Mathematics 1990 Paper 2 Answer

Form 5

HKCEE 1990 Mathematics II 90 1.

(a2n)3 = A. B. C. D. E.

90 2.

B. C. D. E. 90 3.

D. E. 90 5.

y−x x+ y x− y x+ y x y x+y x−y

B. C. D. E.

ax −1 a+x ax −1 a−x 1 − ax a+x 1 − ax a+x ax + 1 a−x

1 If f(n) = n(n − 1), then 2 f(n + 1) − f(n) =

90-CE-MATHS II

f(1) f(n) n 2 1 n

If 2 = 10p, 3 = 10q, express log

1 in 6

terms of p and q.

y y = y y

A. B. C. D. E. 90 6.

B. C. D. E.

90 7.

−p − q 1 pq 1 p+q pq p+q

Let a > b > 0. If a and b are respectively the 1st and 2nd terms of a geometric progression, the sum to infinity of the progression is A.

ab + 1 If x = , then b = a−b A.

90 4.

a6n a8n 3 a 2n 3 a 6n 3 a 8n

x− 1− x+ x+ 1− x− A.

A. B. C.

1 a−b a 1− b ab b−a a2 a+b a2 a−b

a3 + 8a−3 = A. B.

2 2 4 )(a + 2 + 2 ) a a 1 1 (a − )(a2 + 1 + ) 2a 4a 2 (a −

C. D. E.

90 8.

B. C. D. E.

In the figure, the circular cylinder and the circular cone have the same height. The radius of the base of the cylinder is twice that of the cone. If the volume of the cone is 20 cm3, what is the volume of the cylinder?

1 ) 4a 2

4 ) a2 4 ) a2

A. B. C. D. E.

1 32 1 8 1 2 8 32

90 12.

3 : 10 9 : 25 9 : 100 36 : 25 36 : 100

40 cm3 80 cm3 120 cm3 240 cm3 300 cm3

The length, width and height of a cuboid are in the ratios 3 : 2 : 1. If the total surface area of the cuboid is 88 cm2, find its volume. A. B. C. D. E.

If a : b = 3 : 4 and b : c = 2 : 5, then a2 : c2 = A. B. C. D. E.

90 10.

+

If p and q are the roots of the equation x2 − x + 3 = 0, then (2p − 2)(2q − 2) = A.

90 9.

1 1 )(a2 − 2a 2 2 2 (a + )(a − 4 + a 2 (a + )(a2 − 2 + a (a +

6 cm3 48 cm3 48 2 cm3 96 2 cm3 384 cm3

90 13.

If 1 U.S. dollar is equivalent to 7.8 H.K. dollars and 1000 Japanese yen are equivalent to 53.3 H.K. dollars, how many Japanese yen are equivalent to 50 U.S. dollars? A. B. C. D. E.

1463 3417 7317 8315 20 787

In the figure, there are nine circles, each of radius 1. Find the shaded area. A. B. C. D. E.

90 11.

90 14.

Find the amount (correct to the nearest dollar) of $10 000 at 12% p.a., compounded monthly, for 2 years. A.

90-CE-MATHS II

9 − 9π 36 − 9π 40 − 9π 10 − 10π 40 − 10π

10 201

B. C. D. E. 90 15.

12 400 12 544 12 697 151 786

If a flat is sold for $720 000, the gain is 20%. Find the percentage loss if the flat is sold for $540 000. A. B.

E.

D

B

The figure shows a right pyramid with a square base. VAB, VBC, VCD and VDA are equilateral triangles. Find sin ∠VAH. A.

1 2 1 4 1 2 1 3

o

sin θ + cos θ sin θ − cos θ cos θ − sin θ −cos θ − sin θ 2sin θ

If 0 ≤ x < 360 , which of the following equations has only one root?

C. D. E.

C

A

B.

A. B.

90 18.

V

sin(180 + θ ) + sin(θ − 90 ) =

o

2 5

H

o

A. B. C. D. E. 90 17.

−

90 19.

5% 1 6 % 4 10% 1 11 % 9 1 33 % 3

C. D.

90 16.

E.

C. D. E.

3 2

o

sin x = 0 1 sin x = 2 sin x = 2 cos x = 0 cos x = −1

90 20.

