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Chapter 2 Quadratic Equations 20

Chapter 2 Quadratic Equations

WARM-UP EXERCISEExpand the following expressions. (1 3)

1. (a) (x 5)2 (b) (2x 3)2

2. (a) (x 4)2 (b) (3x 7)2

3. (a) (x 9)(x 9) (b) (4x 11)(4x 11)

Factorize the following expressions. (4 11)

4. (a) x2 6x (b) 4x2 12x

5. (a) (x 1)2 (x 1) (b) (2x 5)2 9(2x 5)

6. (a) x2 4x 4 (b) x2 12x 36

7. (a) x2 14x 49 (b) x2 26x 169

8. (a) x2 4 (b) 9x2 16

9. (a) x2 8x 7 (b) 4x2 23x 15

10. (a) x2 10x 16 (b) 6x2 35x 36

11. (a) x2 3x 108 (b) 15x2 2x 8

Solve the following inequalities. (12 16)12. (a) x 4 > 6 (b) 7 x 1

13. (a) 3 5x 28 (b) 9 < 4x 5

14. (a) 7x > 2x 5 (b) 4x > 2(3x 5) 1

15. (a) 3x (2x 5) x (b) 4x (7x 2) 3x

16. (a) 3(2 4x) 14 3x (b) 4 3x 6(x 4) x

21 New Trend Mathematics S4A — Supplement

BUILD-UP EXERCISE[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set, Intermediate Set and Advanced Set. You may choose to complete any ONE set according to your need. ]

Exercise 2A Elementary Set

Level 1Rewrite each of the following quadratic equations into ax2 bx c 0 where a 0. Write down the values of a, b and c. (1 4)

1. x2 9 2x 2. 9x2 5x 4

3. 3x2 7x 4. 3 2x2 6x

Solve the following equations. (5 10)5. x(x 1) 0 6. x(3x 1) 0

7. (x 2)(x 4) 0 8. (x 3)(x 10) 0

9. (2x 7)(5x 1) 0 10. (3x 2)(4x 7) 0

Solve the following equations by factorization. (11 25)

11. x2 7x 0 12. x2 9 0

13. x2 8x 15 0 14. x2 x 12 0

15. x2 16x 64 0 16. x2 17x 60 0

17. 2x2 7x 9 0 18. 2x2 9x 4 0

19. 25x2 20x 4 0 20. 75x2 12 0

21. 18x2 2x 0 22. 10 40x2 0

23. 48 14x x2 0 24. 2x2 9x 35

25. 33x 5x2 18

Level 2Solve the following equations by factorization. (26 34)

26. 27. 4(x2 1) 17x

28. 4x(8x 3) 27 29.

30. (x 8)(x 12) 4 31. x(2x 7) 3x(x 1) 0

32. (x 4)(x 9) 2(x 6) 0 33. (x 3)2 (x 3) 0

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34. 3x(3x 1) 2(3x 1)

Intermediate Set

Level 1Rewrite each of the following quadratic equations into ax2 bx c 0 where a 0. Write down the values of a, b and c. (35 38)

35. x2 16 3x 36. 7x2 4x 10

37. 4x2 9x 38. 2 5x2 8x

Solve the following equations. (39 41)39. x(x 3) 0 40. (x 4)(x 7) 0

41. (2x 1)(x 3) 0


42. 4x2 49 0 43. x2 2x 15 0

44. x2 18x 72 0 45. x2 22x 121 0

46. 3x2 10x 8 0 47. 6x2 5x 4 0

48. 12x2 35x 8 0 49. 49x2 42x 9 0

50. 98x2 288 0 51. 27x2 126x 0

52. 144 100x2 0 53. 6 13x 2x2 0

54. 8x2 22x 5 55. 41x 18x2 10


56. 57. 5(x2 2) 27x

58. 27 2x(15 4x) 59.

60. (x 13)(x 19) 9 61. 4x(2 x) 5x(2x 3) 0

62. (3x 7)(3x 2) 6(3x 4) 0 63. (x 6)2 4(x 6) 0

64. 4x(2x 1) 7(1 2x) 65. (5x 11)2 (5x 11)(2x 1)

66. x(10x 3) (5x 3)(3x 4)


Advanced Set

Level 1Solve the following equations. (67 69)67. x(2x 1) 0 68. (3 x)(x 2) 0

69. (4 x)(5 2x) 0


70. 3x2 192 0 71. 3x2 14x 8 0

72. 5x2 6x 8 0 73. 20x2 84x 27 0

74. 36x2 132x 121 0 75. 324x2 100

76. 55x2 20x 77. 567 175x2 0

78. 21 5x 6x2 0 79. 60x2 10 x

80. 11x 15 12x2


81. 82. 7(x2 3) 46x

83. 35 4x(3x 13) 84.

