Mathematics 1988 Paper 2

Form 5

HKCEE 1988 Mathematics II 88 1.

2 n + 4 − 2(2 n ) Simplify 2( 2n + 3 ) A.

C. D. E.

B. C. D. E.

88 3.

C. D.

A. B. C. D. E.

2

1 x−5 x−2 ( x + 2)( x − 5) 1 x+5 1 x x−3 ( x + 3)( x − 5)

88-CE-MATHS II

88 6.

1 + α

−2 1 − 2 1 − 4 1 2 2

Let f(x) = ax2 + bx + c. When f(x) is divided by (x − 1), the remainder is 10. When f(x) is divided by (x + 1), the remainder is 8. Find the value of b. A. B. C. D. E.

x − 2 x x − 2 x − 15 = × 2 x 3 − 25 x x + x−6 2

E. 88 5.

x −1 x 1+ x 1− x x +1 x −1 x −1 x +1 1− x 1+ x

x2 − 8x − 4 = 0, then the value of

A. B.

1+ y If x = , then y = 1− y A.

If α and β are the two roots of

1 is β

7 8 7 4 1 − 2n+1 1 2n+4 − 8 2n+1

B.

88 2.

88 4.

−4 −2 2 4 It cannot be found

1 1 = 2 + 2 2x − x x + x−6 A. B. C. D. E.

3 x(2 − x )( x + 3) −3 x( x + 2)( x − 3) 6− x x(2 − x )( x + 2)( x − 3) x−6 x(2 − x )( x + 2)( x − 3) 2x + 3 x(2 − x )( x + 3)

88 7.

Which of the following is an identity/are identities? 1 1− x −1= x x II. (ax + b)(x − b) = ax2 − b2 III. 2x2 − 3x + 1 = 0

C. D.

I.

A. B. C. D. E. 88 8.

88 9.

x2 − (a2 − b)x + 2a = 0 x2 + (a2 − b)x + 2a = 0 x2 + 2ax − a2 + b = 0 x2 + 2ax + a2 − b = 0 x2 − 2ax + a2 − b = 0

Which of the following is a G.P./are G.P.’s? I. II.

5, 0.5, 0.05, 0.005, 0.0005 log 5, log 50, log 500, log 5000, log 50000 III. 5, 5sin70o, 5(sin70o)2, 5(sin70o)3, 5(sin70o)4 A. B. C. D. E.

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

88 A solid iron sphere of radius r is melted 10. and recast into a circular cone and a circular cylinder. If both of them have the same height h and the same base radius r, find h in terms of r. A. B.

88 11.

P

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

If the roots of a quadratic equation are a + b and a − b , then the equation is A. B. C. D. E.

E.

1 r 2 9 r 16

88-CE-MATHS II

2 r 3 3 r 4 r 12

T

12

Q

d

S

V

R

In the figure, PQRS is a rectangle with PQ = 24 and PS = d. T is the mid-point of PQ. V is a point on SR and area of PTVS = 2. SV = area of TQRV A. B. C. D. E.

14 . 16 . 18 . 20 . 22 .

88 Find the difference between simple 12. interest and compound interest (compounded annually) on a loan of $1000 for 4 years at 6% per annum. (The answer should be correct to the nearest dollar.) A. B. C. D. E.

$22 $196 $540 $760 $1022

88 Last year, the cost of a school magazine 13. consisted of: cost of paper ………$8 cost of printing ……… $5 cost of binding ……… $3 This year, the cost of paper will increase by 25% and the cost of printing will increase by 40% while the cost of binding will remain unchanged. The cost of a school magazine will increase by A. B. C. D. E.

III. ∠BEH A. B. C. D. E.

88 5 2 sin A − 3 cos A 16. If tan A = − 4 , then 3 sin A + 2 cos A = A.

20% 25% 27.5% 32.5% 65%

B. C.

88 Given that sinθ cosθ > 0, which of the 14. following is/are true? I. 0o < θ < 90o II. 90o < θ < 180o III. 180o < θ < 270o A. B. C. D. E.

θ θ

D

F

G H

In the figure, ABCDEFGH is a cube. Which of the following is a right angle/are right angles? I. II.

∠DHG ∠AHG

88-CE-MATHS II

D

C

A

22 7 22 − 23 2 − 23 2 23 22 7 −

88 17.

B

E

D. E.

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

88 15.

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

A

B

In the figure,

AC = AB

A. B. C. D. E.

2 tan θ tan 2θ tan θ sin 2θ sin θ cos 2θ cosθ

C

88 18.

88 20.

