Mathematics 1984 Paper 2

Form 5

HKCEE 1984 Mathematics II 84 1.

A.

1 ( x − 1) 2 ( x + 1) x+2 ( x − 2)( x + 1)( x − 1) x + 10 ( x − 2)( x + 1)( x − 1) x − 10 ( x − 2)( x + 1)( x − 1)

B. C. D. E.

84 2.

B. C. D. E.

84 4.

2b(2 y − x) , then y = x − 3y

84 6.

1 22n − 1 2

2n + 2n 2 2 n −2 n 2−2n + 1

If x + 2 is a factor of x2 + ax + b, then 2a − b + 3 =

84-CE-MATHS II

7 3 1 −1 −5

If ( 3 − 2 )x = 1, then x = A. B. C. D. E.

84 7.

−7 −1 0 1 7

If α and β are the roots of 3x2 − x − 1 = 0, 1 1 then 2 + 2 = β α A. B. C. D. E.

a + 2b x 3a + 4b a − 2b x − 3a + 4b a + 2b − x 3a + 4b 3a + 4b x a + 2b − 3a + 4b x a − 2b

(2n + 1)2 × (2−2n − 1) ÷ 4n = A. B. C. D. E.

84 5.

x 2 − 3 x − 10 ( x − 2)( x + 1)( x − 1) 2

If a = A.

84 3.

A. B. C. D. E.

4 3 − 2 ( x − 2)( x + 1) x −1

3+ 1 3+ 1 + 3 1 − 3

2 2 1 2 1 2

3− 2 3+ 2

What is/are the root(s) of 5 − x = x − 3? A. B. C. D. E.

4 only 1 and 4 only −1 and −4 only −4 and 4 only −4, −1, 1 and 4

84 8.

The sum of the first ten terms of an arithmetic progression is 120. If the common difference is 4, then the first term is A. B. C. D. E.

84 9.

−12 . −6 . −2 . 2. 6.

$10 000 is invested for 2 years at 10% per annum, compounded half-yearly. The compound interest, correct to the nearest dollar, is A. B. C. D. E.

$12 155 . $2155 . $2100. $2000 . $1025 .

84 The equation x2 + kx + k = 0 has equal 10. roots (k being a constant). k = A. B. C. D. E. 84 11.

If A. B. C. D. E.

4 only −4 only 0 or 4 0 or −4 4 or −4 3x + 2 y = 1, then x + 5y

x+ y: x− y =

1: 5 3:2 5: 6 5:1 7:2

84 A is 25% taller than B. B is 25% 12. shorter than C. A’s height : C’s height = A. B. C. D.

1:1 5:4 3:4 5:3

84-CE-MATHS II

E.

15 : 16

84 A rectangular box, without a lid, is 13. 40 cm long, 30 cm wide and 10 cm height. The area of the external surface of the box is A. B. C. D. E.

2600 cm2 . 3400 cm2 . 3500 cm2 . 3800 cm2 . 12 000 cm2 .

84 A man drives a car at 30 km/h for 3 14. hours and then at 40 km/h for 2 hours. His average speed for the whole journey is A. B. C. D. E.

14 km/h . 30 km/h . 34 km/h . 35 km/h . 70 km/h .

84 A alone can complete a job in 8 hours. 15. B alone takes 12 hours and C alone takes 6 hours. After A and B have worded together on the job for 3 hours, C joins them. How much longer will they take to complete the job? A. B. C. D. E.

1 hour 1 1 hours 2 2 hours 1 2 hours 2 3 hours

84 The marked price of a book is 20% 16. above the cost price. If the book is sold at a discount of 10% off the marked price, what is the gain per cent based on the cost price? A. B. C. D. E.

8% 10% 12% 18% None of the above.

84 17.

E.

tan 2 θ + cos2θ = 1 + tan 2 θ A. B. C. D. E.

3 4

84 20.

1 1 + cos2θ 2 cos2θ 1+ tan2θ 1 + cos2θ

H

h A

84 18.

A

30

35

B

70

A. B. C.

In the figure, BCD is a straight line. ∠ ADC = 90o and BC = 10. AD = A. B. C. D. E.

10 cos 70o 10 sin 70o 10 tan 70o 10 sin 20o sin 55o 10 tan 20o sin 55o

D. E.

1 tan 30o 1 tan 30o h tan 30o h tan 30o

84 21.

A x

84 19.

