Mathematics 1980 Paper 2

Form 5

HKCEE 1980 Mathematics II 80 1.

A. B. C. D. E. 80 2.

C. D. E.

80 6.

1 1 = a + b and = a − b, y x then x + y = A. B. C.

4 p2 = 9q 2

1 4 9 9 4 4 ( )2 9 9 ( )2 4

D. E.

80 7.

If x = A. B.

If n = 10 , then log10n = C. 10a 10n na an a

x −2 − y −2 = x −1 − y −1

80-CE-MATHS II

D. E.

80 8.

x−1 + y−1 x−1 − y−1 x−3 − y−3 1 x− y 1 x+ y

If

a

A. B. C. D. E. 80 5.

E.

625a + b 625ab 125a + 3b 5a + 3b 53a + b

If 4p = 9q, then A. B.

80 4.

(a − b)2 (−a − b)2 (−a + b)2 −(a + b)2 −(a − b)2

125a ⋅ 5b = A. B. C. D. E.

80 3.

A. B. C. D.

2ab − a2 − b2 =

2 a a2 − b2 a a2 − b2 − a 2a 2 a − b2 − 2a a2 − b2 y + (n − 1) z , then n = n +1 x− y+z z x+ y−z z y−x−z x+z y−x−z x−z y+x−z x−z

5n + 2 − 35(5n −1 ) = 18(5n +1 )

A. B. C. D. E. 80 9.

B. C. D. E.

1 18 1 15 1 5 5 5n

80 If the length of a rectangle is increased 13. by 10% and the width decreased by 10%, which of the following is true?

A. B.

x>4

C.

3 >x 4 3 − > x or x > 4 4 3 x> − 4

D. E.

3 4

−

80 When the hour hand has turned through 10. an angle xo, what is the angle through which the minute hand has turned? A. B. C. D. E. 80 11.

A. B. C. D. E.

Solve the inequality (4x + 3)(x − 4) > 0

4>x> −

18 4 3 2 1

80 A man solid a car for $35 000 at a loss 12. of 30% on the cost price. What would have been the loss or gain percent if he had sold it for $50 500? A.

A gain of 10%

80-CE-MATHS II

Its area remains the same Its area is decreased by 1% Its area is increased by 1% Its area is decreased by 10% Its area is increased by 10%

80 The length of a side of a rhombus is 14. 10 cm. If its shorter diagonal is of length 12 cm, what is the area of the rhombus in cm2? A. B. C. D. E.

60 96 100 120 192

80 If the bearing of B from A is S30oW, 15. then the bearing of A from B is

6xo 12xo 60xo 360xo 3 600xo

The first term of an arithmetic progression is 6 and its tenth term is three times its second term. The common difference is A. B. C. D. E.

A gain of 1% No gain nor loss A loss of 10% A loss of 1%

N30oE N60oW N60oE S30oW S30oE

A. B. C. D. E. 80 16.

1 1 −1 sin θ A. B. C. D. E.

−

1 1 = +1 sin θ

2 tan θ 2 tan2 θ 2 tan 2 θ 2 sin θ cos 2 θ 2 sin 2 θ cosθ

80 If cos θ = x and 0o < θ < 90o, then tan θ 17. A.

A. B.

1 1 − x2

B.

1 − x2

C. D.

1 − x2 x x

E.

1− x x

C. D. E.

3 2 x 4 3 2 x 2 1 2 x 4 1 2 x 2 3 x2

2

±

80 21.

1 − x2

D

80 If 0o ≤ θ < 360o, which of the following 18. equations has exactly one root? A. B. C. D. E.

E

sin θ = −1 1 sin θ = − 2 sin θ = 0 1 sin θ = 2 sin θ = 2

A

B

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

80 19. 30 a

60

C

o

b

80 22.

35o 36o 40o 54o 72o S

A

B

P

o

c In the figure, a : b : c = A. B. C. D. E.

3 : 2 :1 9:4:1 2: 3:1 3: 2:1 3:2:1

80 What is the area, in cm2, of an 20. equilateral triangle of side x cm?

80-CE-MATHS II

R D

Q

C

In the figure, ABCD is a square with AB = 5. AP = BQ = CR = DS = 1. What is the area of PQRS? A. B. C.

