Mathematics 1994 Paper 2

Form 5

HKCEE 1994 Mathematics II 94 1.

If f(x) = x2 + 2x, then f(x − 1) = A. B. C. D. E.

94 2.

x2 . x2 − 1 . x2 + 2x − 1 . x2 + 2x − 3 . x2 + 4x − 1 .

1+ 3y 2 1+ 2y 2+ y 1+ 2y 2− y 1− 2y 2+ y 1− 2y 2− y

B. C. D. E.

94 4.

A. B. C. D. E.

0

1

2

3

4

5

x

In the figure, (x, y) is a point in the shaded region (including the boundary) and x, y are integers. Find the greatest value of 3x + y .

.

A. B. C. D. E.

. . .

94 6.

2 , then a −

0. 2 2 . 2 3 . 3 − 2 . 2 3 2 . + 3 2

94-CE-MATHS II

x - 2 y +3=0

1

.

3 +

5

2

x−1. (x − 1)4(x + 1)(x2 + x + 1) . (x − 1)2(x + 1)(x2 + x + 1) . (x − 1)2(x + 1)(x2 − x + 1) . (x − 1)(x + 1)(x2 + x + 1) .

If a =

2 x + y -7=0

3

The L.C.M. of (x − 1)2, x2 − 1 and x3 − is A. B. C. D. E.

y 4

2x − 1 , then x = x+2

If y = A.

94 3.

94 5.

1 = a

If x(x + 1) < 5(x + 1), then A. B. C. D. E.

94 7.

7 8 9.2 10 10.5

x<5 x < −5 or x > 1 . x < −5 or x > 1 . −5 < x < 1 . −1 < x < 5 .

Which of the following is/are an identity/identities? I. (x + 2)(x − 2) = x2 − 4 II. (x + 2)(x − 2) = 0 III. (x + 2)3 = x3 + 8 A. B. C. D. E.

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

94 8.

3α 2 − hα − b = 0 If α ≠ β and  2 , 3β − hβ − b = 0

94 12.

then α + β = A. B. C. D. E.

94 9.

94 10.

A.

b . 3 h. h − . 3 h . 3

B. C. D. E.

1 . 3

1 . 3 4 − . 3 4 . 3 4 − . 9

$ 94 400 $ 100 700 $ 104 900 $ 115 100 $ 116 600

3h

h In the figure, the paper cup in the form of a circular cone contains 10 ml of water. How many ml of water must be added to fill up the paper cup?

$ 2 304 $ 2 400 $ 2 880 $ 3 000 $ 3 456

The bearing of A from B is 075o. What is the bearing of B from A? A. B. C. D. E.

−

94 13.

A wholesaler sells an article to a retailer at a profit of 20 %. The retailer sells it to a customer for $ 3 600 at a profit of $ 720. Find the original cost of the article to the wholesaler. A. B. C. D. E.

94 11.

b . 3

−

Mr. Chan bought a car for $ 143 900. If the value of the car goes down by 10% each year, find its value at the end of the third year. (Give your answer correct to the nearest hundred dollars.) A. B. C. D. E.

81 and 4 its second term is −9, the common ratio is If the sum to infinity of a G.P. is

o

015 075o 105o 195o 255o

94-CE-MATHS II

A. B. C. D. E. 94 14.

20 80 90 260 270 A

B

D

C

In the figure, ABCD is a rectangular field of length p metres and width q meters. The path around the field is of width 2 metres. Find the area of the path. A. B. C. D. E.

(4p + 4q) m2 (2p + 2q + 4) m2 (2p + 2q + 16) m2 (4p + 4q + 16) m2 (pq + 4p + 4q + 16) m2

94 15.

C. D. E. 94 17.

0 2tan θ −2tan θ

Which of the following figures shows the graph of y = 1 + sin x A.

y 2

A 0

r O

B.

π 3

C

0

C. D. E.

94 16.

C. 2

π 3 2  − r 6  4   π 1 2  −  r  6 4 π 3 2  −  r 3 2   π 1 2  −  r  3 2 π 3 2 r− r 3 4

B.

2 cosθ 2 − cosθ

94-CE-MATHS II

x

y

0 D.

2π

x

y 0

2π

x

-2 E.

y 1 0

cosθ cosθ − = sin θ + 1 sin θ − 1 A.

2π

-2

In the figure, OACB is a sector of π radius r. If ∠AOB = , find the area 3 of the shaded part.

B.

x

y

B

A.

2π

-1 94 18.

sin(180o + θ ) = cos(90o − θ ) A.

tan θ

2π

x

−tan θ 1 tan θ 1 −1

B. C. D. E.

B.

