Form 5 HKCEE 1992 Mathematics II 92 1. A. B. C. D. E. 92 2. B. C. D. E. C. D. E. 92 5. 1 , then b = 1 b E. 1 1 . 1 a 1 1 . 1 a 1 1+ . 1 a 1 1+ . 1 a 1 1 + . 1 a 92 6. 1 only 2 only Any value Any value except 1 Any value except 2 92-CE-MATHS II 1 log10a, then b = 2 10 a . 10 + a . 5a . a . 2 a 1+ . 2 5b a 5b + a ab 13b 5a 13a 5b a c k and a, b, c, d are positive, b d then which of the following must be true? B. C. D. E. 1 1 5 2 If A. 5 1 5 1 5 1 5 1 5 Which of the following is a factor of 4(a + b)2 9(a b)2 ? A. B. C. D. E. 92 7. 0 1 2 3 If log10 b = 1 + A. B. C. D. For what value(s) of x does the equality ( x 1)( x 2) = x + 1 hold? x2 A. B. C. D. E. 92 4. ab ab ab ab 1 ab 2 ab 1 ab If a = 1 A. 92 3. A. B. 1 1 a b ac k bd ab = cd = k ac = bd = k a=c=k ac k bd C. D. E. A. B. C. D. E. 92 10. A. B. C. D. E. nn 2 n n2 n2 n 2 1 $16 $400 $416 $800 $816 92 13. 1m 1.2 m If a and b are greater than 1, which of the following statements is/are true? I. II. III. A sum of $ 10 000 is deposited at 4% p.a., compounded yearly. Find the interest earned in the second year. 0.3 m 1.8 m The figure shows a solid platform with steps on one side and a slope on the other. Find its volume. ab a b 1 (a + b1)1 = a + b a2b3 = (ab)6 A. B. C. D. E. I only II only III only I and II only None of them 0.75 m3 0.84 m3 0.858 m3 1.008 m3 1.608 m3 92 14. If a : b = 2 : 3, a : c = 3 : 4 and b : d = 5 : 2, find c : d. P A. B. C. D. E. 92 11. 1:5 16 : 45 10 : 3 20 : 9 5:1 O Q T In the figure, TP and TQ are tangent to the circle of radius 3 cm. Find the length of the minor arc PQ. 15.2% 20% 50% 72.8% 80% A. B. C. D. E. 92-CE-MATHS II 3 cm 60o Suppose x varies directly as y2 and inversely as z. Find the percentage increase of x when y is increased by 20% and z is decreased by 20%. A. B. C. D. E. 0.2 m A. B. 92 9. 92 12. ntimes n n n Simplify . n n n n terms 0.3 m 92 8. 2 3 cm 2 cm 3 cm 2 cm cm 2 92 15. A 92 17. E h F B 3h 2h D H Find the ratio of the volume of the tetrahedron ACHD to the volume of the cube ABCDEFGH in the figure. In the figure, a cone of height 3h is cut by a plane parallel to its base into a smaller cone of height h and a frustum. Find the ratio of the volume of the smaller cone to the volume of the frustum. A. B. C. D. E. A. B. C. D. E. C G 1:8 1:6 1:4 1:3 1:2 92 16. 92 18. A E 4 cm 4c m C O E 2 3 In the figure, the equilateral triangle ACE of side 4 cm is inscribed in the circle. Find the area of the inscribed regular hexagon ABCDEF. 4 In the figure, find cos . 8 3 cm2 8 2 cm2 4 3 cm2 4 2 cm2 16 cm2 A. B. C. D. 92-CE-MATHS II 5. 3. 1. 0. 1 . 92 19. D A. B. C. D. E. The greatest value of 1 2sin is A. B. C. D. E. 4c m B 1 : 27 1 : 26 1:9 1:8 1:7 3 1 4 11 16 3 4 7 8 E. 92 20. 77 9 In which two quadrants will the solution(s) of sin cos < 0 lie? A. B. C. D. E. In quadrants I and II only In quadrants I and III only In quadrants II and III only In quadrants II and IV only In quadrants III and IV only I only II only III only I and II only II and III only 92 24. 36o 92 22. o If A + B + C = 180o, then 1 + cos A cos (B + C) = A. B. C. D. E. 0. sin2 A . 1 + cos2 A . 1 + sinA cos A . 1 sinA cos A . In the figure, O is the centre of the circle. Find . 