281 EQUATION - 197notesbag.com/wp-content/uploads/2018/01/Quadractic...33. I. 6x + 7y = 52 II. 14x +...

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LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

EQUATION - 197Directions (Q. 1-5): Two equations (I) and (II) are given in each question. On the basis of these

equations you have to decide the relation between ‘x’ and ‘y’ and give answer.(1) if x > y (2) if x < y (3) if x y(4) if x y (5) if x = y or no relation can be established between ‘x’ and ‘y’.

1. I. 6x2 - 19x + 15 = 0 II. 10y2 - 29y + 21 = 02. I. 12x2 + 11x - 56 = 0 II. 4y2 - 15y + 14 = 03. I. 3x2 + 13x + 12 = 0 II. y2 + 9y + 20 = 04. I. 8x2 - 15x + 7 = 0 II. 2y2 - 7y + 6 = 05. I. 7x - 3y = 13 II. 5x + 4y = 40

Directions (Q. 6-10): In the following questions, two equations numbered I and II are given.You have to solve both the equations and give answer

(1) if x > y (2) if x < y (3) if x y(4) if x y (5) if x = y or no relation can be established between x and

6. I. 2x2 - 11x + 15 = 0 II. 21y2 - 23y + 6 = 07. I. 5x2 - 16x + 11= 0 II. 5y2 - 3y - 2 = 08. I. x2 + 11x + 28 = 0 II. 2y2 + 13y + 20 = 09. I. 6x2 + 29x + 35 = 0 II. 3y2 + 19y + 30 = 010. I. 2x + 5y = 6 II. 5x + 11y = 9

Directions (Q. Nos. 11-15) In the following questions two equations numbered I and II aregiven. You have to solve both the equations and—

Give answer(1) if x > y (2) if x > y (3) if x < y(4) if x < y (5) if x = y or the relationship cannot be established

11. I. 1225x 4900 0 II. (81)1/4 y + (343)1/3 = 0

12. I. 2 2 218 6 12 8

xx x x II. y3 + 9.68 + 5.64 = 16.95

13. I.5 3

3(2) (11) x6

II. 4y3 = - (589 4) + 5y3

14. I. 12x2 + llx + 12 = 10x2+22x II. 13y2 - 18y + 3 = 9y2 - 10y15. I. (x7/5 9) = 169 y3/5 II. y1/4 y1/4 7 = 273 y1/2

Directions (Q. 16 - 20): Two equations (I) and (II) are given in each question. On the basis ofthese equations you have to decide the relation between x and y and give answer

(1) if x > y (2) if x < y (3) if x y (4) if x y(5) if x = y, or no relation can be established between x and y.

16. I. x = 4 2401 II. 2y2 - 9y - 56 = 0

17. I. 5x2 + 3x - 14 = 0 II. 2y2 - 9y + 10 = 018. I. 8x2 + 31x + 21 = 0 II. 5y2 + 11y - 36 = 0

19. I. 3x - y = 12 II. y = 108920. I. 15x2 + 68x + 77 = 0 II. 3y2 + 29y + 68 = 0

Directions (Q. 21-25): Two equations (I) and (II) are given in each question. On the basis ofthese equations, you have to decide the relation between x and y and give answer

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21. I. 2x2 + x - 1 = 0 II. 6y2 - 13y + 5 = 022. I. 21x2 - 122x + 165 = 0 II. 3y2 - 2y - 33 = 023. I. 5x2 - 29x + 36 = 0 II. 10y2 - 3y - 27 = 024. I. 7x + 4y = 3 II. 5x + 3y = 325. I. 7x2 - 54x + 99 = 0 II. 4y2 - 16y + 15 = 0



26. I. 5x2 - 87x + 378 = 0 II. 3y2 - 49y + 200 = 027. I. 10x2 - x - 24 = 0 II. y2 - 2y = 028. I. x2 - 5x + 6 = 0 II. 2y2 - 15y + 27 = 029. I. 3x + 2y = 301 II. 7x - 5y = 7430. I. 14x2 - 37x + 24 = 0 II. 28y2 - 53y + 24 = 0

Directions (Q. 31-35): In each of these questions, two equations (I) and (II) are given. Youhave to solve both the equations and give answer

(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or relationship between x and y cannot be established

