Mathematics 1991 Paper 2

Form 5

HKCEE 1991 Mathematics II 91 1.

(a2a)(3a4a) A. B. C. D. E.

91 2.

3a6a (3a)6a 3a8a 4a6a (34a)(a6a)

1 1 = 2 − (1 + x) 2 1− x A.

91 5.

2 (1 − x )(1 + x 2 )

C.

(1 − y 2 ) . m(1 + y 2 )

D.

m( y 2 − 1) . ( y 2 + 1)

E.

( y 2 − 1) . m( y 2 + 1)

1 + x2 1 + x

1 y2 = 1 y

2

B.

2x2 (1 − x 2 )(1 + x 2 )

C.

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

D. E.

91 3.

Which one of the following is a factor of x3 − 4x2 + x + 6? A. B. C. D. E.

91 4.

If y = A. B.

(x + 1)(x − 2) (x + 1)(x + 2) (x − 1)(x + 2) (x − 1)(x − 3) (x − 1)(x + 3) 1 + mx , then x = 1 − mx m( y − 1) . y +1 y −1 . m( y + 1)

91-CE-MATHS II

A. B. C. D. E.

91 6.

+ + + − −

1 y2 1 xy 2 xy 2 xy 1 xy

1 y2 1 + 2 y 1 + 2 y 1 + 2 y +

The L.C.M. of x, 2x2, 3x3, 4x4, 5x5 is A. B. C. D. E.

91 7.

1 x2 1 x2 1 x2 1 x2 1 x2

x. 5x5 . 60x5 . 120x5 . 120x15 .

In which of the following cases the equation f(x) = 0 cannot be solved by the method of bisection?

A.

91 8.

y y = f(

x)

B.

A. B. C. D. E.

x

0

91 9.

y y = f(

x)

Solve the following equations : x−1=y+2=x+y−5

1 and partly as x. x When x = 1, y = 5 and when x = 4, 25 y= . Find y when x = 2 . 2 Let y vary partly as

A. x

0 C.

y

y = f(

B. C.

x) D. E. x

0

D.

91 10. y y = f(

x

0

91 11. E.

y y = f( 0

91-CE-MATHS II

x) x

5 2 4 25 4 7 17 2

1 1 : = 2 : 3 and a : c = 4 : 1, then a b a:b:c= If

A. B. C. D. E.

x)

x = 1, y = −2 x = 1, y = 4 x = 4, y = 1 x = 7, y = −2 x = 7, y = 4

12 : 8 : 3 . 8:3:2. 4:6:1. 2:3:8. 2:3:4.

A blanket loses 10% of its length and 8% of its width after washing. The percentage loss in area is A. B. C. D. E.

18.8% . 18% . 17.2% . 9% . 8% .

91 12.

D

C. D. E.

C Q

P

6−π 4−π 2(4 − π )

91 14. M

b

A

N B

a

An equilateral triangle and a square have equal perimeters. Area of the triangle = Area of the square

In the figure, ABCD is a square of side a and MNPQ is a square of side b. The four trapeziums are identical. The area of the shaded region is A. B. C. D. E.

3b + a 4 2 3b − a 2 2 2 5b + a 2 4 2 5b − a 2 4 ( a − b) 2 4 2

A.

2

.

B.

. C. . D. . E. 2

+b .

91 13.

91 15. T

P

A

O

12 − π 8−π

91-CE-MATHS II

A man borrows $10 000 from a bank at 12% per annum compounded monthly. He repays the bank $2000 at the end of each month. How much does he still owe the bank just after the second repayment? A. B. C. D. E.

B

In the figure, TB touches the semi-circle at B. TA cuts the semi-circle at P such that TP = PA. If the radius of the semicircle is 2, find the area of the shaded region. A. B.

9 3 . 16 3 . 4 3 . 3 4 3 . 9 1.

91 16.

$6181 $6200 $6201 $8304 $8400

 1   cosθ + tan θ  (1 − sin θ ) =   A. B. C. D. E.

sin θ cos θ cos2 θ 1 + sin θ sin θ tan θ

91 17.

A. B. C. D. E.

91 18.

91 20.

sin(θ − 90o ) = tan(θ + 180o )

C

In the figure, ∠A = 30o and ∠B = 120o. The ratio of the altitudes of the triangle ABC from A and from B is A. B. C. D. E.

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

91 21. a

91 19.

Y A

A o

X

b

45

2

o

A. B. C. D. E.

Z

91 22.

b 2b 180o − b 360o − b 360o − 2b A B O 70 E

91-CE-MATHS II

c

In the figure, O is the centre of the circle. Find a + c.