85. (3x 5)(3x 17) 36 86. 7x(2 3x) 2x(3 5x) 0

87. (4x 5)(4x 3) 2(7x 11) 0 88. (3x 5)2 9(3x 5) 0

89. 9x(1 4x) 2(4x 1) 90. (4x 9)2 (4x 9)(6x 13)

91. (2x 1)(x 8) x(25 x) 92. (9x 10)(2x 3) x(11x 6)

93. (2x 5)2 (x 4)2 94. (3x 2)2 (2x 3)2

95. Solve the following equations where p and q are real constants. Express the answers in terms of p and q.(a) x2 p2 q2 2pq 0 (b) x2 (p q)x (p 1)(q 1) 0

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Exercise 2B Elementary Set

Level 1Find the value of a of each of the following identities. (1 4)

1. x2 14x a (x 7)2 2. x2 18x a (x 9)2

3. 2x2 4x 2 2(x a)2 4. 2x2 ax 8 2(x 2)2

Solve the following equations. (5 20)

5. (x 5)2 4 6. (x 1)2 4

7. (3x 5)2 4 8. (2x 5)2 9

9. 10.

11. 36(x 2)2 1 12. 4(x 6)2 49

13. 9(x 5)2 25 14. 64(3x 1)2 4

15. (x 6)2 13 16. (3x 5)2 32

17. (2x 7)2 18 18. (17 6x)2 245

19. 4(x 5)2 44 20.

Level 2

Solve the following equations by completing the square. (21 31)

21. x2 4x 12 0 22. x2 3x 40 0

23. x2 10x 27 0 24. x2 6x 15 0

25. x2 7x 11 0 26. 2x2 x 7 0

27. 3x2 x 1 0 28. 2x2 3 5x

29. 4(3x 1) 9x2 30.

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Intermediate Set

Level 1Find the value of a of each of the following identities. (32 35)

32. x2 20x a (x 10)2 33. x2 30x a (x 15)2

34. 4x2 16x 16 4(x a)2 35. 5x2 ax 45 5(x 3)2

Solve the following equations. (36 49)

36. (x 2)2 16 37. (x 5)2 9

38. (5x 4)2 16 39. (4x 3)2 25

40. 41.

42. 50(x 3)2 2 43. 9(2x 3)2 16

44. 16(3x 4)2 49 45. (x 3)2 20

46. (5x 7)2 80 47. (4x 11)2 75

48. 3(5x 1)2 36 49.

Level 2


50. x2 6x 16 0 51. x2 5x 84 0

52. x2 12x 40 0 53. x2 10x 36 0

54. x2 13x 21 0 55. 3x2 7x 3 0

56. 5x2 2x 3 0 57. 3x2 10x 8

58. 4x2 5(4x 5) 59.

60. 61.

62. 2(x 3)2 6x 17

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Advanced Set

Level 1Solve the following equations. (63 76)

63. (x 4)2 16 64.

65. (3x 7)2 64 66. (7x 5)2 36

67. 68.

69. 50(4x 1)2 72 70. 108(5x 2)2 147

71. (x + 10)2 45 72. (4x 9)2 150

73. 4(3x 7)2 80 74.

75. 76.

Level 2


77. x2 8x 33 0 78. x2 9x 52 0

79. x2 16x 72 0 80. x2 14x 42 0

81. x2 11x 14 0 82. 4x2 5x 2 0

83. 6x2 7x 3 0 84. 5x2 19x 12

85. 3(8x 3) 16x2 86.

87. 88.

89. 90. 3(x 1)2 20x 9

91. 5(x 2)2 2(8 5x)

92. Consider the equation x2 4px 2q 0 where p and q are real constants.(a) By completing the square, show that (x 2p)2 4p2 2q.(b) Hence show that .(c) Use the above results to solve the following equations.