31. I. 11x + 5y = 117 II. 7x + 13y = 15332. I. 6x2 + 51x + 105 = 0 II. 2y2 + 25y + 78 = 033. I. 6x + 7y = 52 II. 14x + 4y = 3534. I. x2 + 11x + 30 = 0 II. y2 + 12y + 36 = 035. I. 2x2 + x - 1 = 0 II. 2y2 - 3y + l = 0

Directions (Q.36-40) In the following questions three equations numbered I, II and III aregiven. You have to solve all the equations either together or separately, or two together and oneseparately, or by any other method and give answer If

(1) x < y = z (2) x < y < z (3) x < y > z (4) x = y > z(5) x = y = z or if none of the above relationship is established

36. I. 7x + 6y + 4z = 122 II. 4x + 5y + 3z = 88 III. 9x + 2y + z = 7837. I. 7x + 6y =110 II. 4x + 3y = 59 III. x + z = 15

38. I. x = 1/2 1/4(36) (1296) II. 2y + 3z = 33 III. 6y + 5z = 71

39. I. 8x + 7y= 135 II. 5x + 6y = 99 III. 9y + 8z = 12140. I. (x + y) 3= 1331 II. x - y + z = 0 III. xy = 28


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or relationship between x and y cannot be established

41. I. 7x2 - 9x + 2 = 0 II. y2 - 4y + 3 = 042. I. x2 = 64 II. 2y2 + 25y + 72 = 043. I. x2 + x - 20 = 0 II. 2y2 - 19y + 45 = 044. I. 7x + 3y = 26 II. 2x + 17y = -4145. I. 3x2 - 20x + 33 = 0 II. 2y2 - 11y + 15 = 0

Directions (Q. 46-50): In each of these questions, two equations (I) and (II) are given. You

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have to solve both the equations and give answer(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or relationship between x and y cannot be established.

46. I. 4x2 - 43x + 105 = 0 II. 7y2 - 29y + 30 = 047. I. x2 + 13x + 40 = 0 II. y2 + 7y + 10 = 0

48. I. 3x 2197 II. 2y2 - 54y + 364 = 0

49. I. 5x2 - 27x + 36 = 0 II. y2 - 2y + 2 = 050. I. 13x - 8y + 81 = 0 II. 15x + 5y + 65 = 0



51. I. 15x2 - 19x + 6 = 0 II. 6y2 - 5y + 1 = 0

52. I. x 172 II. y2 - 29y + 210 = 053. I. 3x2 - 20x + 32 = 0 II. 2y2 - 19y + 44 = 054. I. 3x + 8y = -2 II. 4x + 18y = l55. I. 2x2 - 15x + 28 = 0 II. 10y2 - y - 119 = 0


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or relationship between x and y cannot be established.

56. I. 676x2 - l = 0 II. 3

1y13824

57. I. 8x + 13y = 62 II 13x - 17y + 128 = 058. I. 7x2 + 2x = 120 II. y2 + 11y + 30 = 059. I. x2 = 7x II. (y + 7)2 = 060. I. 2x2 + 5x - 33 = 0 II. y2 - y - 6 = 0


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or the relationship between x and y cannot be established.

61. I. x2 + 12x + 36 = 0 II. y2 + 15y + 56 = 062. I. x2 = 35 II. y2 + 13y + 42 = 063. I. 2x2 - 3x - 35 = 0 II. y2 - 7y + 6 = 064. I. 6x2 - 29x + 35 = 0 II. 2y2 - 19y + 35 = 065. I. 12x2 - 47x + 40 = 0 II. 4y2 + 3y - 10 = 0


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between ‘x’ and ‘y’.

66. I. x2 + 3x - 28 = 0 II. y2 - 11y + 28 = 067. I. 6x2 - 17x + 12 = 0 II. 6y2 - 7y + 2 = 0

68. I.256x576

II. 3y2 + y-2 = 0

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69. I. x2 = 64 II. y2 = 9y70. I. x2 + 6x - 7 = 0 II. 41y + 17 = 140


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or a relationship between x and y cannot be established.

71. I. x2 + 3x = 28 II. y2 + 16y + 63 = 0

72. I. x = 3 2197 II. y2 = 169

73. I. 8x2 - 49x + 45 = 0 II. 8y2 - y - 9 = 074. I. 42x - 17y = -67 II. 7x + 12y = -2675. I. x2 - 8x + 15 = 0 II. 2y2 - 21y + 55 = 0

Directions (Q. 76-80): In each of these questions two equations (I) and (II) are given. Youhave to solve both the equations and give answer

(1) if p > q (2) if p q (3) if p < q (4) if p q(5) if p = q or no relation can be established between p and q.

