Mathematics 1983 Paper 2

Form 5

HKCEE 1983 Mathematics II 83 1.

A. B. C. D. E.

83 2.

1 ( x − 2)( x − 3) 1 ( x + 2)( x + 3) 1 ( x + 2)( x − 3) 1 ( x − 2)( x + 3) x − 27 ( x − 2)( x + 3)( x − 3)

83 4.

B. C. D. E.

1 1 + 2 2 a b 1 1 + + 2 a ab 1 1 + 2 − a ab a2 − ab + b2 a2 + ab + b2 y2

If x = A. B. C. D.

a 2 + bz 1 b 1 b

y4 − a2) 2 x x2 ( 4 − a2) y

1 2 x2 (a − 4 ) y b 1 2 y4 (a − 2 ) b x

83-CE-MATHS II

1 b2 1 b2

83 6.

1

x2 y 2 x

−

1 2

y−2

1 a+1 a−1 a2 + 1 a2 − 1

When f(x) is divided by (2x + 1), the remainder is A. B. C. D. E.

83 7.

xy xy−1 xy−3

The H.C.F. of a3 − 1 and a4 − 1 is A. B. C. D. E.

, then z =

(

1

E. 83 5.

1 2 x2 (a − 2 ) y b

(x2y−1) ÷ ( x 2 y−1)2 = A. B. C. D.

1 1 + 3 3 a b = 1 1 + a b A.

83 3.

E.

6 5 − 2 2 x −9 x + x−6

f(2) f(1) f(−1) 1 f( ) 2 1 f( − ) 2

If α and β are the roots of 2x2 − 3x − 4 = 0, then α2 + 3αβ + β 2 A. B. C. D.

1 4 1 4 4 5 1 8 4

E. 83 8.

2x − 3a − 4 > 3x + 5a + 6 is equivalent to A. B. C. D. E.

83 9.

13

−14 −10 10 50 54

2:3:5 5:3:2 6 : 10 : 15 15 : 10 : 6 25 : 9 : 4

83 11.

4x

3 2x 1 x

In the figure, all the corners are rightangled. If the perimeter of the figure is 40, then x = 0.25

83-CE-MATHS II

2 cm2 4 cm2 8 cm2 16 cm2 It cannot be determined

83 A hollow cylindrical metal pipe, 1 m 13. long, has an external radius and an internal radius of 5 cm and 4 cm respectively. The volume of metal is 90π cm3 100π cm3 180π cm3 900π cm3 1800π cm3

83 Two men cycle round a circular track 14. which 3 km long. If they start at the same time and at the same spot but go in opposite direction with speeds 6km/h and 9 km/h respectively, for how long must they cycle before they meet for the first time?

x

A.

A. B. C. D. E.

A. B. C. D. E.

83 If 2x = 3y = 5x, then x : y : z = 10. A. B. C. D. E.

2 2.5 4 4.5

83 If the lengths of the diagonals of a 12. rhombus are 2 cm and 4 cm respectively, what is the area of the rhombus?

x > −8a − 10 x > 2a − 10 x < −8a − 10 1 x < (2a + 2) 5 1 x > (2a + 2) 5

The sixth term and the eleventh term of an A.P. are 10 and 30 respectively. The first term is A. B. C. D. E.

B. C. D. E.

A. B. C. D. E.

12 minutes 15 minutes 18 minutes 24 minutes 60 minutes

83 A man marks his good at a price that 15. will bring him a profit of 25% on the cost price. If he wants to sell his goods to a friend at the cost price, the percentage discount on the marked price should be A. B.

25% . 20% .

C. D. E.

83 19.

2 %. 3 15% . 12% . 16

A a

A. B. C. D. E. 83 17.

sin4θ cos4θ −sin4θ −cos4θ sin2θ cos2θ

D. E.

83 18.

In the figure, ∠ABC = ∠ACD = ∠BDC = 90o. AC = a. CD = A. B. C. D. E.

cos θ −cos θ sin 2 θ − cosθ cos 2 θ − sin θ sin 2 θ cosθ A

83 20.

p

E.

2 cm A

q

(q − p) sin θ (q − p) cos θ (q − p) tan θ q− p sin θ q− p cosθ

In the figure, OAB is a sector of a circle. Radius OA is 3 cm long and arc AB = 2 cm. The area of the sector is

C

A. B. C. D. E.

3 cm2 . 6 cm2 . 9 cm2 . 3π cm2 . 6π cm2 .