P A

T

48

o

Q

B

4 and θ lies in the second 3 quadrant, then sin θ − 2 cos θ = If tan θ = −

A. B. C. D.

2 −2 11 5 2 5

90-CE-MATHS II

C In the figure, TQ is the tangent to the tangent to the circle at A. If arc AC = arc BC and ∠PAQ = 48o, then ∠QAC = A. B. C.

42o 48o 66o

71o 84o

D. E.

90 23.

90 21.

A

B

R O

44

E F

o

M Q

C

P In the figure, O is the centre of the circle. If OR // PQ and ∠ROQ = 42o, find ∠RMQ. 21o 42o 63o 84o 126o

A. B. C. D. E.

In the figure, ABCDE is a regular pentagon. Find ∠AFD. A. B. C. D. E.

120o 112o 110o 108o 100o

90 If the mean of the numbers 3, 3, 3, 3, 4, 24. 4, 5, 5, 6, x is also x, which of the following is/are true?

90 22.

A

I. Mean = Median II. Mode = Range III. Median = Mode

D F B

G

E

C

In the figure, AC // DE, FG // BC and AD : DF : FB = 1 : 2 : 3. If BE = 10, find FG. A. B. C. D. E.

D

5 6 8 9 10

90-CE-MATHS II

A. B. C. D. E.

I and II only I and III only II and III only None of them All of them

90 Ten years ago, the mean age of a band 25. of 11 musicians was 30. One of them is now leaving the band at the age of 40. What is the present mean age of the remaining 10 musician? A. B. C. D. E.

40 39 37 30 29

90 There are 7 bags, 3 of which are empty 26. and the remaining 4 each contains a ball. An additional ball is now put into one of the bags at random. After that a bag is randomly selected. Find the probability of selecting an empty bag. A. B. C. D. E.

90 29.

l2 l1 x O

2 7 3 7 6 49 12 49 18 49

(5, 6) (6, 7) (7, 8) (8, 9) (9, 10)

l3 l4

In the figure, the slopes of the straight lines l1, l2, l3, and l4 are m1, m2, m3, m4 respectively. Which of the following is true? A. B. C. D. E.

90 ABCD is a line segment. AB : BC : CD 27. = 3 : 2 : 1. If A = (4, 5), D = (10, 11), find C. A. B. C. D. E.

y

90 30.

m1 > m2 > m3 > m4 m2 > m1 > m3 > m4 m1 > m2 > m4 > m3 m2 > m1 > m4 > m3 m4 > m3 > m2 > m1 y

(4, 5)

90 x y 28. If the line y = mx + b and a + b = 1 are perpendicular, find m. A. B. C. D. E.

a b b a ab a − b b − a

90-CE-MATHS II

O

x

In the figure, a circle cuts the x-axis at tow points 6 units apart. If the circle has centre (4, 5), then its equation is A. B. C. D. E.

(x − 4)2 + (y − 5)2 = 25 (x − 4)2 + (y − 5)2 = 34 (x − 4)2 + (y − 5)2 = 52 (x + 4)2 + (y + 5)2 = 34 (x + 4)2 + (y + 5)2 = 25

90 31.

y

B.

O

C. D. E.

x

The graph of y = ax2 + bx + c is given as shown. Which of the following is/are true? I. a<0 II. b < 0 III. c < 0 A. B. C. D. E. 90 32.

90 33.

x

Sign of f(x)

1.22 1.23 1.24 1.25 1.245

+ + + − +

A.

1 1− 5

90-CE-MATHS II

1 2+ 3

A. B. C. D. E.

A. B. C. D. E.

2k k 0 −k −k − 1

+

1 2 3 4 5

90 If a < b < 0, which of the following 36. must be true?

1.20, correct to 2 decimal places 1.24, correct to 2 decimal places 1.25, correct to 2 decimal places 1.245, correct to 3 decimal places 1.2475, correct to 4 decimal places

1 + 1+ 2 1 = 4+ 5

90 Let f(x) = 3x3 − 4x + k. If f(x) is 34. divisible by x − k, find the remainder when f(x) is divided by x + k.

90 Let m be a constant. Find the value of 35.  x 2 + x + 1 = m x such that  x − 1 = 26  m

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

From the table, a root of the equation f(x) = 0 must be A. B. C. D. E.

