Mathematics 1989 Paper 2

Form 5

HKCEE 1989 Mathematics II 89 1.

A. B. C. D. E. 89 2.

2

3n −1 2 9 n −1 32n 62n 92n

89 3.

B. C.

5 x + 2 y = 10

(3x − 2)2 9x2 − 4 9x2 + 4 9x2 − 6x + 4 9x2 + 6x + 4

O

D. E.

3

x x x

1 2

B. C. D.

 x + 2y ≥ 6  5 x + 2 y ≥ 10  y≥0 

B.

 x + 2y ≤ 6  5 x + 2 y ≤ 10  x≥0 

C.

 x + 2y ≥ 6  5 x + 2 y ≤ 10  x≥0 

D.

 x + 2y ≤ 6  5 x + 2 y ≥ 10  y≥0 

E.

 x + 2y ≥ 6  5 x + 2 y ≤ 10  y≥0 

1 − 4

If f(x) = A.

A.

1 4

−

3 4

x 1 , then f( ) = 1− x x

1 x −1 1 1− x x x −1 x 1− x

. . . .

89-CE-MATHS II

x

Which of the following systems of inequalities is represented by the shaded region in the figure?

x4 x

y x +2 y =6

x = x A.

1− x . x

89 5.

27 x 3 − 8 = 3x − 2 A. B. C. D. E.

89 4.

E.

3n − 1 × 3n + 1

89 6.

Let f(x) = ax2 − 5 and g(x) = 27x3 − 18x + 4. If both expressions leave the same remainder when divided by 3x + 1, then a =

A. B. C. D. E. 89 7.

B. C. D. E.

x y x y x y x y x y

>−

A. B. C. D. E.

3 . 2

2 . 3 2 < . 3 2 >− . 3 2 <− . 3 >

Given that r is the only real root of x5 + x − 1 = 0, which of the following ranges contains r? A. B. C. D. E.

89 9.

If both 2 and 3 are factors of m, then 6 is also a factor of m. II. If 15 is a factor of n, then both 3 and 5 are factors of n. III. If p is a multiple of both 4 and 6, then p is also a multiple of 24.

If 3x > −2y and y < 0, then A.

89 8.

I.

−74 . 0. 36 . 76 . 126 .

−2 < r < −1 −1 < r < 0 0
89 11.

E

A. B. C.

A. B.

D.

C. D. E.

z2 is a constant xy

89 Which of the following is/are true? 10.

89-CE-MATHS II

D

C

A

B

In the figure, ABCD is a square and AE Area of AED = BE. = Area of ABCD

If z varies inversely as x and directly as y, then xyz is a constant. xz is a constant y yz is a constant x xz 2 is a constant y

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

E.

1 2 3 8 1 3 1 4 1 8

89 12.

A

D

B

89 The least value of 9cos2θ − 6cosθ + 1 is 15. E

C

Aright conical vessel placed on horizontal ground contains some water as shown in the figure. If AD : DB = 2 volume of empty space : 3 , then = volume of water A. B. C. D. E.

A. B. C. D. E.

4 . 9 8 . 19 8 . 27 8 . 117 8 . 125

89 16.

−4 . 0. 1. 4. 16 .

1 1 − = 1 1 −1 +1 cosθ cosθ A. B. C. D. E.

2 tan 2 θ 2 tan θ 2 tan2θ 2 cosθ sin 2 θ 2 cos 2 θ sin θ

89 17.

y y = cos 2

x

89 If A is greater than B by 20% and B is 13. smaller than C by 30%, then A. B. C. D. E.

A is smaller than C by 16% A is smaller than C by 6% A is greater than C by 6% A is greater than C by 10% A is greater than C by 16%

89 At the beginning of a year, a man 14. borrows $1000 from a bank at 5% per annum, compounded yearly. He promises to repay $300 at the end of each year. How much will he still owe the bank just after the second repayment? A. B. C. D. E.

O

$402.5 $450 $487.5 $500 $502.5

89-CE-MATHS II

A

D

B

x

C

The figure shows the graph of y = cos 2x, where 0 ≤ x ≤ π. The area of the rectangle ABCD is A. B. C. D. E.

π . 2 π . 4 π. 3π . 2 2π .

89 Given that 0 ≤ θ ≤180 , how many 18. roots has the equation (sin θ + 1)(tan θ + 3) = 0? o

A. B. C. D. E.

o

D. E.

In the figure, ABCDE is a regular pentagon and ABYE is a rhombus. Find ∠CAY.

D

A. B. C. D. E.

φ

θ x

D

A

C

y

sin φ sin θ cos φ cosθ tan φ tan θ cosθ cos φ tan θ tan φ

89 20.

x y

27o 24o 21o 18o 15o

89 22.