(i) x2 8x 14 0(ii) x2 16x 6 0

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Exercise 2C Elementary Set

Level 1Solve the following equations by using the quadratic formula. (1 16)

1. x2 3x 2 0 2. x2 6x 7 0

3. x2 6x 9 0 4. x2 8x 12 0

5. 2x2 5x 3 0 6. 3x2 26x 16 0

7. 9x2 42x 49 0 8. 25x2 30x 9 0

9. x2 2x 9 0 10. 5x2 4x 2 0

11. 2x2 4x 5 0 12. 4x2 5 3x

13. 9x 2(x2 3) 14. x2 4(x 2)

15. 3(x2 1) 14(2 x) 16. 4x(x 3) 20 x


17. 18.

19. 20. (x 3)(x 3) 6x 1

Intermediate Set


21. x2 3x 10 0 22. 3x2 5x 2 0

23. 5x2 27x 10 0 24. 16x2 56x 49 0

25. 121x2 132x 36 0 26. x2 4x 9 0

27. 7x2 12x 3 0 28. 3x2 9x 8 0

29. 6x2 7 8x 30. 10x 3(x2 1)

31. 2x2 3(4x 1) 32. 6x(x 4) 42 5x

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33. 34.

35. 36.

37. 38. (2x 1)(2x 1) 1 x

39. (4x 5)2 6x(2x 3) 40. (3x 4)(x 6) 5(2x 1) 1

Advanced Set


41. 4x2 19x 5 0 42. 6x2 31x 30 0

43. 162x2 72x 8 0 44. 64x2 144x 81 0

45. x2 6x 6 0 46. 10x2 22x 9 0

47. 5x2 8x 4 0 48. 3 5x 9x2

49. 6x 5(x2 3) 50. 11x 35 6x(x 6)


51. 52.

53. 54.

55. 56.

57. (5x 1)2 8x(2x 3) 58.

59. (a) Solve 30x2 x 14 0 by using the quadratic formula.(b) Hence solve 30(x 2)2 (x 2) 14 0.

60. A root of the quadratic equation ax2 20x 3 0 is 3.(a) Find the value of a.(b) Hence find the other root of the equation by using the quadratic formula.


Exercise 2D Elementary Set

Level 11. Find the value of the discriminant of each of the following quadratic equations.

(a) x2 6x 4 0 (b) x2 3x 10 0(c) 2x2 6x 5 0 (d) 3x2 5x 4 0(e) x2 8x 3 (f) 4x2 25 20x

2. Determine the nature of the roots of each of the following quadratic equations.

(a) x2 x 4 0 (b) 2x2 6x 9 0(c) 4x2 36x 81 0 (d) 2x2 9x 5

3. (a) Find the discriminant of the quadratic equation x2 8x k 0.(b) Find the range of values of k if the equation x2 8x k 0 has real roots.

4. (a) Find the discriminant of the quadratic equation 3x2 16x 4k 0.(b) Find the range of values of k if the equation 3x2 16x 4k 0 has real roots.

5. (a) Find the discriminant of the quadratic equation kx2 12x 4 0.(b) Find the range of values of k if the equation kx2 12x 4 0 has no real roots.

6. (a) Find the discriminant of the quadratic equation 5kx2 24x 6 0.(b) Find the range of values of k if the equation 5kx2 24x 6 0 has no real roots.

7. (a) Find the discriminant of the quadratic equation 2x2 8x k 1 0.(b) Find the value of k if the equation 2x2 8x k 1 0 has two equal real roots.

8. (a) Find the discriminant of the quadratic equation .

(b) Find the value of k if the equation has two equal real roots.

9. Find the value of k if each of the following quadratic equations has two equal real roots.(a) kx2 10x 1 0 (b) 3x2 8x 4k 0

10. Find the range of values of k if each of the following quadratic equations has two unequal real roots.(a) 2x2 x k 0 (b) kx2 2x 4 0

11. Find the range of values of k if each of the following quadratic equations has no real roots.(a) x2 9x k 0 (b) kx2 4x 5 0

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Level 212. Find the range of values of k if the quadratic equation x2 4x k 1 0 has two unequal

real roots.

13. Find the range of values of k if the quadratic equation 2x(x 3) x k has no real roots.

14. Find the range of values of k if the quadratic equation x2 5x 3k 1 0 has real roots.

15. Find the values of k if the quadratic equation x2 kx 4k 0 has a double root.

Intermediate Set


(a) x2 8x 2 0 (b) x2 6x 3 0(c) 3x2 7x 2 0 (d) 4x2 3x 11 0(e) 2x2 5x 6 (f) 24x 16x2 9

17. Determine the nature of the roots of each of the following quadratic equations.(a) x2 10x 25 0 (b) 2x2 x 5 0(c) 4x2 x 8 0 (d) 3 2x 5x2

18. (a) Find the discriminant of the quadratic equation 2x2 10x k 0.(b) Find the range of values of k if the equation 2x2 10x k 0 has real roots.