A'

B

M

B' 2r

θ

A

C

θ

θ O r

4r

C'

C. D. E.

88 19.

o

θ

Q

In the figure, M is the mid-point of PQ and ∠PSQ = 30o. Find tan θ. A. B. C.

7π r2 7 2 πr 2 7 2 πr 4 7 2 πr 6 7 π r2 12

30

S

In the figure, AOC’ is a straight line. OAA’, OBB’ and OCC’ are 3 sectors. If OA = 4r, OB = 2r and OC’ = r, find the total area of the sectors in terms of r. A. B.

P

D. E.

0.268 3 6 3 2 3 4 3 8

88 21.

A

C 12

B

30

o

In the figure, the area of ∆ABC is 15 cm2 and ∠A = 30o. AC is longer than AB by 4 cm. AC = A. B. C. D. E.

6 cm 8.8 cm 10 cm 11.5 cm 14 cm

88-CE-MATHS II

C

A

O

5

B

In the figure, O is the centre of the circle of radius 5. AB is a tangent and AO = 12. AC = A. B. C. D. E.

13 17 219 244 269

88 22.

88 24.

C

R

θ A

Q 70

B

O

o

P

50

o

T In the figure, TP and TQ are tangents to the circle PQR. If ∠RPQ = 70o and ∠ PTQ = 50o, then ∠RQP =

In the figure, O is the centre of the circle of diameter 13. AC = 12. sin θ A. B. C. D. E.

88 25.

J

F A

20

44

o

a

B. C. D. E.

5a 11 a sin 50o a sin 70o sin50o a sin 50o sin70o a sin 50o sin20o

88-CE-MATHS II

o

C

G E

A

In the figure, BC = a. AB = A.

I

H

88 23.

B

20o 45o 50o 60o 70o

A. B. C. D. E.

5 12 3 13 313 13 12 13 13 12

2 B

6

C

D

In the figure, ABEF, BCGH and CDIJ are three squares. If AB = 2 and BC = 6 and F, H, J lie on a straight line, then CD = A. B. C. D. E.

8 10 12 16 18

88 The line y = mx + c is perpendicular to 26. the line y = 3 − 2x. Find m. A. B. C. D.

2 1 2 −2 1 2 −

E.

−

1 3

88 Which of the following circles has the 27. lines x = 1, x = 5, y =4 and y =8 as its tangents? A. B. C. D. E.

(x − 1)2 + (y − 4)2 = 4 (x − 5)2 + (y − 8)2 = 4 (x − 3)2 + (y − 6)2 = 4 (x − 1)2 + (y − 8)2 = 4 (x − 5)2 + (y − 4)2 = 4

88 28.

y

O

79.3 cm 79.7 cm 80 cm 80.3 cm 80.7 cm

88 31.

A B

C ( c , 1) x

In the figure, A(5, 3), B(b, 1) and C(c, 1) are the vertices of a triangle. If AB = AC, then b + c = 3 5 6 8 10

88 The maximum load a lift can carry is 29. 600 kg. 11 men with a mean weight of 49 kg are already in the lift. If one more man is to enter the lift, his weight must not exceed A. B. C. D. E.

A. B. C. D. E.

A (5, 3)

B ( b , 1)

A. B. C. D. E.

88 The mean length of 30 rods is 80 cm. If 30. one of these rods of length 68 cm is taken out and replaced by another rod of length 89 cm, then the new mean length is

49 kg 50 kg 51 kg 59 kg 61 kg

88-CE-MATHS II

The figure shows 3 paths joining A and B. A man walks from A to B and another man walks from B to A at the same time. If they choose their paths at random, what is the probability that they will meet? A. B. C. D. E.

1−

1 9

1 3 1 3 1 1 × 2 3 1 1 × 3 3

1−

88 32.

E.

100

88 If log a > 0 and log b < 0, which of the 35. following is/are true?

80

Cumulative frequency

60

I.

a >0 b II. log b2 > 0 III. 1 log > 0 a

40 20 0 130

140 150 Height (cm) (less than)

160

170

180

The figure shows the cumulative frequency polygon of the heights of 100 persons. If one person is selected at random from the group, find the probability that his height is less than 170 cm but not less than 150 cm. A. B. C. D. E.

1 5 2 5 3 10 1 2 7 10

88 Which of the following expressions 33. CANNOT be factorized? A. B. C. D. E.

x3 − 125 4x2 − 9y2 x3 + 125 4x2 + 9y2 3x2 + 6xy + 3y2

88 If f(x) = 3 + 2x, then f(2x) − f(x) = 34. A. B. C. D.

2x(2x − 1)

2x 23x 3 + 2x 2x(2x + 1)

88-CE-MATHS II

A. B. C. D. E.

log

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

88 36.

y x - y +2=0 II I

IV

x - 3 y =0

III x

O x + y -4=0

In the figure, which region represents the solution to the following inequalities?  x − 3y ≤ 0  x − y + 2 ≥ 0 x + y − 4 ≥ 0  A. B. C. D. E.