92 25. 2 42o 36o 24o 21o 18o A. B. C. D. E. y O 6 A x 6 E 2 5 The figure shows the graph of the function 92 23. tan(x + ) . tan(x ) . tan x . + tan x . tan x . 2 cos2 sin2 = 1 2 cos2 sin2 = 2 2 cos2 sin2 = 1 92-CE-MATHS II C In the figure, ABCD is a square with side 6. If BE = CE = 5, find AE. Which of the following equations has/have solutions? I. II. III. B 5 D A. B. C. D. E. O 42 92 21. A. B. C. D. E. 4 A. 61 B. C. D. E. 9 10 6 3 109 92 26. C. D. E. R 92 29. Q P In the figure, the circle is inscribed in a regular pentagon. P, Q and R are points of contact. Find . 30o 32o 35o 36o 45o A. B. C. D. E. 92 27. 92 30. S If 0 < k < h, which of the following circles intersect(s) the y-axis? I. II. III. (x h)2 + (y k)2 = k2 (x h)2 + (y k)2 = h2 (x h)2 + (y k)2 = h2 + k2 A. B. C. D. E. I only II only III only I and II only II and III only If the line y = mx + 3 divides the circle x2 + y2 4x 2y 5 = 0 into two equal parts, find m. A. C P 3x B. C. D. B A E. 2x D 92 31. T In the figure, ST is a tangent to the smaller circle. ABC is a straight line. If TAD = 2x and DPC = 3x, find x. A. B. C. D. E. 92 28. 92 32. If the two lines 2x y + 1 = 0 and ax + 3y 1 = 0 do not intersect, then a = A. B. (3.5, 3) (3, 2) (3, 1) (1.5, 2) (1, 2) The table shows the mean marks of two classes of students in a Class A Class B 6 . 2 . 92-CE-MATHS II 1 4 1 0 5 4 2 The mid-points of the sides of a triangle are (3, 4), (2, 0) and (4, 2). Which of the following points is a vertex of the triangle? A. B. C. D. E. 30o 36o 40o 42o 45o 2. 3. 6. 5 Number of students 38 42 Mean mark 72 54 A student in Class A has scored 91 marks. It is found that his score was wrongly recorded as 19 in the calculation of the mean mark for Class A in the above table. Find the correct mean mark of the 80 students in the two classes. III. c.f. 100 O A. B. C. D. E. 92 33. B. C. D. E. A. B. C. D. E. 3 25 3 20 4 25 6 25 3 10 92 35. II. III. c.f. A. B. C. D. E. 100 O II. 1 92 36. 100 92-CE-MATHS II 1 a, b, c form an arithmetic progression. a, b, c form a geometric progression. b Both roots are . a I only II only III only I and II only II and III only x c.f. O I, II, III I, III, II II, I, III II, III, I III, I, II If the quadratic equation ax2 2bx + c = 0 has two equal roots, which of the following is/are true? I. I. x The figure shows the cumulative frequency curves of three distributions. Arrange the three distributions in the order of their standard deviations, from the smallest to the largest. Two cards are drawn randomly from five cards A, B, C, D and E. Find the probability that card A is drawn while card C is not. A. 92 34. 61.65 62.55 63 63.45 63.9 1 x 6 Which of the following intervals must contain a root of 2x3 x2 x 3 = 0? I. II. III. 1 < x < 1 0<x<2 1<x<3 A. B. C. I only II only III only D. E. 92 37. I and II only II and III only C. D. E. How many integers x satisfy the inequality 6x2 7x 20 0? A. B. C. D. E. 92 38. 92 41. 0 1 2 3 4 x2. x+2. x1. x+2. x. y O 92 42. x From the figure, if x , then A. B. C. D. E. 92 43. ax + (b m)x + (c k) 0 . ax2 + (b m)x + (c k) < 0 . ax2 + (b m)x + (c k) = 0 . ax2 + (b m)x + (c k) > 0 . ax2 + (b m)x + (c k) 0 . 