19. (a) Find the discriminant of the quadratic equation 3kx2 18x 8 0.(b) Find the range of values of k if the equation 3kx2 18x 8 0 has no real roots.

20. (a) Find the discriminant of the quadratic equation 2x2 6x (k 2) 0.(b) Find the value of k if the equation 2x2 6x (k 2) 0 has two equal real roots.

21. Find the value of k if each of the following quadratic equations has two equal real roots.(a) 4kx2 60x 25 0 (b) 5x2 16x 8k

22. Find the range of values of k if each of the following quadratic equations has two unequal real roots.(a) x2 3x 4k 0 (b) kx2 5x 2 0

23. Find the range of values of k if each of the following quadratic equations has no real roots.(a) x2 14x 6k 0 (b) kx2 6x 15 0

Level 224. Find the range of values of k if the quadratic equation 5x2 3x 2(k 1) 0 has two

unequal real roots.

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25. Find the range of values of k if the quadratic equation 5x 4 3(x2 k) has no real roots.

26. Find the range of values of k if the quadratic equation 3(2x2 1) 2(4x k) has real roots.

27. Find the values of k if the quadratic equation x2 2kx k 2 0 has a double root.

28. (a) Find the value of k if the quadratic equation 3kx2 kx 2 0 has a double root.(b) Using the result of (a), solve the equation 3kx2 kx 2 0.

29. (a) Find the value of k if the quadratic equation kx2 (3k 1)x (2k 1) 0 has a double root.

(b) Using the result of (a), solve the equation kx2 (3k 1)x (2k 1) 0.

Advanced Set


(a) x2 2x 12 0 (b) 2x2 7x 4 0(c) 5x2 6x 1 0 (d) 6x2 2x 5 0(e) 3x 2x2 2 (f) 42x 9 49x2

31. Determine the nature of the roots of each of the following quadratic equations.(a) x2 2x 4 0 (b) 3x2 6x 3 0(c) 7x2 4x 5 0 (d) 5 11x 6x2

32. Find the value of k if each of the following quadratic equations has two equal real roots.(a) kx2 7x 28 0 (b) 10x2 12x 3k

33. Find the range of values of k if each of the following quadratic equations has two unequal real roots.(a) 3x2 4x 2k 0 (b) 2kx2 3x 4 0

34. Find the range of values of k if each of the following quadratic equations has no real roots.(a) x2 12k 18x (b) kx2 10 16x

Level 235. Find the range of values of k if the quadratic equation (k 1)x2 5x 2 0 has two

unequal real roots.

36. Find the range of values of k if the quadratic equation (k 1)x2 12 4x has no real roots.

37. Find the range of values of k if the quadratic equation k(1 2x) 3 2x kx2 has real roots.

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38. Find the values of k if the quadratic equation k(x2 1) 6x has a double root.

39. (a) Find the value of k if the quadratic equation has a double root.

(b) Using the result of (a), solve the equation .

40. (a) Find the value of k if the quadratic equation 2kx2 (4k 3)x (2k 5) 0 has a double root.

(b) Using the result of (a), solve the equation 2kx2 (4k 3)x (2k 5) 0.

41. (a) Express the discriminant of the quadratic equation 4kx2 (k 8)x 2 0 in terms of k.(b) Hence prove that the quadratic equation 4kx2 (k 8)x 2 0 has real roots for all

real values of k.

Exercise 2E Elementary Set


1. (x 1)2 2(x 1) 3 0 2. (x 4)2 7(x 4) 18 0

3. 4. 6(x 2)2 5(x 2) 1 0

5. 6.

7. 8. (x2)2 3(x2) 54 0

9. 10. (3x)2 6(3x) 27 0


11. 12. x4 26x2 25 0

13. 14.

15.

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Intermediate Set


16. (2x 3)2 2(2x 3) 8 0 17. 2(3x 1)2 3(3x 1) 5 0

18. 19.

20. (x2)2 2(x2) 48 0 21.

22. 23. (4x)2 20(4x) 64 0


24. 25. x4 22x2 72 0

26. 27.