1 5 −1 1+ 5 1− 5 −1 + 5

1 3+ 4

+

A. B. C. D. E.

−a < −b a <1 b a2 < b2 10a < 10b a−1 < b−1

90 The H.C.F. and L.C.M. of three 37. expressions are xyz2 and x3y5z4 respectively. If two of the expressions are x2y3z3 and x3yz2, find the third expression. A. B. C. D. E.

x2y3z3 x2y5z3 xy3z3 xy5z4 xy3z4

B. 90 Let a, x1, x2, b and a, y1, y2, y3, b be two 38. x2 − x1 = arithmetic progressions. y3 − y2 A. B. C. D. E.

3 4 3 4 1 4 5 5 4

C. D. E.

3 3 2 2π − 3 3 2π −

90 41.

90 39.

Three equal circles of radii 1 touch each other as shown in the figure.

C

A. 6

5 5 A

B.

5 M

B

C. D.

In the figure, AM = MB = MC = 5 and BC = 6. The area of triangle ABC = A. B. C. D. E.

3 3 2 2π − 3 4π −

12 16 24 30 48

90 40.

E.

1−

π 2

π 2 π 2 3 − 2 π 3 − 6 π 2 3 − 6 3 −

90 If A is 30% greater than B and B is 30% 42. less than C, then A. B. C. D. E.

A is 9% less than C C is 9% less than A A=C A is 9% greater than C C is 9% greater than A

90 Which of the following graphs shows 43. that y is partly constant and partly varies inversely as x? In the figure, an equilateral triangle is inscribed in a circle of radius 1. The circumference of the circle is greater than the perimeter of the triangle by A.

4π − 3 3

90-CE-MATHS II

A.

90 If sin θ and cos θ are the roots of the 44. equation x2 + k = 0, then k =

y

A. B. C. O B.

x D.

y E.

90 45. O C.

−1 1 − 2 1 − 4 1 4 1 2

y

x

y

x

O

P

O D.

The figure shows the graph of y = 3 sin2x. The point P is

x

A.

y

B. C. D. O E.

x E.

y

4π 3 3π ( 4 4π ( 3 3π ( 4 3π ( 2 (

90 46.

, −3) , −3) , −1) , −1) , −1) A

D

3 60 O

x

B

o

4

C

In the figure, ABCD is a parallelogram. BD =

90-CE-MATHS II

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

5 7

B. C. D. E.

13 27 37

90 47.

1 2 1

2

90 49.

NORTH

H

2 3

E 120

D

o

C A

A

B

30

C

B

In the figure, A, B and C are three points on the same horizontal plane. A is due north of B, C is due east of B and H is a point vertically above A. Which of the following angles is/are 90o?

In the figure, AC = CD, ∠ABC = 30o AB and ∠CED = 120o. = DE

I. ∠HAC II. ∠ABC III. ∠HBC A. B. C. D. E.

A.

1 2 1 3 2 3

B.

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

90 48.

C. D. E.

2

90 50.

C

A P

A

30

o

o

48

o

B

B

C

In the figure, AB is a diameter and ∠ BAC = 30o. If the area of ∆ABC is 3 , then the radius of the circle is

90-CE-MATHS II

In the figure, PA and PC are tangents to the circle ABC. If ∠P = 48o, then ∠ ABC = A. B. C.

84o 96o 106o

D. E.

114o 132o

90 51.

90 53.

C

A

B

D E

E

D

B

A

A. B. C. D. E.

30o 40o 45o 50o 60o

90 52.

3 4 5 6 7

90 54.

A

R S

E

Q

F C

D

B

U

In the figure, if CD = CF, CE = BE and DA = DB, then ∠C = A. B. C. D. E.

30o 36o 40o 45o 60o

T P

In the figure, ∆PTQ, ∆SQR and ∆RUT are equilateral triangles. Which of the following is/are true? I. ∆UPT ≅ ∆RQT II. PU = QS III. PQSU is a parallelogram A. B. C. D. E.

90-CE-MATHS II

C

In the figure AB, AC and BC are three tangents touching the circle at D, E and F respectively. If AC = 24, BC = 18 and ∠ACB = 90o, find the radius of the circle.

In the figure, TA and TB are tangents to the circle ABC. If TA ⊥ TB and BD ⊥ AC, find ∠CBD. A. B. C. D. E.

F

All of them None of them I and II only I and III only II and III only

90-CE-MATHS II