2 B. C. n is any positive integer n is any positive odd integer n is any positive even integer n is any multiple of 3 n is the square of any positive integer D. E. 92 44. The L.C.M. of P and Q is 12ab3c2. The L.C.M. of X, Y and Z is 30a2b3c. What is the L.C.M. of P, Q, X, Y and Z? A. B. 360a3b6c3 60a2b3c2 92-CE-MATHS II 100 =5 x 1 100 100 x 1 x 100 100 x x 1 100 100 x 1 x 100 100 x x 1 =5 =5 =5 =5 By selling an article at 10% discount off the marked price, a shop still makes 20% profit. If the cost price of the article is $19 800, then the marked price is A. 7 22n 22n 22n 3 22n 2 22n 2 If the price of an orange rises by $1, then 5 fewer oranges could be bought for $100. Which of the following equations gives the original price $x of an orange? A. Under which of the following conditions must the mean of n consecutive positive integers also be an integer? A. B. C. D. E. Find the (2n)th term of G.P. 1 , 1, 2, 4, … 2 A. B. C. D. E. y = ax2 + bx + c 92 40. If a polynomial f(x) is divisible by x 1, then f(x 1) is divisible by A. B. C. D. E. y = mx + k 92 39. 60ab3c2 6a2b3c 6ab3c $21 600 . B. C. D. E. 92 45. $26 136 . $26 400 . $27 225 . $27 500 . 92 47. E A. B. 135 o D. 3 E. In the figure, find tan . C. D. E. C 92 48. 1 3 1 8 3 8 2 7 1 2 2 . 2 3 3 sin . 2 4 1 sin . 2 3 2 sin = 3 3 sin = 4 sin O N 2 cm B. B 2 In the figure, if is the angle between the diagonals AG and BH of the cuboid, then A. D 4 A C. 2 F 2:3 5:6 6:5 3:2 55 : 34 92 46. G 4 Coffee A and coffee B are mixed in the ratio x : y by weight. A costs $50/kg and B costs $40/kg. If the cost of A is increased by 10% which that of B is decreased by 15%, the cost of the mixture per kg remains unchanged. Find x : y. A. B. C. D. E. H M C 2 cm A 2 2 cm 2c m B In the figure, OA is perpendicular to the plane ABC. OA = AB = AC = 2 cm and BC = 2 2 cm. If M and N are the midpoint of OB and OC respectively, find the area of AMN. A. B. 92-CE-MATHS II 8 1 cm2 2 1 cm2 E. 92 49. 92 51. A E ? 2 cm2 3 cm2 2 3 cm2 C. D. 40o B B a 30o A a D In the figure, EB and EC are the angle bisectors of ABC and ACD respectively. If A = 40o, find E. 30o C A C In ABC, A = 30o, c = 6. If it is possible to draw two distinct triangles as shown in the figure, find the range of values of a. A. B. C. D. E. C 6 c= c= 6 B A. B. C. D. E. 0<a<3 0<a<6 3<a<6 a>3 a>6 20o 25o 30o 35o 40o 92 52. B O A 92 50. D A C B 24o In the figure, O is the centre of the circle. If the diameter AOB rotates about O, which of the following is/are constant? E C In the figure, the two circles touch each other at C. The diameter AB of the bigger circle is tangent to the smaller circle at D. If DE bisects ADC, find . A. B. C. D. E. 24o 38o 45o 52o 66o 92-CE-MATHS II D 9 I. II. III. + A. B. C. D. E. I only II only III only I and II only II and III only AC + BD AC BD 92 53. B 9 D F 4 16 8 E G 2 ? C A In the figure, AB = 16, CD = 8, BF = 9, GD = 4, EG = 2. Find GC. A. B. C. D. E. 92 54. 4.5 5 6 8 10 F a A B a E D C In the figure, ABCD is a square of side a and BDEF is a rhombus. CEF is a straight line. Find the length of the perpendicular from B to DE. A. B. C. 1 a 2 2a 3 a 2 D. E. 3 a 2 a 92-CE-MATHS II 10
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