28. 29.

30. 31.

32.

Advanced Set


33. (1 5x)2 5(1 5x) 4 0 34.

35. 36. 4(x2)2 11(x2) 7 0

37. 38. 2(22x) 5(2x) 2 0


39. 40. 2x4 23x2 45 0

41. 42.

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43. 44.

45. 46.

47.

48. (a) Solve the equation y2 2y 3 0.(b) Hence solve the equation (x2 2x)2 2(x2 2x) 3 0.

49. (a) Solve the equation 4t2 21t 26 0.

(b) Hence solve the equation .

50. (a) Solve the equation .

(b) Hence solve the equation .

51. Consider the equation …………. (1).

(a) Let . Show that 2u2 3u 2 0………… (2).(b) Solve equation (2) and hence solve equation (1).

Exercise 2F Elementary Set

Level 11. The height of a moving balloon above the ground after x seconds is given by

. When is the balloon 50 m above the ground?

2. The price of a plate is where x cm is the radius of the plate. If the price of the plate is $46, what is its radius?

3. The cost of making a bookshelf is where x cm is the height of the bookshelf. Find the height of the bookshelf if the cost is $687.

4. In the figure, the area of rectangle ABCD is . Find the value of x.A D

B C

x cm

(x 5) cm

5. The product of two consecutive numbers is 156.

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(a) If the smaller number is x, find the larger number in terms of x.(b) Find the two numbers.

6. The product of two consecutive odd numbers is 143.(a) If the smaller number is x, find the larger number in terms of x.(b) Find the two numbers.

7. The sum of two numbers is 30.(a) If one of the numbers is x, find the other number in terms of x.(b) Given that the product of the two numbers is 216, find the two numbers.

8. The difference between two numbers is 4.(a) If the larger number is x, find the smaller number in terms of x.(b) Given that the product of the two numbers is 480, find the two numbers.

9. In the figure, the sum of the areas of square ABCD and square PQRS is 130 cm2. Find the value of x.A D

B C

x cm (x 2) cm

R

S

Q

P

10. It is given that where n is a positive integer.

If 1 2 3 n 300, find n.

11. The perimeter of a rectangle is 30 cm.(a) If the length of the rectangle is x cm, find its width in terms of x.(b) Given that its area is , find the dimensions of the rectangle.

12. In the figure, the rectangular photo frame is 40 cm long and 24 cm wide. The border of the frame is made of wood with width x cm. If the area of the photo displayed is , find the value of x.

24 cm

40 cm

x cm

x cm

x cm

x cmPhoto

13. In the figure, ABC is a right-angled triangle. AB x cm, BC (2x 1) cm and AC (3x 7) cm. Find the value of x.

(3x 7) cm

(2x 1) cm

x cm

C B

A

14. It is given that a rectangular paper is 10 cm long and 8 cm wide. Four squares with sides x cm are cut away from the four corners of the paper (as shown in the figure) in order to

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make an open rectangular box with base area . Find the value of x.x cm

x cm

8 cm

10 cm

Level 215. John is 2 years older than his brother Tom, but 28 years younger than his mother Jenny.

Three years later, Jenny’s age will be exactly the product of her sons’. How old is Jenny now?

16. A man bought some toy cars for $2 200 but twelve of them were broken. He sold each of the remaining toy cars at $8 more than the cost. Finally, he obtained a profit of $440. Let $x be the cost of each toy car.(a) Express the number of toy cars sold in terms of x.(b) Find the cost of each toy car.

17. A fast food restaurant charges $480 to a group of students for the food of a Christmas party. If 10 students cannot join the party, each of the remaining students has to pay $4 more to cover the charge. Find the original number of students who planned to join the party.

Intermediate Set

Level 118. The distance between a coastline and a ship after x hours is given by .

When are the ship and the coastline 52 km apart?

19. A ball is thrown vertically upwards. Its height above the ground after x seconds is given by . When is the ball 4 m above the ground?


21. The difference between two numbers is 7.(a) If the smaller number is x, find the larger number in terms of x.(b) Given that the product of the two numbers is 450, find the two numbers.


(a) If 1 2 3 n 630, find n.(b) If 31 32 33 n 1 488, find n.

23. A square with sides 4 cm is cut away from each corner of a tin sheet with dimensions x cm (x 15) cm. Then the sheet is folded up to form an open box. If the capacity of the box is , find the length and the width of the box.