2

θ

C

In the figure, ∆ABC lies in a horizontal plane. ∠BAC = 90o. HA is vertical and HA = h. tan θ =

o

D

C

o

θ

B

o

45

o

30

o

2x

B

3

C 4 In the figure, cos θ = A. B. C. D.

1 4 1 − 2 1 4 1 2 −

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In the figure, AB = x and AC = 2x. The area of ∆ABC is 16. x (correct to 2 decimal places) is A. B. C. D. E.

2.83 . 4.00 . 4.30 . 5.66 . 6.08 .

84 The sum of the interior angles of a 22. convex polygon is greater than the sum of the exterior angles by 360o. How many sides has the polygon? A. B. C. D. E.

3 4 5 6 8 56

2

P

O

45

o

T

In the figure, O is the centre of the circle. TA and TB touch the circle at A and B respectively. OA = 2. The length of the arc APB is

o

xo

A.

o

26

o

122

B.

In the figure, x = ?

C.

31 34 40 48 It cannot be determined.

D. E.

B 56

o

A. D B.

A

C

In the figure, AB and AC touch the circle at B and C respectively. ∠A =

C. D.

o

30 40o 50o 80o 85o

E.

 2 x1 + x2 2 y1 + y2  ,   3 3    x1 + 2 x2 y1 + 2 y2  ,   3 3   2 x − x 2 y − y  1 2 2  , 1   3 3    x1 − 2 x2 y1 − 2 y2  ,   3 3   x + x y + y  1 2 1 2  ,   . 3   3

. . . .

84 The line x + y + k = 0 (k being a 27. constant) passes through the centre of the circle x2 + y2 − 2x + 4y − 6 = 0. k = A.

84-CE-MATHS II

π . 4 π . 2 3π . 4 3π . 2 3π .

84 The point P divides AB internally so 26. that AP : PB = 2 : 1. The coordinates of A and B are (x1, y1) and (x2, y2) respectively. The coordinates of P are

84 24.

A. B. C. D. E.

A

B

84 23.

A. B. C. D. E.

84 25.

−2

B. C. D. E.

D.

−1 0 1 2

84 The equation of a circle is 28. x2 + y2 − 2x + 5y − 7 = 0 . Which of the following is/are true?

y

x

0 E.

y

I.

The circle passes through the point (−1, 1) . II. The centre of the circle lies in the second quadrant. III. The circle intersects the x-axis at two points. A. B. C. D. E.

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

84 If a, b and c are positive real numbers, 29. which of the following graphs could represent the line ax + by + c = 0? A.

0

84 The probability that John will win a 30. 1 game is and the probability that he 3 2 will lose is . What is the probability 3 that, in three games, he will win any two games and lose one game? A.

y

B. C. x

0 B.

y

0 C.

D. E.

x

y B. C.

84-CE-MATHS II

x

4 27 2 27 1 27 2 9 1 9

84 Two dice are thrown. What is the 31. probability of getting a sum of 8? A.

0

x

D.

1 12 1 11 5 36 1 6

E.

2 9

84 The standard deviation of the five 32. numbers a − 2d, a − d, a, a + d, a + 2d, is A. B. C. D. E.

0. d. 2d. 5d. 10 d .

C.

B. C. D. E.

3 3 or x ≥ − . 2 2 3 3 ≤x≤ − . 2 2 3 3 − ≤x≤ . 2 2 3 3 x ≥ − or x ≤ . 2 2 3 3 x ≤ − or x ≥ . 2 2 x ≥

D. E.

A. B. C. D. E.

−3 −2 −1 0 1

84 If a and b are non-zero real numbers 35. and a > b, which of the following must be true? a2 > b2 1 1 > a b III. a3 > b3 I. II.

84-CE-MATHS II

log10 2 − 1 x +1 log10 x 10( x + 1) log10 x x +1 log10 10 x x +1 log10 − 2x x

84 If a ≠ ±1, then 1 + a2 + a4 + … + a2n = 37. A. B. C.

2

84 The graph of y = x + ax + b (a and b 34. being constants) cuts the x-axis at (2, 0) and (h, 0), and cuts the y-axis at (0, −2). h =

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

84 If f(x) = (log102x) − x, 36. then f(x + 1) − f(x) = A. B.

84 4x2 − 9 ≥ 0 is equivalent to 33. A.