9 15 16

D. E.

17 18

80 25.

A Y

80 23.

D

yo

C

xo

X

E B In the figure, circle AXB passes through the centre of circle AYB. y = A

A. B. C. D.

B

In the figure, ABCD is a square and ABE is an equilateral triangle. ∠ADE =? A. B. C. D. E. 80 24.

E.

o

72 74o 76o 78o None of the above

80 26.

E

A

D III

I II

E

A yo

xo

C

B

A. B. C. D. E.

In the figure, the two circles intersect at A and B. CAE and CBD are straight lines. ∠CED =

I and II I and III II and III II and IV III and IV

80 27.

A D

B

80-CE-MATHS II

C

In the figure, ABCD is a rectangle ∠ BEF = 90o. Which two of the triangles I, II, III and IV must be similar?

D

yo 180o − yo 180o −xo − yo 180o −xo + yo 360o −xo − yo

F IV

B

A. B. C. D. E.

2x 180 − 2x 180 − x 1 (90 − x) 2 1 (180 − x) 2

C

D.

In the figure, the inscribed circle of ∆ ABC touches AC at D. If AB = 7, AC = 5 and AD = 2, then BC = A. B. C. D. E.

9.5 9 8.5 8 7.5

E.

80 The 2nth term of the geometric 31. progression, 8, −4, 2, −1, …, is A.

C. D. E.

m+n m+n 2 1 1 + m n m+n mn mn m+n

1 2n+ 2

80 A certain sum of money is just sufficient 28. to pay the wages of one man for m days or the wages of one boy for n days. For how many days will this sum be just sufficient to pay the wages of one man and one boy together? A. B.

1 8 1 16

B. C. D. E.

80 32.

2 −1 22 n + 2 1 2n −3 2 1 2n − 4 2 −1 22n − 4

y

x

80 If the value of y2 + 3y + 7 is 2, what is 29. the value of 2y2 + 6y − 3? A. B. C. D. E.

−13 −7 7 13 It cannot be found from the information given

80 A, B, C are three spheres. If 30. Surface area of A = 4 and Surface area of B Volume of B = 2, then Volume of C Volume of A = Volume of C A. B. C.

16 8 2

80-CE-MATHS II

The figure above shows the graph of y = ax2 + bx + c. Determine whether a and c are positive or negative. A. B. C. D. E.

a > 0 and c > 0 a < 0 and c < 0 a > 0 and c < 0 a < 0 and c > 0 It cannot be determined from the given data

80 $P amounts to $Q in n years at simple 33. interest. The rate per annum is A. B.

100n(Q − P ) % P 100(Q − P ) % n

C. D. E.

100(Q − P ) % nP 100(Q − P ) % nQ 1

Q 100[ ( ) n − 1]% P

80 1 2 If 0 < x < 1, which of x, x , , x is 34. x the smallest? Which is the largest? A. B. C. D. E.

A. B. C. D. E.

80 If x and y are real numbers, what is the 38. minimum value of the expression (x + y)2 − 1 ? A. B. C. D. E.

x is smallest, x2 is largest 1 is smallest, x2 is largest x 1 x is smallest, is largest x 1 x2 is smallest, is largest x x2 is smallest, x is largest

k = −6 only k = 6 only −6 ≤ k ≤ 6 k = 6 or −6 only k ≤ −6 or k ≥ 6

−5 −1 0 3 It cannot be determined

80 39.

B

C

A

80 The Highest Common Factor of two 35. unequal Positive integers a and b is 8. Which of the following must be true? The difference between a and b is divisible by 8 II. (a + b) is divisible by 16 III. ab is divisible by 64

In the figure, the areas of the surfaces A, B, C of the cuboid are 10 cm2, 14 cm2 and 35 cm2 respectively. What is the volume of the cuboid?

A. B. C. D. E.

A. B. C. D. E.

I.