76. I. 2.3p - 20.01 = 0 II. 2.9q - p = 0

77. I. p = 1764 II. q2 = 1764

78. I. p2 - 26p + 168 = 0 II. q2 - 25q + 156 = 079. I. p2 - 13p + 42 = 0 II. q2 + q - 42 = 080. I. 6p - 5q = -47 II. 5p + 3q = 11

Directions(Q. 81-85): In each of these questions, two equations (I) and (II) are given. Youhave to solve both the equations and give answer

(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between ‘x’ and ‘y’.

81. I. 2x2 + 13x - 7 = 0 II. 2y2 - 5y + 3 = 082. I. 2x2-15x + 28 = 0 II. 4y2 - 16y + 15 = 083. I. x2 + 8x + 16 = 0 II. y2 = 1684. I. x2 - 2x - 24 = 0 II. y2 + 8y = 0

85. I. x2 + 4x = 0 II. y2 + 10y + 25 = 0

Directions (Q. 86-90): In each of these questions two equations (I) and (II) are given. Youhave to solve both the equations and give answer

(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between x and y

86. I. 2x2 + x – 1 = 0 II. 2y2 + 13y + 15 = 087. I. x2 + 12x + 32 = 0 II. 2y2 + 15y + 27 = 088. I. 6x2 – 17x + 12 = 0 II. 7y2 – 13y + 6 = 089. I. x2 – 82x + 781 = 0 II. y2 = 504190. I. 6x2 – 47x + 80 = 0 II. 2y2 – 9y + 10 = 0


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation between ‘x’ and ‘y’ can be established.

91. I. 3x2 – 7x – 20 = 0 II. y2 – 8y + 16 = 092. I. x2 – 72 = 0 II. y2 – 9y + 8 = 093. I. 9x2 – 114x + 361 = 0 II. y2 = 3694. I. 13x + 17y = 107 II. x – 11y = – 41

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95. I. 9x2 + 18x + 9 = 0 II. y2 – 3y + 2 = 0Directions (Q. 96-100) : In each of these questions, two equations (I) and (II) are given. You

have to solve both the equations and give answer .(l) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between ‘x’ and y.

96. I. 4x + 7y = 42 II. 3x - 11y = – l97. I. 9x2 – 29x + 22 = 0 II. y2 – 7y + 12 = 098. I. 3x2 – 4x – 32 = 0 II. 2y2 – 17y + 36 = 099. I. 3x2 – 19x – 14 = 0 II. 2y2 + 5y + 3 = 0100. I. x2 + 14x + 49 = 0 II. y2 + 9y = 0

Directions (Q. 101-105): In each of these questions, two equations (I) and (II) are given. Youhave to solve both the equations and give answer ,

(1) if x < y (2) if x y(3) if x = y, or no relation can be established between x and y(4) if x > y (5) if x y

101. I. 9x2 = 1 II. 4y2 + 11y - 3 = 0102. I. 3x2 + 5x - 2 = 0 II. 2y2 - 7y + 5 = 0103. I. 6x2 + 13x + 5 = 0 II. 3y2 + 11y + 10 = 0104. I. 7x - 4y = 29 II. 5x + 3y - 50 = 0105. I. x2 - 5 = 0 II. 4y2 - 24y + 35 = 0

Directions (Q. 106-110) : In each of these questions, two equations (I) and (II) are given. Youhave to solve both the equations and give answer

(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between x and y

106. I. 35x2 - 53x + 20 = 0 II. 56y2-97y + 42 = 0107. I. x = 3 4913 II. 13y + 3x = 246108. I. x2 - 5x - 14 = 0 II. y2 + 7y + 10 = 0109. I. x2 - 3481 = 0 II. 3y2 = 3 216000110. I. 5x2 + 2x - 3 = 0 II. 2y2 + 7y + 6 = 0


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relationship can be established.

