Mathematics 1987 Paper 2

Form 5

HKCEE 1987 Mathematics II 87 1.

2

A. B. C. D. E. 87 2.

If A. B. C. D. E.

87 3.

2 x 4 x 2 x2 4 x2 0

B. C. D. E.

87 5.

−1 . 1. 5. 6. 1 . 6

C. D. E. 87 6.

b + 3cd , then c = b − 3cd a . 6d b . 3d b(a − 1) . 6d b(a + 1) . a −1 b(a − 1) . 3d (a + 1)

The radii of two solid spheres made of the same material are in the ratio 2 : 3. If the smaller sphere weight 16 kg, then the larger one weighs

87-CE-MATHS II

87 7.

−1 1 − 3 3 1 3 1

If x +

1 1 = 1 + 2 , then x2 + 2 = x x

A. B. C. D. E.

1. 3. 1+2 2 . 2+2 2 . 3+2 3 .

If 32k + 1 = 32k + 6, then k = A. B. C. D. E.

87 8.

24 kg . 36 kg . 48 kg . 54 kg . 60 kg .

Given that x ≠ 0 and −x, x, 3x2 are in G.P., find x. A. B.

1 1 1 1 1 − = , and x = , z = , then y = x y z 2 3

If a = A.

87 4.

A. B. C. D. E.

2

 x + 1  x − 1   −  =  x   x 

1 . 4 1 − . 2 1 . 4 1 . 2 3. −

When the expression x2 + px + q is divided by x + 1, the remainder is 4. Find the value of 2p − 2q + 1.

A. B. C. D. E.

87 12.

−3 −5 −7 −9 It cannot be determined.

87 9.

y

O

2

y=x

A

x

b c −b −c b − c

87 11.

A. B. C. D. E.

B. C. D. E.

1 cm 2 cm 4 cm A solid rectangular iron block, 4cm × 2 cm × 1 cm, is melted and recast into a cube. The decrease in the total surface area is

x2 . x2 − 1 . x2 + 2 . x2 − 2x . x2 − 2x + 2 .

y = 10

156 cm2 169 cm2 216 cm2 312 cm2 338 cm2

87 13.

A. B. C. D. E.

If log10x, log10y, log10z are in A.P., then A.

C

18 cm

ABCD is a trapezium in which AB // DC, AB = 8 cm, DC = 18 cm, AD = BC = 13 cm. Find the area of the trapezium.

87 If f(x) =x2 + 1, then f(x − 1) = 10. A. B. C. D. E.

B

13 cm

D

In the figure, the graph of y = x2 + bx + c cuts the x-axis at A and B. OA + OB = A. B. C. D. E.

8 cm

13 cm

+ bx + c

B

A

x+ z 2

.

1 cm2 . 2 cm2 . 3 cm2 . 4 cm2 . 5 cm2 .

87 14.

x+z . 2 y2 = x + z . y2 = xz . y = 10 xz . y=

Figure a

87-CE-MATHS II

Figure b

Figure a shows a circular measuring cylinder 4 cm in diameter containing water. Three iron balls, each of diameter 2 cm, are dropped into the cylinder as shown in Figure b. What is the rise in the water level? A. B. C. D. E.

D. E. 87 18.

1 cm 4 1 cm 3 1 cm 2 1 cm 2 cm

A. B. C. D. E.

2 %. 3 20% . 60% . 2 116 % . 3 120% .

A. B. C.

1 cm . 3 cm . 2 2 cm .

87-CE-MATHS II

75

o

o

C

In the figure, ∠A = 75o, ∠B = 45o and BD CD bisects ∠ACB. CD A. B. C. D. E.

2 . 3 1 2 2 2 3 3 2

. . . .

87 A rectangle is 6 cm long and 8 cm wide. 19. The acute angle between its diagonals, correct to the nearest degree is

16

87 The circumference of a circle is 17. 6π cm. The length of an arc of the 1 circle which subtends an angle of 3 radian at the centre is

45

B

$3200 $3605 $3686 $13 200 $13 686

87 If the selling price of 5 pens is the same 16. as the cost price of 6 pens, the percentage profit in selling a pen will be

A D

87 Find, correct to the nearest dollar, the 15. compound interest on $10 000 at 8% p.a. for 4 years, compounded halfyearly. A. B. C. D. E.