A. B. C. D. E.

D. E.

1 − a 2n 1− a 1 − a 2n 1 − a2 1 − a 2 n +1 1− a 1 − a 2 n +1 1 − a2 1 − a 2n + 2 1 − a2

84 Which of the following must be 38. geometric progression(s)? I. log103, log109, log1027, log1081 II. 0.9, 0.99, 0.999, 0.9999 III. 1, −3, 9, −27 A. B. C. D. E.

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

84 a, b, c are positive numbers such that 39. a b = = k (k being a constant), which b c of the following must be true? b2 = k2 a+b =k b+c III. a = k2 c I. II.

A. B. C. D. E.

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

84 Last year, a man saved 10% of his 40. income. By how much per cent must his income be increased if his expenditure increased by 20% and he wants to save 20% of his income? A. B. C. D. E.

50% 35% 30% 20% 15%

84 The external and internal radii of a 41. hollow metal sphere are 4cm and 3 cm respectively. Volume of metal = Volume of the enclosed empty space A. B. C. D. E.

1 27 1 3 4 3 37 27 64 27

84-CE-MATHS II

84 A solid metal sphere of volume 252 cm3 42. is melted and recast into 3 smaller solid spheres whose radii are in the ratio 1 : 2 : 3. The volume of the smaller sphere is A. B. C. D. E.

5 cm3 . 7 cm3 . 14 cm3 . 18 cm3 . 28 cm3 .

84 The base radii of two right circular 43. cylinders are in the ratio 2 : 3. If the two cylinders have the same height, what is the ratio of their curved surface area? A. B. C. D. E.

2:3 4:9 8 : 27 8 : 27 None of the above.

84 3 44. The greatest value of 4 + 2 cosθ is A. B. C. D. E.

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

84 If 0o ≤ θ < 360o, the number of roots of 45. the equation 1 2 sin θ + = 3 is sin θ A. B. C. D. E.

0 1 2 3 4

84 46.

E.

A

180 A πr 2

84 48.

p

A

θ B

q

C

D

P

In the figure, ∠B = 90o and BCD is a straight line. If AB = p and BC = q, then cos θ = A.

a

θ B

p q

Q

C

R

In the figure, PQRS is a square inscribed in ∆ABC. AB = AC and PQ = a. AB =

B.

p

C.

p2 + q2 q

D.

p2 + q2 −p

C.

E.

p2 + q2 −q p2 + q2

D.

84 47.

S

A.

1 cos θ) 2 1 a(sin θ + sin θ) 2 1 1 a( + ) sin θ 2 cosθ 1 1 a( + ) cosθ 2 sin θ 2a sin θ a(sin θ +

B.

P

E.

Q

r xo

84 49.

q

A

B

A In the figure, the radius of the sector is r and ∠POQ = xo. If the area of the sector is A, then x = A. B. C. D.

2A r2 360 A r2 360 A πr 2 180 A r2

84-CE-MATHS II

70 D

o

50 p

o

C

In the figure, AB // DC. AB = q and DC = p. BC = A. B. C.

( p + q) sin 50o 2 sin 70o ( p + q) sin 70o 2 sin 50o ( p − q) sin 70o sin 60o

D.

84 In ∆ABC, BC = a, AC = b, AB = c and 52. a > b > c. Which of the following must be true?

( p − q) sin 70o sin 50o ( p − q) sin 50o sin 70o

E.

84 50.

I. ∠A > ∠B > ∠C II. b + c > a III. ∠B + ∠C > ∠A

A

X

A. B. C. D. E.

Y

B

C

In the figure, XY // BC. AX : XB = 2 : 1. If the area of the trapezium BCYX = 20, then the area of ∆ABC = A. B. C. D. E. 84 51.

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

84 53.

P 8 A

80 60 45 40 36

Q In the figure, AB is a diameter of the circle. AP = AQ. AB = 10 and BP = 8. PQ =

y 2

A. B. C. D. E.

1 0 -1

π 4

π 2

3π 4

π

θ

5 6 6.4 8 9.6

84 54.

-2

P

The figure shows the graph of y = a sin kθ. What are the values of the constant a and k? A. B. C. D. E.

B

10

a = 1 and k = 1 a = 1 and k = 2 1 a = 1 and k = 2 a = 2 and k = 2 1 a = 2 and k = 2

84-CE-MATHS II

40 A

o

D 30

B

C

o

Q

In the figure, the chords BA and CD, when produced, meet at P . The chords AD and BC, when produced, meet at Q. ∠B = A. B. C. D. E.

35o 40o 45o 50o 55o

84-CE-MATHS II