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

80 x, y and z are three consecutive positive 36. integers. Which of the following is true? A. B. C. D. E.

x + y + z must be odd x + y + z must be even xyz must be odd xyz must be even x2 + y2 + z2 must be even

80 If x2 − kx + 9 ≥ 0 for all real values of x, 37. what is the value of k?

80-CE-MATHS II

49 cm3 70 cm3 140 cm3 350 cm3 4 900 cm3

80 x is a positive integer such that 40. x2 + 2x + 7 is even. What are the possible values of x? A. B. C. D. E.

x can be any positive integer x can be any positive even number x can be any positive odd number x must be an even number greater than 10 000 x must be an positive odd number greater than 10 000

80 The perimeter of a sector is 16 and its 41. angle is 2 radians. What is the area of the sector? A. B. C. D. E.

p + q tan θ q p+ tan θ p + q cos θ −p + q tan θ q −p + tan θ

A. B. C. D. E.

16 32 64 16π 32π

80 42.

80 44.

B

A

C E 73

2

π 10

In the figure, diameter AB = 2. π ∠CAB = rad. Minor arc BC = 10

B. C. D. E.

π 10 π 5 3π 10 4π 5 9π 10

Q r

α

O X

C

p

D

B

In the figure, ∠B = ∠C = 90o. If AB = p and BC = q, then CD =

80-CE-MATHS II

P

Y

In the figure, O is the centre of the circle and its radius is r. XY touches the circle at P. Find the distance of Q from XY. q

θ

C

50o 68o 74o 78o 80o

A. B. C. D. E. 80 45.

80 43.

A

D

o

In the figure, AD and BE bisect ∠A and ∠B respectively. ∠C =

A

A.

80

B

o

A. B. C. D. E.

r(1 − sin α) r(1 + sin α) r(1 − cos α) r(1 + cos α) r(2 − sin α)

80 Which of the following is the graph of y 46. = 2 sin θ, where 0 ≤ θ ≤ 2π ?

A.

In the figure, AB = BC = CD. ∠AED =

y 2 1

π

-1 -2

B.

2π

θ

50o 65o 75o 90o 105o

80 48.

y 2 1

π

-1 -2

C.

A. B. C. D. E.

2π

A B

θ

C

y

R

2 1

π

-1 -2

D.

2π

In the figure, RS is a tangent to the circle at C. BA is any chord parallel to RCS. Which of the chords AB, BC and CA must be equal in length?

θ

A. B. C. D. E.

y 2 1

π

-1 -2

E.

S

2π

θ

AB and BC only AC and BC only AB and AC only All of them No two of them

80 49. B

y 2 1

A

C π

-1 -2

80 47.

C

2π

A

80-CE-MATHS II

D In the figure, AB = AC, D is the midpoint of arc BC. Which of the following is/are true?

D

I. AD bisects ∠BAC II. BC ⊥ AD III. AD is a diameter of the circle

B 25

θ

o

E

A. B. C.

I only II only III only

D.

I and II only

80-CE-MATHS II

E.

II and III only

80 50.

80 52.

C

P

Q

a b c A

D

d

O

θ

B

A

A. B. C. D.

a only a and b only a, b and c only a, b, c and d None of them

E.

80 51.

C

In the figure, AC and BC are diameters of two semi-circles touching each other internally at C. PQC is a straight line. If AB = 1, then PQ =

In the figure, AOB is a diameter of the circle, centre O. CD is the perpendicular bisector of OA. Which of the angles a, b, c, d is/are equal to 30o? A. B. C. D. E.

1 B

cos θ sin θ tan θ 1 sin θ 1 cosθ

80 53.

A

x

d

a O B

X

C

In the figure, circle O is inscribed in ∆ ABC, touching BC at X. Which of the following must be true?

With the notation in the figure, express a + b + c + d in terms of x. A. B. C. D. E.

I. OX ⊥ BC II. OA bisect ∠A III. AO produced bisect BC A. B. C. D. E.

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

c

b

80 54.

x − 180o x 540o − x 360o − x 180o − x P

A

θ

B y

x O

80-CE-MATHS II

In the figure, O is the centre of the circle. PAB is a straight line. x + y = A. B.

2θ 90o + θ

80-CE-MATHS II

C. D. E.

180o − θ 180o − 2θ 180o