93 5.

A. B. C. D. E.

n [2a + (n − 1)d], then d = 2 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)

x4 + 1 x4 − x2 + 1 x4 + x2 + 1 x4 − 3x2 − 2 3 x − 1 x4 + 3x3 − 2 3 x2 + 3 x − 1

a + a− b A. B. C.

a . a+ b

1 a− b a + 2 ab − b a−b b+ a 2 a

93-CE-MATHS II

b + 2 ab − a a−b a+b a−b

If 3x2 + ax − 5 ≡ (bx − 1)(2 − x) − 3, then

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 1

A

Simplify (x2 − 3 x + 1)(x2 + 3 x + 1). A. B. C. D. E.

93 4.

D.

O

1

2

3

4

5

x

3 x +2 y =0 Find the greatest value of 3x + 2y if (x, y) is a point lying in the region OABCD (including the boundary). A. B. C. D. E.

15 13 12 9 8

93 7.

A. B. C. D. E.

y 2

y = ax

-1

0

1

2 3

4

+ bx

93 11.

x

A. B. C. D. E. 93 8.

The

−1, 1 −1, 2 0, 1 0, 3 1, 3

93 12.

A. B.

D.

E.

p=q=1. q p= . q −1 q p= . q +1 q +1 p= . q q −1 p= . q

E. 93 13.

A. B. C. D. E.

1 4 12 16 18

If 3, a, b, c, 23 are in A.P., then a+b+c=

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C. D. E. 93 14.

a abc abc ab2c3 a2b3c4

−3 −1 1 − 3 2 3 3

If the simultaneous equations  y = x2 − k have only one solution,   y=x find k. A. B.

The expression x2 −2x + k is divisible by (x + 1). Find the remainder when it is divided by (x + 3).

L.C.M. abc ab2c3 a2b3c4 abc abc 2 3 4

If α and β are the roots of the quadratic equation x2 − 3x − 1 = 0, find the value 1 1 of + . α β

If log(p + q) = log p + log q, then

D.

93 10.

A. B. C. D. E.

A. B. C.

C.

93 9.

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

y = cx + d The diagram shows the graphs of y = ax2 + bx and y = cx + d. solutions of the equation ax2 + bx = cx + d are

13 . 26 . 33 . 39 . 65 .

−1 1 − 4 −4 1 4 1 r h

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.

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.

$25 $28.80 $31.50 $36 $54

A. B. C. D. E.

24 : π 3:1 6:π 3:π 8 : 3π

93 17.

93 15.

h r

2 rad

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

1.5 cm Find the perimeter of the sector in the figure. A. B. C. D. E.

volume of cylinder : volume of hemisphere

2.25 cm 3 cm π   + 3  cm  60  4.5 cm 6 cm

A. B. C. D. E. 93 18.

93 16.

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.

h

r 93 19.

12.5% 15% 22.5% 35% 37.5%

cosθ 1 − cos 2 θ ⋅ = 1 − sin 2 θ sin θ A. B.

93-CE-MATHS II

1:3 2:3 3:4 3:2 3:1

sin θ cos θ

tan θ 1 sin θ 1 cosθ

C. D. E.

93 23.

A

B

93 cos4θ − sin4θ + 2 sin2θ = 20. A. B. C. D. E.

θ C

0 1 (1 − sin2θ)2 (1 − cos2θ)2 (cos2θ − sin2θ)2

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

93 21.

1 4 1 3 1 2 1 3

B B. x

C. 8 D.

C

5

In the figure, cosA = − A. B. C. D. E.

A E. 4 . Find a. 5

153 137 89 41 25

93 The largest value of 3sin2θ + 2cos2θ − 1 22. is A. B. C. D. E.

1. 3 . 2 2. 3. 4.

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P

3 2

93 24.

C 15 2x-

D

o

4x+ 5o 2 x - 10

A

o

B

In the figure, points A, B, C and D are concyclic. Find x. A. B. C. D. E.

20o 22.5o 25o 27.5o 30o

93 25.

A

38

B

θ

o

C

E

o

72 D

In the figure, BA // DE and AC = AD. Find θ. A. B. C. D. E.

34o 54o 70o 72o 76o

93 26.

20

C

o

B

100o 110o 120o 135o 140o

93 If the points (1, 1), (3, 2) and (7, k) are 27. on the same straight line, then k = A. B. C. D. E.

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

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.

In the figure, AB is a diameter. Find ∠ ADC. A. B. C. D. E.