In the figure, XPY and YQZ are semicircles with areas A1 and A2 respectively. ∠YXZ = 60o and ∠YZX = 45o. The ratio A1 : A2 = 2: 3 . 2:3. 2:3. 2: 3 . 3: 2 .

O

Q

1

60

A. B. C. D. E.

o

30

A

1 2 3 4 5

P

o

120

cos θ −cos θ cos 2 θ sin θ cos 2 θ − sin θ 1 sin θ

For 0 ≤ θ < 2π, how many roots does the equation tan θ + 2 sin θ = 0 have? A. B. C. D. E.

B

o

D

C

In the figure, O is the centre of the circle BCD. ABC and EDC are straight lines. BC = DC and ∠AED = 70o. Find ∠BOD. 40o 70o 80o 90o 140o

A. B. C. D. E. 91 23.

A. B. C. D. E.

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

91 25.

A E

C

F

B

X

C

B D

G

Y A E

D

Z

In the figure, ABCDE and ABXYZ are two identical regular pentagons. Find ∠AEZ. 15o 18o 24o 30o 36o

A. B. C. D. E.

Q

T

R

P

A

In the figure, TPA and TQB are tangents to the circle at P and Q respectively. If PQ = PR, which of the following must be true? I. ∠APR = ∠QRP II. ∠QTP = ∠QPR III. ∠QPR = ∠APR

3 4 6 8 12

91 The circle x2 + y2 + 4x + ky + 4 = 0 26. passes through the point (1, 3). The radius of the circle is

B

91-CE-MATHS II

In the figure, E and F are the midpoints of AB and AC respectively. G and H divide DB and DC respectively in the ratio 1 : 3. If EF = 12, find GH. A. B. C. D. E.

91 24.

H

A. B. C. D. E.

68 . 48 . 17 . 6. 3.

91 Let A and B be the points (4, −7) and (− 27. 6, 5) respectively. The equation of the line passing through the mid-point of AB and perpendicular to 3x − 4y + 14 = 0 is A. B.

3x − 4y − 1 = 0 . 3x + 4y + 7 = 0 .

C. D. E.

91 31.

4x − 3y + 1 = 0 . 4x + 3y − 7 = 0 . 4x + 3y + 7 = 0 .

P

91 PQRS is a parallelogram with vertices P 28. = (0, 0), Q = (a, b) and S = (−b, a). Find R. A. B. C. D. E.

(−a, −b) (a, −b) (a − b, a − b) (a − b, a + b) (a + b, a + b)

91 29.

I. The mean of P < the mean of Q. II. The mode of P > the mode of Q. III. The inter-quartile range of P < the inter-quartile range of Q. B

35

o

o

75

E

O

In the figure, A and B are the positions of two boats. The bearing of B from A is A. B. C. D. E.

x The graph shows the frequency curves of two symmetric distributions P and Q. Which of the following is /are true?

N

A

A. B. C. D. E.

A.

C. D.

91 The mean and standard deviation of a 30. distribution of test scores are m and s respectively. If 4 marks are added to each score of the distribution, what are the mean and standard deviation of the new distribution?

A. B. C. D. E.

m+4 m+4 m+4 m m

91-CE-MATHS II

Standard Deviation s s+2 s+4 s+2 s+4

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

91 A fair die is thrown 3 times. The 32. probability that “6” occurs exactly once is

B.

N55oE . N70oE . N20oE . S35oE . S75oE .

Mean

Q

E.

1 . 3 3 1   . 6 1 1 × . 3 6 2  1  5     .  6  6  2

 1  5  3    .  6  6 

91 If ( 3 + 1) x = 2, then x = 33. A. B. C. D. E.

2− 3 . 3−1. 1. 2(2 − 3 ) . 4− 3 .

91 34.

If log x : log y = m : n, then x = A. B. C. D. E.

my . n (m − n)y . m−n+y.

91 Which one of the following shaded 38. regions represents the solution of 2 ≤ x + y ≤ 6   0≤x≤4 ?  0≤ y≤4  A.

m

yn . m log y . n

6 4 2

91 1 1 35. If f(x) = x − x , then f(x) − f  x  =   A. B. C. D. E.