π cm . 2π cm .

A. B. C. D. E.

37o . 41o . 49o . 74o . 83o .

87 20.

B

C

104

o

A D

In the figure, chords AC and BD meet at E and AB // DC. If ∠CED = 104o, find ∠ABD. A. B. C. D. E.

87 23.

C a A

A

B

a

C. D. E.

D

b

C

s ( a + b) a s ( a + b) b s ( a + b) 2 a2 s ( a + b) 2 b2 s (a 2 + b 2 ) a2

87 The real number π is 22. A. B. C. D. E.

22 . 7 3.1416 . the ratio of the area of a circle to the square of its diameter. the ratio of the circumference of a circle to its radius. the ration of the circumference of a circle to its diameter.

87-CE-MATHS II

B

In the figure, AB, BC and CD are three equal chords of a circle. If ∠BAC = a, then ∠AED = A. B. C. D. E.

In the figure, BD = a. DC = b and the area of ∆ABC.

B.

D

76o 52o 38o 14o It cannot be determined.

87 21.

A.

E

87 24.

2a . 3a . 90o − a . 180o − 2a . 180o − 3a . D

C F

A

E

B

In the figure, ABCD and ABEF are Area of ABCD parallelograms. = Area of ABEF A. B. C. D. E.

AD . AF BC . BF BC . EF AD 2 . AF 2 BC 2 . EF 2

87 25.

E.

C

A

87 Two perpendicular lines kx + y − 4 = 0 28. and x − 2y + 3 = 0 intersect at the point (h, k). Find h and k.

I

γ

II

α

β

B

III

In the figure, I, II and III are equilateral triangles. Area of I : Area of II : Area of III = A. B. C. D. E.

α:β:γ. sin α : sin β : sin γ . sin2 α : sin2 β : sin2 γ . cos α : cos β : cos γ . cos2 α : cos2 β : cos2 γ .

87 Which of the following straight lines 26. divide(s) the circle (x − 1)2 + (y + 1)2 = 1 into two equal parts? I. x−y−2=0 II. x + y + 2 = 0 III. x − y + 2 = 0 A. B. C. D. E.

II and III only

A. B. C. D. E.

h = −7, k = −2 1 h = −2, k = 2 h = 1, k = 2 1 h = −4, k = − 2 h = −3, k = 2

87 If the median of the 5 different integers 29. 2, 7, 10, x, 2x − 3 is 7, then x = A. B. C. D. E.

3. 4. 5. 6. 8.

87 The figures show the histograms of the 30. three frequency distributions. Arrange their standard deviations in ascending order of magnitude. I.

f 30

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

87 The equation of a circle is 27. x2 + y2 − 4x + 2y + 1 = 0. Which of the following is/are true?

20 10 -3 II.

I only II only III only I and II only

87-CE-MATHS II

1

3

1

3

x

f 30

I. The centre is (−2, 1). II. The radius is 2 units. III. The circle intersects the y-axis at two distinct points. A. B. C. D.

-1

20 10 -3

-1

x

III.

87 A die is thrown twice. Find the 33. probability that the number obtained at the first throw is greater than that at the second throw.

f 30 20

A.

10 -3 A. B. C. D. E.

-1

1

3

x

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

87 One letter is taken from each of the 31. words “MAN” and “ART” at random. Find the probability that the two letters are not the same. A. B. C. D. E.

1 9 1 3 4 9 2 3 8 9

B. C. D. E.

1 4 1 3 1 2 2 3 5 6

C. D. E.

87 If a : b = 3 : 2, b : c = 4 : 3, then a + b : 34. b + c = A. B. C. D. E.

7 : 10 . 5:7. 1:1. 7:5. 10 : 7 .

87 Peter bought an article for $ x. He sold 35. it to Mary at a profit of 20%. Mary then sold it to John for $90 at a loss of 25%. Find x.

87 Four persons A, B, C, D sit randomly 32. around a round table. The probability that A sits next to B is A.

B.

1 6 5 12 1 2 7 12 5 6

. . . . .