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4 cm4 cm

x cm

(x + 15) cm

24. After cutting a small rectangle with width x cm from a corner of a large rectangle with an area of , two of the sides of the remaining figure become 10 cm and 4 cm as shown. If the length of the small rectangle is 2 times its width, find the dimensions of the small rectangle.

4 cm

10 cmx cm

25. In the figure, ABCD is a trapezium where CD AD. If the area of ABCD is , find the length of AD.

(x 2) cm

(4x 5) cm

(2x 3) cmC

D A

B

26. The product of two consecutive even numbers is 224. Find the two numbers.

Level 2

27. The sum of the reciprocals of two consecutive even numbers is . Find the two numbers.

28. Find the dimensions of a rectangle with a perimeter of 45 m and an area of .

29. Owen is 28 years younger than his mother and 35 years younger than his father. Five years later, the sum of the ages of Owen’s parents will be exactly the square of the age of Owen. How old is Owen now?

30. 84 students are evenly divided into groups. If the number of groups decreases by 3, the number of students in each group increases by 9. What is the original number of groups?

31. A man bought some calculators for $4 600 but four of them were broken. He sold each of the remaining calculators at $40 more than the cost. Finally, he obtained a profit of $980. Find the cost of each calculator.

32. Sammy had bought a number of compact discs from a shop for $40 last week. The shop reduced the price of each compact disc by $0.4 this week. If Sammy bought the compact discs this week, she would buy 10 more compact discs by paying extra $2. What was the original price of each compact disc?

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33. A company charges $600 to a group of students for the rental of a coach for a picnic. If 5 students cannot join the picnic, each of the remaining students has to pay $4 more to cover the charge. Find the original number of students who planned to join the picnic.

Advanced Set

Level 134. The cost of making a model is where x cm is the height of the

model. If the cost of the model is $1 464, what is its height?


36. The difference between two numbers is 11.(a) If the smaller number is x, find the larger number in terms of x.(b) Given that the product of the two numbers is 476, find the two numbers.

37. The largest number of three consecutive odd numbers is x and the product of the other two numbers is 7x 8. What are the three numbers?


(a) If 1 2 3 n 351, find n.(b) If 2 4 6 2n 650, find n.

39. In the figure, ABCD is a rectangular floor with length 9 m and width 6 m. PQRS is a rectangular carpet put on the floor and leaves a uniform width x m of the floor uncovered all round the room.

6 m

9 m

x m

x m

D C

A B

Q

R

P

S

x m

(a) Find the area of PQRS in terms of x.(b) If the area of PQRS is one third the area of ABCD, find the dimensions of PQRS.

40. The sum of the areas of two squares is and the sum of their perimeters is 52 cm.(a) If the side of one of the squares is x cm long, find the length of each side of another

square in terms of x.(b) Hence find the lengths of each side of the two squares.

41. In the figure, ABCD is a trapezium. AB 2x cm, BC 5x cm, CD (7x 1) cm and AD (x 2) cm.

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D C

B

(7x 1) cm

(a) Find the values of x.(b) Hence find the possible area of trapezium ABCD.

42. The product of two consecutive odd numbers is 2 499. Find the two numbers.

Level 2

43. The sum of the reciprocals of two consecutive odd numbers is . Find the two numbers.

44. Find the dimensions of a rectangle with a perimeter of 92 m and an area of .

45. Bruce is 32 years older than his son and 35 years older than his daughter. Three years later, the product of the ages of Bruce’s children will be exactly the age of Bruce. How old is Bruce now?

46. A man bought some bowls for $1 500 but twelve of them were broken. He sold each of the remaining bowls at $6 more than the cost. Finally, he gained a profit of 69.2%. Find the cost of each bowl.

47. Joan had bought a number of pens from a shop for $300 last week. The shop reduced the price of each pen by $0.3 this week. If Joan bought the pens this week, she would buy 20 more pens by paying extra $8. How much was a pen after the price reduction?

48. A travel company charges $7 200 to a group of students for a trip. If 12 more students take the trip, each student will pay $20 less to cover the charge. Find the original number of students who planned to join the trip.

49. The distance between Kenny’s home and town A is 72 km. He drives from home to town A at an average speed of x km/h and returns in the same route at an average speed of 3 km/h less. The difference in time travelled between two trips is 1.2 hours. Find the value of x.