0. 2x . 2 − . x 1  2 x −  . x  1  2 − x . x 

91 If p(x2 − x) + q(x2 + x) = 4x2 + 8x, find 36. p and q. A. B. C. D. E.

A. B. C. D. E.

0 B.

x+y x−y y−x xy y x

4

6

2

4

6

2

4

6

2

4

6

x

y

4 2 0 C.

x

y 6 4 2 0

D.

x

y 6 4 2 0

91-CE-MATHS II

2

6

p = 4, q = 8 p = −8, q = 4 p = −2, q = 6 p = 2, q = 6 p = 6, q = −2

91 If x < 0 < y, then which one of the 37. following must be positive?

y

x

E.

y

91 P sold an article to Q at a profit of 43. 25%. Q sold it to R also at a profit of 25%. If Q gained $500, how much did P gain?

6 4 2 0

2

4

6

A. B. C. D. E.

x

91 If (x − 2)(x − 3) = (a − 2)(a − 3), solve 39. for x. A. B. C. D. E.

x = 0 or 5 x = 2 or 3 x = a or 2 x = a or 3 x = a or 5 − a

91 44.

From a rectangular metal sheet of width 3 cm and length 40 cm, at most how many circles each of radius 1 cm can be cut?

I. x + 3, y + 3, z + 3 are in G.P. II. 3x, 3y, 3z are in G.P. III. x2, y2, z2 are in G.P. A. B. C. D. E.

A. B. C. D. E.

16 18 54 70 It cannot be found.

91 If x, y, z are in G.P, which of the 41. following must be true?

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

0.3 kg 0.6 kg 0.75 kg 1.5 kg

91-CE-MATHS II

20 21 22 23 24

DIRECTIONS: Question 45 and 46 refer to the figure below, which shows a cuboid ABCDEFGH with AE = 2a, EF = 2b and FG = 2c. AC and BD intersect at X. 91 45.

D

C

X

A

2a

91 3 kg of a solution contains 40% of 42. alcohol by weight. How much alcohol should be added to obtain a solution containing 50% of alcohol by weight? A. B. C. D.

$250 $320 $333 $400 $500

40 cm

91 If the sum to n terms of an A.P. is 40. n2 + 3n, find the 7th term of the A.P. A. B. C. D. E.

3.75 kg

3 cm

E.

B

H

G 2c

E

2b

XE = A.

a 2 + b2 + c2

F

B.

a 2 + b 2 + ( 2c) 2

C.

a 2 + (2b) 2 + c 2

D.

(2a ) 2 + b 2 + c 2

E.

A. B. C. D. E.

2 a 2 + b2 + c2

91 If the angle between XE and the plane 46. EFGH is θ, then tan θ = A.

91 49.

P

a . b 2a . b

B. C.

45 30

E.

In the figure, the height of the vertical pole PO is A. B. C. D. E.

π 3π + cos π + cos + cos 2π + …+ 2 2 cos 10π = 0 1 −1 10 −10

91 48.

East

O

o

South

cos

A. B. C. D. E.

o

30 m

( 2a ) 2 + c 2 . b a . b2 + c2 2a . 2 b + c2

D.

91 47.

y = 2 cos x . y = 2 − sin x . y = 2 + sin x . y = 2 − cos x . y = 2 + cos x .

91 50.

7.5 m . 15 m . 15 2 m . 15 3 m . 45 m . A B 30

y 45

4

D

3

10 cm

o

75

o

o

C

In the figure, find the length of AB, correct to the nearest cm.

2 1 0

π 2

π

3π 2

2π

x

The figure shows the graph of the function

91-CE-MATHS II

A. B. C. D. E.

14 cm 15 cm 16 cm 17 cm 18 cm

91 51.

A D

E 95

o

In the figure, M is the mid-point of BC and AD = 2DB. AM and CD intersect area of ∆ADK at K. Find . area of ∆AKC A.

45

B

o

B.

C

In the figure, ABC and CDE are equilateral triangles. Find ∠ADE .

C. D.

A. B. C. D. E.

o

15 35o 40o 45o 50o

E.

1 2 2 3 3 4 4 5 1

91 In the figure, which of the pairs of 54. triangles must be congruent?

91 52.

E

I.

15

16 o

50

16

D

13

15

30o 36o 60o 72o 120o

13 o

50

III.

60 50

o

60

o

16

91 53.

A. B. C. D. E.

A

D K B

91-CE-MATHS II

M

o

15

In the figure, arc AB : arc BC : arc CD : arc DE : arc EA = 1 : 2 : 3 : 4 : 5. Find θ. A. B. C. D. E.

50

15

C

B

o

II.

θ

A

50

C

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

50

o

16

o