Q

12

P

4

5

3

Referring to the figure, find the length of the line segment joining P and Q. 40

xo

o

xo+ yo 2yo Referring to the figure, find y. A. B. C. D. E.

E

C

In the figure, AD ⊥ BC. Find

C.

B

Y

B

B.

A

0 1 2 3 4

89 19.

A.

89 21.

20 30 40 50 80

89-CE-MATHS II

A. B. C. D. E.

25 10 5 18 8 5 194

89 23.

C

I. median < mean II. range = 3 III. mode = 3

G F

A

D

B

E

O

In the figure O is the centre of two Concentric circles. ADOEB and CGFB are straight lines. Which of the following is/are true? I. AC // DG II. BF = CG III. A, E, F and C are concyclic A. B. C. D. E.

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

89 24.

θ

o

C

In the figure, TC is a tangent to the circle at C and AB // DC. If ∠BCT = 48o, then θ = 48o 72o 84o 90o 96o

89 Referring to the data 1, 1, 1, 1, 1, 2, 2, 25. 2, 3, which of the following is/are true?

89-CE-MATHS II

B.

E.

D

A. B. C. D. E.

A.

D.

48

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

89 A BIASED die is thrown. Suppose the 26. probabilities of getting 1, 2, 3, 4, and 4 1 1 1 1 1 are respectively , , , and . 2 4 8 16 32 What is the probability of getting 6?

C.

B

A

A. B. C. D. E.

1 64 1 36 1 32 1 12 1 6

89 A bag contains 4 red, 3 green and 2 27. white balls. Three men A, B and C each draw one ball in turn from the bag at random without replacement. If A draws first, B second and C third, what is the probability that the balls drawn by B and C are both white? A. B. C. D. E.

1 36 1 28 4 81 25 72 11 28

89 The equation of the straight line 28. perpendicular to 2x + y − 3 = 0 and passing through (1, −1) is A. B. C. D. E.

x + 2y + 1 = 0 . x − 2y − 3 = 0 . −x + 2y − 1 = 0 . 2x + y − 1 = 0 . 2x − y − 3 = 0 .

89 29.

A. B. C. D. E.

7 4 7 2 7 8 14

89 31.

y A

y

C

(3, 4)

B

2θ

ax - 2 y + 5 = 0

x

O x

O

In the figure, C is the centre of the circle x2 + y2 − 6x − 8y + 21 = 0. OA and OB are tangents. If ∠AOB = 2θ, find sinθ.

In the figure, the line ax − 2y + 5 = 0 passes through the point (3, 4). What is the area of the shaded part? A. B. C. D. E.

89 30.

A.

6 25 4 10 12 25 2

B. C.

4 5 3 5

D.

y

E.

A

2 21 2 5

89 32.

C

O

21 5

B

B C

x

In the figure, C is the centre of the circle x2 + y2 − 8x − 7y + 12 = 0. If the circle cuts the x-axis at A and B, find the area of ∆CAB.

A

D

X

Y

W

89-CE-MATHS II

Z

10(b − a ) . 11 a + 9b . 10 a + 10b . 11

C. In the figure, ABCD and WXYZ are sectors of equal radii. If arc BCD : arc XYZ = s : t, then which of the following is/are true? I. II. III.

A. B. C. D. E.

BD s = XZ t area of sector ABCD s = area of sector WXYZ t ∠BAD s = ∠XWZ t I only II only III only I and III only II and III only

89 33.

B A

D. E.

89 1 35. Given that y ∝ x , if x increased by 25%, find the percentage change in y. A. B. C. D. E.

89 The costs of two kinds of coffee A and 36. B are $12/kg and $20/kg respectively. In what ratio by weight should A and B be mixed so that the mixture will cost $15/kg? A. B. C. D. E.

O

In the figure, O is the centre of two concentric circles. AB is tangent to the smaller circle. If AB = 2, find the area of the shaded part. A. B. C. D. E.

π 2 π 2π 4π It cannot be found.

89 If 10 arithmetic means are inserted 34. between a and b, then the last one is A. B.

10a + b . 11 9a + b . 10

89-CE-MATHS II

Decreased by 20% Decreased by 25% Decreased by 80% Increased by 20% Increased by 25%

4:3 5:2 5:3 3:2 5:4

89 37.

A 8

5 E

E B

15 C

In the figure, D and E are points on AB and AC respectively such that ∠ABC = ∠AED, AD = 8, AE = 5 and EC = 15. If the area of ∆ADE is 16, then the area of the quadrilateral BCED is A. B. C. D. E.

200 . 100 . 96 . 84 . 40 .