83 21.

A 44

B

83-CE-MATHS II

B

O

B

In the figure, AB = p, DC = q and ∠A = ∠D = 90o. BC = A. B. C. D.

a sin2θ a sin2θ a tanθ a sin θ cos θ a cosθ sin θ

3 cm

θ D

C

D

cos(90o − θ ) = tan(180o − θ ) A. B. C.

θ

B

83 sin2θ − (sin2θ cos4θ + sin4θ cos2θ ) = 16.

o

C

In the figure, AB = AC. If the area of ∆ ABC is 64 cm2, then AB = A. B. C. D. E.

83 24.

42

o

A. B. C. D. E.

B

In the figure, D is a point on BC and AC = AD = BD. ∠CAD = A. B. C. D. E.

o

D

In the figure, chords AB and CD intersect at P. BP = DP. ∠CAD = D

35

o

B

C

A

100

P

32 cm 16 2 cm 16 cm 8 2 cm 4 cm

83 22.

A

C

58o 86o 88o 92o 142o

83 25.

A o

30

o

20 25o 30o 35o 40o

Q

P

B

83 23.

70 P

o

C

In the figure, the three sides of ∆ABC touch the circle at the points P, Q and R. ∠PQR =

The sum of the six marked angles in the figure is A. B. C. D. E.

360o . 540o . 600o . 720o . 900o .

83-CE-MATHS II

A. B. C. D. E.

30o 50o 55o 70o 75o

83 If the line 2x − 3y + c = 0 passes 26. through the point (1, 1), then c = A. B. C. D. E.

−2 −1 0 1 2

83 The equation of the line passing 27. through (1, −1) and perpendicular to the x-axis is A. B. C. D. E.

x−1=0. x+1=0. y−1=0. y+1=0. x+y=0.

E.

83 There are 12 boys and 8 girls in a class. 31. 1 1 of the boys and of the girls wear 4 4 glasses. What is the probability that a student chosen at random from the class is a boy not wearing glasses or a girl wearing glasses? A.

83 A circle has its centre at (3, 4) and 28. passes through the origin. Its equation is A. B. C. D. E.

0 a2 + b2 2(a2 + b2) (a − b)2 2(a − b)2

83 30.

C. D. E.

5 20 9 20 11 20 15 20 9 100

83 32.

P Q Frequency

A. B. C. D. E.

B.

x2 + y2 − 6x − 8y + 25 = 0 x2 + y2 − 3x − 4y = 0 x2 + y2 − 6x − 8y = 0 x2 + y2 + 6x + 8y = 0 x2 + y2 − 6x − 8y + 25 = 0

83 If d is the distance between the point 29. (a, b) and (b, a), then d2

5 hours

R

Other Activities Sleeping 120

Height

45

o

90

o

o

Watching Television Playing

Studying

The pie chart shows how a boy spends the 24 hours of a day. If the boy spends 4 hours playing, how much time does he spend watching television? A. B. C. D.

1 hour 2 hours 3 hours 4 hours

83-CE-MATHS II

In the figure, P, Q and R are curves showing the frequency distributions of the heights of students in three schools, each having the same number of students. Which distribution has the greatest standard deviation and which the smallest? A. B. C. D. E.

Greatest P P Q R R

83 1 1 33. If x + x = 2 + 2 , then x =

Smallest Q R R P Q

A. B. C. D. E.

B. C. D. E.

2 only −2 only 1 only 2 −2 or 2 1 or 2 2

83 A function f(x) is called an even 37. function if f(x) = f(−x). Which of the following function is/are even functions?

83 12 − x − x2 < 0 is equivalent to 34. A. B. C. D. E. 83 35.

I.

1 x II. f2(x) = x2 III. f3(x) = x3

x < −4 . x>3. −4 < x < 3 . x < −3 or x > 4 . x < −4 or x > 3 .

A. B. C. D. E.

y y = mx + c

0

x

In the figure, the equation of the straight line is y = mx + c. Which one of the following is true? A. B. C. D. E.

m > 0 and c > 0 m > 0 and c < 0 m < 0 and c > 0 m < 0 and c < 0 m > 0 and c = 0

83 If a and b are positive numbers, which 36. of the following is/are true? I. II.

log10(a + b) = log10a + log10b a log10 = log10a − log10b b III. log10 a a = log10 b b A.