111. I. 20x2 - 67x + 56 = 0 II. 56y2 - 67y + 20 = 0112. I. x4 = 65536 II. y = 3 4096113. I. 2x2 + 11x - 40 = 0 II. 4y2 - 27y + 44 = 0114. I. 7x = 4y + 85 II. y = 3 17576

115. I. x2 = 14641 II. y = 14641Directions (Q. 116-120): In each of these questions, two equations (I) and (II) are given. You

have to solve both the equations and give answer(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or if there is no relation between ‘x’ and ‘y’.

116. I. x2 + 42 = 13x II. 4y 1296

117. I. x2 + x - 2 = 0 II. y2 + 7y + 12 = 0118. I. 3x2 - 23x + 40 = 0 II. 2y2 - 23y + 66 = 0119. I. 15x2 - 46x + 35 = 0 II. 4y2 - 15y + 14 = 0120. I. x2 + 5x - 6 = 0 II. 2y2 - 11y + 15 = 0

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(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or If there is no relation between ‘x’ and ‘y’.

121. I. 2x2 – 21x + 54 = 0 II. y2 – 14y + 49 = 0122. I. x2 – 19x + 70 = 0 II.2y2 – 17y + 35 = 0123. I. 3x2 + 5x – 8 = 0 II. y2 – 4y + 3=0124. I. 12x2 – 16x + 5 = 0 II. 18y2 – 45y + 25 = 0125. I. 3x2 + 11x + 8 = 0 II. 3y2 + 20y + 32 = 0


(l) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relationship can be established between ‘x’ and ‘y’.

126. I. x = 3 357911 II. y = 5041127. I. 5x + 7y = -43 II. 9x – 17y = 41128. I. x2 + 11x + 30 = 0 II. y2 + 9y + 20 = 0129. I. 4x2 + 3x – l = 0 II. 6y2 – 5y + l = 0130. I. 3x2 + 15x + 18 = 0 II. 2y2 + 15y + 27 = 0

Directions (Q. 131-135) : In the following questions, two equations numbered I and H aregiven. You have to solve both the equations and give answer—

(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or relationship cannot be established

131. I. 4x + 3y = (1600)1/2 II. 6x – 5y = (484)1/2

132. I. 22x (4 13)x 2 13 0 II. 210y (18 5 13)y 9 13 0

133. I. (6x2 + l7) – (3x2 + 20) = 0 II. (5y2 – 12) – (9y2 – 16) = 0

134. I. 1/2(169) x 289 134 II. 1/2 2(361) y 270 1269

135. I. 82lx2 – 757x2 = 256 II. 3 3196 y 12y 16


(l) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between x and y.

136. I. 5x - 7y = -24 II. 13x + 3y = 86137. I. x2 - 13x + 40 = 0 II. y2 + 3y - 40 = 0138. I. 8x2 - 26x+15 = 0 II. 2y2 - 17y + 30 = 0139. I. x2 = 484 II. y2 - 45y + 506 = 0

140. I. 13x - 21=200 - 4x II. y = 3 2197Directions (Q. 141-145) : In each of these questions, two equations (I) and (II) are given. You

have to solve both the equations and give answer(1) if p > q (2) if p q (3) if p < q (4) if p q(5) if p = q or there is no relation between ‘p’ and ‘q’.

141. I. (p + q) 2 = 3136 II. q + 2513 = 2569142. I. 4p2 - 16p +15 = 0 II. 2q2 + 5q - 7 = 0143. I. p2 = 49 II. q2 +15q + 56 = 0144. I. 2p2 + 5p - 12 = 0 II. 2q2 - q - 1 = 0145. I. p2 - 12p + 35 = 0 II. q2 - 25 = 0

Directions (Q. 146–150) : In each of these questions, two equations (I) and (II) are given. Youhave to solve both the equations and give answer

(1) if x > y (2) if x < y (3) if x y (4) if x y(5) if x = y or there is no relation between ‘x’ and ‘y’.

287


146. I. 3x2 + 7x + 2 = 0 II. 2y2 + 9y + 10 = 0147. I. x2 + x – 2 = 0 II. y2 – 3y + 2 = 0148. I. 20x2 – 51x + 27 = 0 II. 15y2 – 16y + 4 = 0149. I. 7x2 + 16x – 15 = 0 II. y2 – 6y – 7 = 0150. I. x2 = 729 II. y2 + 58y + 840 = 0

Directions (Q. 151-155) : In the following questions two equations numbered I and II aregiven. You have to solve both the equations and give answer if

(l) x > y (2) x y (3) x < y (4) x y(5) x = y or the relationship between ‘x’ and ‘y’ cannot be established.