50. If the rate of the current is 3 km/h, Johnny spends 4.8 hours in rowing downstream of 6 km and returning upstream of 6 km. Find his rowing speed in still water.

51. Mary takes 5 days more than Susan to finish a piece of work. When they do it together, they take 4 days less than Susan alone to finish the work. How long does Susan take to finish the work alone?

CHAPTER TEST (Time allowed: 1 hour)

Section A


1. Solve the equation . (3 marks)

2. Solve the equation (x 2)(x 6) 20. (3 marks)

3. A root of the quadratic equation is k.(a) Find the values of k. (2 marks)(b) Using the result of (a), solve the equation . (2 marks)

4. Find the range of values of k if the quadratic equation has real roots.(4 marks)

5. (a) Find the value of k if the quadratic equation has two equal real roots. (3 marks)

(b) Using the result of (a), solve the equation . (2 marks)

6. In the figure, AF (x 1) cm, AB 12 cm, BC 14 cm and CD (x 1) cm.(a) Find EF in terms of x. (1 mark)(b) If the area of ABCDEF is , find

the length of DE. (5 marks)

Section B7. (a) The quadratic equation has a double root.

(i) Find the value of k.(ii) Hence solve the equation . (5 marks)

(b) (i) For the equation if k is the value obtained in

(a)(i) , does the equation have a double root? Explain your answer.

(ii) Hence solve the equation . (5 marks)

8. Lily takes 10 minutes to drive from her home to a supermarket at an average speed of

x km/h. Then she takes minutes more than the previous trip to drive from the

supermarket to a beach at an average speed of (x 22) km/h.

(a) Find the distance between Lily’s home and the supermarket in terms of x. (2 marks)(b) Find the distance between the supermarket and the beach in terms of x. (3 marks)(c) If the total distance travelled by Lily is 29 km, find the time required for the whole

trip. (5 marks)

Multiple Choice Questions (3 marks each)

9. Which of the following equations has

roots 4 and ?

A.B.

FA

E

D

C

B

(x 1) cm

(x 1) cm

14 cm

12 cm


C.D.

10. If 4 is a root of , find the values of a.A. 2 or 6B. 2 or 4C. 2 or 2D. 2 or 4

11. If (x 2)(x 3) (a 2)(a 3), then x A. a.B. a or a 1.C. a or 1 a.D. a or a 1.

12. Find the value of the discriminant of the equation .A. 33B. 17C. 16D. 39

13. Which of the following equations has no real roots?I.II.III.

A. I and II onlyB. I and III onlyC. II and III onlyD. I, II and III

14. If , then x

A. 4.

B. .

C. 4 or .

D. 4 or .

15. If , then x

A. 1.

B. 3.

C. 9.

D. 1 or 9.

16. Solve the equation

.

A. 1 or 7

B. 1 or 7

C. 2 or 10

D. 4 or 4

17. The sum of two numbers is 20. If one of the numbers is x and the product of the two numbers is 99, which of the following equations can be used to solve the problem?A.B.C.D.

18. If the speed of a car increased by 10 km/h in a journey of 300 km, 1 hour would have been saved. Find the original speed of the car.A. 40 km/hB. 50 km/hC. 60 km/hD. 70 km/h

H INTS (for questions with in the textbook)


Revision Exercise 235. (b) Key information

The result obtained in (a).Analysis A quadratic equation has real roots means that it has two equal or unequal real roots,

and therefore the value of the discriminant of the equation is greater than or equal to zero.

As the discriminant is expressed in terms of constants a and b and their values are unknown, we can be sure that the value of the discriminant is always greater than or equal to zero only if it can be written as a complete square.

MethodTo factorize the discriminant of the equation by applying the identity

.

36. (a) Key informationThe quadratic equation

AnalysisSince the equation is not written in general form, we cannot directly apply the formula for finding the discriminant.MethodRewrite the quadratic equation into general form before applying the formula for finding the discriminant. Then, expand to compare with the expression obtained for the discriminant.

38. (a) Key information Points B and E lie on the straight line y kx. ABCD and DEFG are two squares each with one side lying on the x-axis. The coordinates of A are (2, 0).Analysis Point D lies on the x-axis.

The y-coordinate of D is zero. ABCD is a square.

AB ADThus knowing the length of AB can help find the x-coordinate of D.

MethodSince ABCD is a square, point B has the same x-coordinate as point A. Substitute x 2 into y kx to find the y-coordinate of B and the length of AB can be found.

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