89 38.

89 41. 20

o

O

40

o

In the figure, O is the centre of the circle of radius 6 cm. The area of the shaded part is A. B. C. D. E.

2π cm2 . 4π cm2 . 6π cm2 . 9π cm2 . 12π cm2 .

89 If the sum to infinity of the G.P. 1, −t, 39. 2 t2, −t3, … is , find the fourth term. 3 A. B. C. D. E.

1 16 1 − 8 1 16 1 8 5 8 −

89 x + 3y 3x + y 40. If 2 x + y = 2, find x + 2 y A. B. C. D. E.

2 3 1 2 1 3 6 7

89-CE-MATHS II

(1 − x 2 ) n + (1 − x ) n = (1 − x) 2 n A.

(1 + x) n + 1 (1 − x) n

B.

2 − x − x2 (1 − x) 2

C.

(1 + x) n + 1 (1 − x ) 2

D.

(1 − x) n + 1 (1 + x) n

E.

2 − x n + x 2n 1 − x 2n

89 log4 2 2 = 42. A. B. C. D. E.

3 8 3 4 1 4 2 2

3 4 3 8

89 If x = a + 1 − a , where a > 0, then 43. 1 x+ x A. B. C. D. E.

2. 2 2 2 2(

a . a +1 . a +1 − a . a +1 + a ) .

89 If p is a root of ax2 + bx + c = 0, which 44. of the following is a root of x−3 2 x−3 a( ) + b( )+c = 0? 2 2 A.

2p + 3

B. C. D. E.

89 45.

89 47.

2p − 3 3 − 2p p+3 2 p−3 2 y

2 cm

(2, 18)

A

(5, 0)

x

y = (x − 2)2 + 18 y = −(x − 2)2 + 18 y = (x + 1)(x − 5) y = −2(x + 1)(x − 5) y = 2(x − 1)(x + 5)

89 If 2sin2θ − sinθ cosθ − cos2θ = 0, the 46. tanθ

B. C. D. E.

1 1 or . 2 1 −1 or . 2 1 1 or − . 2 1 −1 or − . 2 1 or −2 .

89-CE-MATHS II

B

2 cm

In the figure, VABCD is a right pyramid of height 3 cm. The base ABCD is a square of side 2 cm. Let θ be the angle between the face VBC and the base. Find tanθ

In figure shows the graph of a quadratic function y = f(x). Given that the graph has vertex (2, 18) and it cuts the x-axis at (5, 0), find the quadratic function.

A.

C

D

O

A. B. C. D. E.

V

A.

1 3

B.

2 3 3 2 3 2 2 3

C. D. E. 89 48.

10

2x

θ 3x In the figure, if cosθ = value of x. A. B. C. D. E.

2 3 4 5 6

3 , find the 4

89 49.

Ray of sunlight

B R

10 m

S 60

89 51.

Ray of sunlight

o

WALL

E

θ

A

1m

O

45

o

Q

o

124

P

N

In the figure, O is the centre of the smaller circle. OAB and PQR are straight lines. Find θ.

A vertical rectangular wall on the horizontal ground, 1 m high and 10 m long, runs east and west as shown in the figure. If the sun bears S60oE at an elevation of 45o, find the area of the shadow of the wall on the ground. A.

5 2 m 2 5 m2 5 2 m2 5 3 m2 10 m2

B. C. D. E. 89 50.

D

56o 108o 112o 118o 124o

A. B. C. D. E. 89 52.

A

C

56

B

o

D

1

α A

C

β B

In the figure, B is the mid-point of arc AC. AC = AD. If ∠ADC = 56o, then ∠BCD =

In the figure, ABCD is a trapezium with AB // DC. If BC = 1, then AD = A. B. C. D. E.

sin β . sin α sin α . sin β sin α sin β . cos β . cosα cosα . cos β

84o . 90o . 96o . 112o . 124o .

A. B. C. D. E. 89 53.

C P

Q A

89-CE-MATHS II

F

D

E

B

In the figure, ABCD is a parallelogram. E and F are the mid-points of AB and DC respectively. BF and ED cut AC at P and Q respectively. If the area of ABCD is 48, find the area of the shaded part. A. B. C. D. E. 89 54.

6 8 9.6 12 16 A

D 2

12 cm 7 cm

2

O 1 cm

B

2

C

In the figure, AC cuts BD at O. The areas of ∆AOB, ∆AOD and ∆BOC are 7 cm2, 12 cm2 and 10.5 cm2 respectively. Find the area of ∆OCD. A. B. C. D. E.

5.5 cm2 8 cm2 8.5 cm2 15.5 cm2 18 cm2

89-CE-MATHS II