5 6 5 3 3 3 5 5 3 9

C. D.

94 19.

D

E. o

120

C

A

94 21.

2

C

5

3

B

π 6

In the figure, ABCD is a cyclic quadrilateral with AB = 5, BC = 2 and ∠ADC = 120o. Find AC A. B. C. D. E.

O

A

In the figure, O is the centre of the π circle. If AC = 3 and ∠BAC = , find 6 the diameter AB.

19 21 6 34 39

A.

94 20.

P

6 30

3 2 6 3 3 2 2 3 3 3

B. C. D. E.

5

A

B

C

o

94 22.

B

θ C

B 28

In the figure, PC is a vertical pole standing on the horizontal plane ABC. If ∠ABC = 90o, ∠BAC = 30o, AC = 6 and PC = 5, find tan θ. A.

3 5

94-CE-MATHS II

A

o

x

P

In the figure, PA is tangent to the circle at A, ∠CAP = 28o and BA = BC. Find x. A. B.

28o 48o

C. D. E.

56o 62o 76o

94 25.

94 23.

A

x

A

B

30

z

o

y

C O 25

In the figure, x, y and z are the exterior angles of ∆ABC. if x : y : z = 4 : 5 : 6, then ∠BAC =

o

B

C

A. B. C. D. E.

In the figure, O is the centre of the inscribed circle of ∆ABC. If ∠OAC = 30o and ∠OCA = 25o, find ∠ABC. A. B. C. D. E.

50o 55o 60o 62.5o 70o

94 24.

48o . 84o . 96o . 120o . 132o .

94 The points A (4, −1), B (−2, 3) and 26. C (x, 5) lie on a straight line. Find x. A. B. C. D. E.

A 80

o

65

B

o

D

94 27.

−5 −4 0 2 5 y

C In the figure, AB = AD and BC = CD. If ∠BAD = 80o and ∠ADC = 65o, then ∠BCD = A. B. C. D. E.

100 . 130o . 145o . 150o . 160o .

x =3

x + y =5

In the figure, the shaded part is bounded by the axes, the line x = 3 and x + y = 5. Find the area. A. B. C. D. E.

94-CE-MATHS II

x

0

o

10.5 12 15 19.5 21

94 AB is a diameter of the circle 28. x2 + y2 − 2x − 2y − 18 = 0. If A is (3, 5), then B is A. B. C. D. E.

(2, 3) . (1, −1) . (−1, −3) . (−5, −7) . (−7, −9) .

C. D. E.

94 A box contains 5 eggs, 2 of which are 31. rotten. If 2 eggs are chosen at random, find the probability that exactly one of them is rotten. A.

94 The equations of two circles are 29. x2 + y2 − 4x − 6y = 0, x2 + y2 + 4x + 6y = 0, Which of the following is/are true? The two circles have the same centre. II. The two circles have equal radii. III. The two circles pass through the origin.

B. C. D.

I.

A. B. C. D. E.

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

$720 $1 800 $12 000

E.

2 5 3 5 3 10 6 25 12 25

94 The mean, standard deviation and 32. interquartile range of n numbers are m, s and q respectively. If 3 is added to each of the n numbers, what will be their new mean, standard deviation and interquartile range? Mean

94 30.

30 % Food

Entertainment 10 % Clothing

40 % Rent 15 % Other

In the figure, the pie chart shows the monthly expenditure of a family. If the family spends $ 4 800 monthly on rent, what is the monthly expenditure on entertainment? A. B.

$240 $600

94-CE-MATHS II

A. B. C. D. E.

m m m+3 m+3 m+3

Standard Deviation

s s+3 s s s+3

Interquartil e Range

q q+3 q q+3 q+3

94 (3x)2 = 33. A. B. C. D. E.

2

3( x ) 3x + 2 32x 6x 92x

94 If log 2 = a and log 9 = b, then log 12 = 34. A.

2a +

b . 3

B. C.

b . 2 2 2 a+ b . 3 3

2a +

D.

a2 + b .

E.

a2 b 2

94 38.

y y=x

94 36.

y=

D. E.

(a − b)(a − b − 1) (a − b)(a − b + 1) (a − b)(a + b − 1) (a + b)(a − b − 1) (a − b − 1)2

2y − x 2y + x 1 2y − x 1 2y + x 1 4y − x

94 P(x) is a polynomial. When P(x) is 37. divided by (5x − 2), the remainder is R. If P(x) is divided by (2 − 5x), then the remainder is A. B. C. D. E.

mx + k

1

2 1 − x y = 4y x − x y A. B. C.