I II III IV V

88 37.

y y = ax + b

c O

x

d

y = kx

2

In the figure, the line y = ax + b cuts the curve y = kx2 at x = c and x = d. Find the range of values of x for which kx2 < ax + b. A. B. C. D. E.

c<xd

88 p, q, r, s are in A.P. If p + q = 8 and r + 38. s = 20, then the common difference is A. B. C. D. E.

3 4 6 7 12

88 y varies inversely as x2. If x is increased 39. by 100%, then y is A. B. C. D. E.

increased by 100% increased by 300% decreased by 25% decreased by 75% decreased by 100%

88 8abc3 is the H.C.F. of 24ab2c3 and 40. A. B. C. D. E.

12a2bc4 30a2bc3 32a2bc5 40ab2c3 48a3bc5

88-CE-MATHS II

88 X sells an article to Y at a profit. Y ten 41. sells it to Z for $60 at a profit of 20%. If X had sold the article directly to Z for $60, how much MORE profit would he have made? A. B. C. D. E.

$10 $12 $48 $50 It cannot be found

88 A car travels from P to Q. If its speed 42. is increased by k%, then the time it takes to travel the same distance is reduced by A. B. C. D. E.

k% 100 % k 100k % 100 + k k % 100 + k k % 100 − k

88 A bag contains n balls of which 60% 43. are red and 40% are white. After 10 red balls are taken out from the bag, the percentage of red balls becomes 50%. Find n. A. B. C. D. E.

20 40 50 60 100

88 The weight of a gold coin of a given 44. thickness varies as the square of its diameter. If the weights of two such coins are in the ratio 1 : 4, then their diameter are in ratio A. B. C. D. E.

1:2 2:1 1:4 4:1 1 : 16

88 45.

r

C. D. E.

2r r

C. D. E.

y

1 -

0% 2 16 % 3 20% 25% 1 33 % 3

88 46.

A. B. C. D. E.

B Q P

2:1 2:1 2 2:1 π:1 4:1

88 If x and y can take any value between 0 47. and 360, what is the greatest value of 2 sin xo − cos yo? A. B.

1 2

88-CE-MATHS II

88 49.

-

π O 4

π 4

x

π 2

y = −tan x y = 1 − tan x y = 1 + tan x y = cos x − sin x y = cos x + sin x A

B

O

C

The figure shows the circumscribed circle P and the inscribed circle Q of the square ABCD. Find area of P : area of Q A. B. C. D. E.

π 2

The figure shows the graph of the function

A

D

5 It cannot be found

88 48.

A cylindrical hole of radius r is drilled through a solid cylinder, base radius 2r and height r, as shown in the figure. The percentage increase in the total surface area is A. B.

3

D

C

ABCD is a square of side 2 cm. O is the mid-point of AD. A sector with centre O is inscribed in the square as shown in the figure. What is the area of the sector? A. B. C. D.

π cm2 2 2 3 π cm2 3 π cm2 2 π cm2 3

E.

4 π cm2 3

88 52.

88 50.

O

A

12

A B

C

D

O

B. C. D. E.

3π rad 2 (π + 1) rad 4 π rad 3 (2π − 1) rad 7 π rad 4

88 51.

A

B. C. D. E.

1 2 2 4 6 8 A

3

B

D B

C

In the figure, ABCD is a parallelogram. AB ⊥ BD, AB = 3 and BC = 5. AC =

ABCD is a cyclic quadrilateral with AB = AD and CB = CD. Find ∠ABC.

88-CE-MATHS II

2

5

C

75o 90o 105o 120o It cannot be found

A.

88 53.

D

A. B. C. D. E.

B

In the figure, O1 and O2 are the centres of the two circles, each of radius r and AB = 12 find r.

In the figure, ABCD is a G-shaped curve, where ABC is an arc of a circle and DC is a radius. If the length of the curve ABCD is the same as that of the complete circle, find, in radians, the angle subtended by the arc ABC at the centre. A.

1

A. B. C. D. E.

10 . 12 . 13 . 26 . 2 13 .

88 54.

A

D

B

θ

C

In the figure if AB = AC and AD = BD = BC, then ∠ACB = A. B. C. D. E.

30o 32o 36o 40o 72o

88-CE-MATHS II