A. B. C. D. E.

93 A circle of radius 1 touches both the 29. positive x-axis and the positive y-axis. Which of the following is/are true?

D

A

I. ∆ABP II. ∆ABQ III. ∆ABR

3. 4. 6. 7. 10 .

93 A(0, 0), B(5, 0) and C(2, 6) are the 28. 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?

93-CE-MATHS II

A. B. C. D. E.

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

93 What is the area of the circle 30. x2 + y2 − 10x + 6y − 2 = 0? A. B. C. D. E.

32π 34π 36π 134π 138π

93 Two fair dice are thrown. What is the 31. probability of getting a total of 5 or 10? A. B. C. D. E.

1 9 5 36 1 6 7 36 2 9

93 A group of n numbers has mean m. If 32. 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.

93 If a : b = 2 : 3 and b : c = 5 : 3, then 35. a + b + c = a−b+c A. B.

1 2 3 6 n−3

93 33.

C. D. E. 93 36.

A

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.

Interquartile range of Interquartile range of B II. Standard deviation of Standard deviation of B III. Mode of A > Mode of B A. B. C. D. E.

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

93 If 9x + 2 = 36, then 3x = 34. A. B. C. D. E.

2 . 3 4 . 3 2. 6 . 9.

93-CE-MATHS II

A

<

A

>

−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.

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 Given that the positive numbers p, q, r, 37. s are in G.P., which of the following must be 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. A. B. C. D. E.

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

93 38.

D

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

C

y

A. A

B

x

B.

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

32 cm 34 cm 7 cm 226 cm 241 cm

C.

E.

93 42.

Q

93 41.

(a2 − b2) is a factor. (a2 + b2) is a factor. (a2 − ab − b2) is a factor. (a2 − ab + b2) is a factor. it cannot be factorized.

A

93-CE-MATHS II

E

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

B. C. C

D. E.

B

R O

A.

a = 5, c = 2 . a = −5, c = 2 . a = 5, c = −2 . a = 1, c = −2 . a = −1, c = 2 . D

P S

93 If the solution of the inequality 40. x2 − ax + 6 ≤ 0 is c ≤ x ≤ 3, then A. B. C. D. E.

3 8 3 4 3 2

D.

93 In factorizing the expression 39. a4 + a2b2 + b4, we find that A. B. C. D. E.

1 4 1 3

4 . 25 2 . 5 9 . 25 16 . 25 21 . 25

93 Which of the following gives the 43. compound interest on $ 10 000 at 6% p.a. for one year, compounded monthly?

A.

The figure shows the graph of the function

0.06 × 12 12 $ 10 000(1.0612 − 1) $ 10 000 ×

B. C. D. E.

 0.6 12   − 1 $10 000 1 + 12   

93 47.

C

93 2 44. Originally 3 of the students in a class failed in an examination. After taking a re-examination, 40% of the failed students passed. Find the total pass percentage of the class. A.

A

2 % 3 1 33 % 3 40% 60% 1 73 % 3

C. D. E.

A. B. C. D.

93 Solve tan4θ + 2tan2θ − 3 = 0 for 45. 0o ≤ θ < 360o.

93 46.

E.

45o, 135o only 45o, 225o only 45o, 60o, 225o, 240o 45o, 120o, 225o, 300o 45o, 135o, 225o, 315o

A. B. C. D. E.

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.

26

B.

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.

12

 0.06  $ 10 000 1 +  12    0.06 12   − 1 $10 000 1 + 12    

12 3(1 + tan30o) 6(1 + tan30o) 1   3 1 + o   tan 30  1   6 1 + o   tan 30 

93 48.

E H

D

3

F

y

C

1

G

A

12

5 0

80

o

-1

93-CE-MATHS II

170

o

260

o

350

o

x

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

A. B. C. D.

A.

3 5 3 12 3 13 3 178

E.

B. C. D. E.

θ . 2 θ − 90o . 180o−θ . 180o− 2θ . 2θ − 180o .

93 51.

153 5

H O

93 49.

M

C

K

b

A

a

A

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

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

A. B. C. D. E.

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

93 50.

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

c

N

93 52.

I only II only III only I and II only I, II and III only H

G C

D P

T

M

A

θ E

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 = θ, then ∠PTQ =

93-CE-MATHS II

B F

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

1:2 1:3

C. D. E.

1: 1:

2 3

2: 3

93 53.

D

F

A

B

E

D

C 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 54.

X

Y

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. B. C. D. E.

8 cm 10 cm 12 cm 14 cm 16 cm

93-CE-MATHS II