87-CE-MATHS II

A. B. C. D. E.

56.25 81 90 100 144

87 If x and y are integers with x > y, which 36. of the following is/are true? x2 > y2 1 1 < y x x III. 10 > 10y I. II.

A. B. C. D. E.

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

C. D. E.

87 Solve the inequality 37. x log100.1 > log1010. A. B. C. D. E.

x > −1 x>1 x > 100 x<1 x < −1

87 If a is 10% less than b and b is 10% 41. greater than c, then a : c =

87 If x2 + y2 = 5 and x + y = 3, then x − y = 38. A. B. C. D. E. 87 39.

1. −1 . 1 or −1 . 1 or −5 . −1 or 5.

A. B. C. D. y = ax

0

2

+ bx + c

x

The figure shows the graph of y = ax2 + bx + c. Which of the following is/are true? I. a>0 II. b > 0 III. c > 0 I only I and II only I and III only I and II only I, II and III

87 Find the H.C.F. of (2x − 1)(x2 − 6x + 9) 40. and (x2 − 3x)(4x2 − 1). A. B.

(x − 3) (2x − 1)

87-CE-MATHS II

A. B. C. D. E.

1:1. 9 : 10 . 10 : 9 . 99 : 100 . 100 : 99 .

87 1 1 1 42. If 3a = 2b = 5c, then a : b : c =

y

A. B. C. D. E.

(x − 3) (2x − 1) x(x − 3)2 (2x − 1) (2x + 1) There is no H.C.F.

E.

3:2:5. 5:2:3. 1 1 1 : : . 3 2 5 1 1 1 : : . 5 3 2 1 1 1 : : . 2 3 5

87 A man walks from place A to place B at 43. a speed of 3 km/h and cycles immediately back to place A along the same road at a speed of 15 km/h. The average speed for the whole trip is A. B. C. D. E.

5 km/h . 6 km/h . 9 km/h . 10 km/h . 12 km/h .

87 Let n be a positive integer. Which of 44. the following number is/are odd? I. 22n + 1 II. 2n + 1 III. 3(2n) A. B. C. D. E.

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

87 45.

a

In the figure, O is the centre of the circle. If AB = 12 and AC = 13, then cos θ =

c

b

A.

O

B. C. In the figure, O is the center of the circle. a + b = A. B. C. D. E.

180o . c. c . 2 180o − c . c 180o − . 2

D. E.

5 12 5 13 12 13 12 25 13 25

87 48.

. . . . . D

F

87 46.

θ

D q

φ

p

A

θ

C

B

In the figure, AD = p, CD = q and ∠B = 90o, BC = A. B. C. D. E.

p sin θ − q sin φ . p sin θ − q cos φ . p cos θ − q sin φ . p sin θ + q cos φ . p cos θ + q sin φ .

87 47.

12

A

A.

C. D. E.

θ O 13

C

o

30

o

E

B

1 . 4 1 . 2 3 . 2 3 . 3 3 . 4

87 How many different values of x 49. between 0o and 360o will satisfy the equation (sin x + 1)(2 sin x + 1) = 0? A. B.

87-CE-MATHS II

60

In the figure, ABCD is a rectangle inclined at an angle of 30o to the horizontal plane ABEF. ∠CBD = 60o. Let θ be the inclination of BD to the horizontal plane. sin θ =

B.

B

A

C

0 1

C. D. E.

2 3 4

E.

87 If 0o ≤ x < 360o, the number of points of 50. intersection of the graph of y = sin x and y = 1 + cos x is A. B. C. D. E.

87 53.

B. C. D. E.

87 52.

C B

1 . 8 1 . 5 3 . 10 9 . 16 3 . 4

35o . 40o . 55o . 65o . 70o . A D

87-CE-MATHS II

E

B o

4

In the figure, O is the centre of the circle of radius 4. The area of the shaded region is 4π −4. 3 4π −8. 3 4π −4 3. 3

R

Q

87 54.

O

C.

o

In the figure, C is the centre of the circle. ABCD is a straight line. AQR touches the circle at Q. If ∠DAR = 20o, then ∠DQR = A. B. C. D. E.

30

B.

20

A

A

A.

D

0. 1. 2. 3. 4.

87 In ∆ABC, if AB : BC : CA = 4 : 5 : 6, 51. then cos A = A.

2π −4. 3 8π −8. 3

D.

B

F

C

In the figure, DE // BC and AB // EF. If AE : EC = 1 : 2, then area of ∆ADE : area of parallelogram BFED = A. B. C. D. E.

1:2. 1:3. 1:4. 1:5. 1:6.