+ bx + c

1 2

94 Factorize a2 − 2ab + b2 − a + b. 35. A. B. C. D. E.

2

R. −R . 2 R. 5 2 . 5 2 − . 5

94-CE-MATHS II

0

α

β

x

In the figure, the line y = mx + k cuts the curve y = x2 + bx + c at x = α and x = β. Find the value of αβ A. B. C. D. E.

−b c m−b k−c c−k

94 If x = 3, y = 2 satisfy the simultaneous 39. ax + by = 2 equations  , find the values bx − ay = 3 of a and b A. B. C. D. E.

a = 0, b = 1 a = 0, b = −1 5 1 a= ,b= − 6 4 1 37 a= − ,b= 13 39 12 5 a= − ,b= 13 13

94 From the table, which of the following 40. intervals must contain a root of f(x) − x = 0 x −2 −1 0 1 2

f(x) 1.2 0.8 0.7 0.2 −0.1

3 A. B. C. D. E.

0.8 A. B. C. D. E.

−2 < x < −1 −1 < x < 0 0<x<1 1<x<2 2<x<3

94 If the product of the first n terms of the 41. sequence 10, 102, 103, … , 10n, … exceeds 1055, find the minimum value of n. A. B. C. D. E.

94 45.

B

D

9 10 11 12 56

A

C

E

In the figure, AD : DB = 1 : 2, AE : EC = 3 : 2. Area of ∆BDE : Area of ∆ABC =

94 If a : b = 2 : 3, a : c = 3 : 4 and 42. a : d = 4 : 5, then b : c : d = A. B. C. D. E.

9 10 11.25 12 12.5

2:3:4. 3:4:5. 3 : 6 : 10 . 18 : 16 : 15 . 40 : 45 : 48 .

A. B. C. D. E.

1:3 2:5 3:4 4 : 25 36 : 65

94 46.

A

94 Let x vary inversely as y . If y is 43. increased by 69%, then x will be A. B. C. D. E.

increased by 23.1% (3 sig. fig.). increased by 30%. decreased by 23.1% (3 sig. fig.). decreased by 30%. decreased by 76.9% (3 sig. fig.).

94 44.

E

P

Q

D

B F

E

A

D

C

In the figure, CDEF is a sector of a circle which touched AB at E. If AB = 25 and BC = 15, find the radius of the sector.

94-CE-MATHS II

B

C

In the figure, area of ∆ABC : area of square BCDE = 2 : 1. Find PQ : BC. A. B. C. D. E.

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

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94-CE-MATHS II

94 For 0o ≤ x ≤ 360o, how many roots does 47. the equation sin x(cos x + 2) = 0 have? A. B. C. D. E.

A. B. C.

0 1 2 3 4

D. E.

94 The largest value of (3cos2θ − 1)2 + 1 is 48. A. B. C. D. E.

94 51.

2. 5. 17 . 26 . 50 .

8 B

D. E.

A. B. C. D. E.

C

In the figure, sin A : sin B : sin C = 4 : 5 : 6 . If AB = 8, find AC.

C.

D o

22

30

x

o

B

A

In the figure, ABCD is a semi-circle, CDE and BAE are straight lines. If ∠ CBD = 30o and ∠DEA = 22o, find x.

A

B.

C

E

94 49.

A.

p sin θ p cos θ p sin θ cos 2 θ p sin 2 θ cosθ p cos2 θ sin θ

1 3 2 6 3 3 9 5 10 12

38o 41o 44o 52o 60o

94 52.

A

B

5

94 50.

C

D

A. B. C. D. E.

D p

θ

C

In the figure, AB = p, ∠ACB = θ. Find CD

94-CE-MATHS II

O

In the figure, OABCD is a sector of a circle. If arc AB = arc BC = arc CD, then x =

A

B

x

105o . 120o . 135o . 144o . 150o .

94 53.

A

B

D

C

In the figure, AB // DC and ∠DAB = ∠ DBC. Which of the following is/are true? I. II. III.

A. B. C. D. E.

AB BD = BD DC AB AD = BD BC AD BD = BD CD I only II only III only I and II only II and III only

94 54.

D

C 6

M

A

9

N 2 B

In the figure, ABCD is a trapezium with AB // DC, ∠ABC = 90o and MN is the perpendicular bisector of AD. If AB = 9, BN = 2 and NC = 6, find CD. A. B. C. D. E.

1 2 3 6 4 7 41 113 4

94-CE-MATHS II