I only

83-CE-MATHS II

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

f1(x) =

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

83 In an arithmetic progression, the first 38. term is 3 and the common difference is 2. If the sum of the first n terms of the arithmetic progression is 143 then n = A. B. C. D. E.

10 11 12 13 14

83 Three positive numbers a, b and c are in 39. geometric progression. Which of the following are true? I.

1 1 1 , , are in geometric a b c progression II. a2, b2, c2 are in geometric progression. III. log10a, log10b, log10c are in arithmetic progression. A. B. C. D. E.

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

83 The scale of a map is 1 : 20 000. On 40. the map the area of a farm is 2 cm2. The actual area of the farm is A. B. C. D. E.

83 44.

400 m2 . 800 m2 . 40 000 m2 . 80 000 m2 . 8 000 000 m2 .

C. D. E.

A. B. C. D. E.

A. B. C. D. E.

C

q

In the figure, ABCD is a trapezium in which AB // DC and ∠C = ∠D = θ. If CD = p and AB = q, then the area of the trapezium is A.

1 (p + q)2 tanθ . 2 1 2 (p + q2) tanθ . 4 1 2 (p − q2) tanθ . 2 1 2 (p − q2) tanθ . 4 ( p2 − q2 ) 4 tan θ

B. C. D. E.

83 A solid sphere is cut into two 45. hemispheres. The percentage increase in the total surface area is

he gained $100 . he gained $50 . he lost $100 . he lost $50 . he lost $48 .

83 Three number are in the ratio 2 : 3 : 5. 43. The ratio of their average to the largest of the three numbers is

θ

D

10% . 2 16 % . 3 25% . 1 33 % . 3 40% .

83 A merchant sold 2 articles each at 42. $1000. For first article, he gained 25% on the cost price. For the second article, he lost 20% on the cost price. Altogether

B

θ

83 It took Paul 40 minutes to walk from 41. Town A to Town B. If the return journey took him 30 minutes, the percentage increase in his speed was A. B.

p

A

A. B.

25% . 1 33 % . 3 50% . 75% . 100% .

C. D. E. 83 46.

A

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

50 20

o

a

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

83-CE-MATHS II

a sin20o a sin 20o sin 70o

o

C

C.

83 50.

a sin 20o sin 50o a sin 50o sin 20o a sin 50o sin 70o

D. E.

y 3 2 1 0

83 47.

-1

C q

θ A

(p + q) sin θ . (p + q) cos θ .

Y

D. E.

p cos θ + q sin θ . p sin θ + q cos θ .

p + q sin θ . 2

A.

2

B. C. D. E.

2. 3. 4. 5. 6.

83 The maximum value of cos23x is 49. A. B. C. D. E.

1. 2. 3. 6. 9.

83-CE-MATHS II

x

xo . 2 y = tan xo . y = tan 2xo . y = tan(x − 90)o . y = tan(x + 90)o . y = tan

83 51.

A

B

C

83 If 0o ≤ θ < 360o, the number of roots 48. of the equation 4 sin2θ cosθ = cosθ is A. B. C. D. E.

360

The figure above shows the graph of a tangent function form 0o to 360o. The function is

In the figure, ABCD is a rectangle. AB = p and BC = q. If ∠BAY = θ, the distance of C from the XAY is A. B. C.

270

-3

p

X

180

-2 B

D

90

D In the figure, ABCD is a quadrilateral with AB = BC and AD = DC. Which of the following is/ar true? I. ∠BAD = ∠BCD II. AC ⊥ BD III. BD bisect AC A. B. C. D. E.

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

83 52.

83 54.

A

A

P

60

Q

o

X C B

C

Y

In the figure, X and Y are points on AB and BC respectively such that AX : XB = 3 : 2 and BY : YC = 4 : 3. If the area of ∆ABC = 70, then the area of ∆AXY = A. B. C. D. E. 83 53.

16 24 30 40 42

124

B

o

E

A

D In the figure, chords AB and CD intersect at E. The length of the minor arc BD is three times the length of the minor arc AC. ∠BAD = 31o 35o 42o 45o 56o

83-CE-MATHS II

B

Y

In the figure, PQ and XY touch the circle at A and B respectively. PQ // XY and ∠QAC = 60o. ∠CBX = A. B. C. D. E.

C

A. B. C. D. E.

X

150o 135o 120o 110o 100o