151. I.1215 9 (x)

x x II. y10 - (36)5 = 0

152. I. 5x + 2y = 96 II. 3(7x + 5y) = 489

153. I.1

2 22(441) x 111 (15) II. 2 3121y (6) 260

154. I. 17x = (13)2 + 196 + (5) 2 + 4x II. 9y - 345 = 4y - 260155. I. 3x2 - 13x + 14 = 0 II. y2 - 7y + 12 = 0


(1) if x > y (2) if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between x and y.

156. I. 2x2 – 15x + 28 = 0 II. 2y2 + 3y-35 = 0157. I. 7x – 5y = 24 II. 4x + 3y = 43

158. I. x = 3 2744 II. y = 487159. I. x2 – 9x + 8 = 0 II. 2y2 – 11y + 5 = 0160. I. 2x2 + 3x + 1 = 0 II. 6y2 + 17y + 12 = 0



161. I. 3x2 – 29x + 56 = 0 II. 3y2 – 5y – 8 = 0162. I. 5x2 + 26x – 24 = 0 II. 5y2 – 34y + 24 = 0163. I. x2 – 7x = 0 II. 2y2 + 5y + 3 = 0164. I. 7x – 4y = 40 II. 8x + 8y = 8165. I. 15x2 – 41x + 14 = 0 II. 2y2 – 13y + 20 = 0

Directions (Q. 166-170) : In each of these questions, two equations (I) and (II) are given. Youhave to solve both the equations and give answer “>


166. I. 2x 8 3x 45 0 II. 2y 2y 24 0

167. I. x 7 2x 24 0 II. y 5 2y 12 0

168. I. 12x2 - 17x + 6 = 0 II. 20y2 - 31y + 12 = 0169. I. 3x2 - 8x + 4 = 0 II. 4y2 - 15y + 9 = 0170. I. x2 -16x + 63 = 0 II. y2 - 2y - 35 = 0


(1) if x > y (2)if x y (3) if x < y (4) if x y(5) if x = y or no relation can be established between x and y.

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171. I. 63x 94 x 35 II. 32y 52 y 21 0

172. I. 2x 7 3x 35 15 5 5x II. 2y 5 5y 30 0173. I. 14x2 + 11x - 15 = 0 II. 20y2 - 31y + 12 = 0

174. I. 25x 16y 41 II. 16x 25y 40

175. I.

152

2

(18)x 0x

II.

92(19)y 0

y

Directions (Q. 176-180) : In each of these questions, two equations (I) and (II) are given.Solve both the equations and give answer

(1) if x > y (2) if x < y (3) if x y (4) if x y(5) if x = y or no relation can be established between ‘x’ and ‘y’.

176. I. 63x 194 x 143 0 II. 99y 255 y 150 0

177. I. 16x2 – 40x – 39 = 0 II. 12y2 – 113y + 255 = 0

178. I. x 7 3x 36 0 II. y 12 2y 70 0

179. I. 2x 7 7x 84 0 II. 2y 5 5y 30 0

180. I. 10x – 6y = 13 II. 45x + 24y = 56Directions (Q. 181-185) : In each of these questions, two equations (I) and (II) are given. You

have to solve both the equations and give answer(1) if x > y (2) if x y (3) if x < y (4) I x y(5) if x = y or no relation can be established between x and y.

181. I. x2 - 2x -15 = 0 II. y2 + 5y + 6 = 0182. I. x2 - x - 12 = 0 II. y2 - 3y + 2 = 0

183. I. x - 169 = 0 II. y2 - 169 = 0

184. I. x2 - 32 = 112 II. y - 256 = 0

185. I. x2 - 25 = 0 II. y2 - 9y + 20 = 0Directions (Q. 186-190): In the following questions, three equations numbered I, II and III

are given. You have to solve all the equations either together or separately, or two together andone separately or by any other method and give answer

(1) if x = y > z (2) if x < y = z (3) if x < y > z(4) if x = y = z or if none of the above relationship can be established. (5) if x y < z

186. I. 3x + 5y = 69 II. 9x + 4y = 108 III. x + z = 12

187. I.1 13 4y (729) (6541) II. 2x + 5z = 54 III. 6x + 4z = 74

188. I. 2x + 3y + 4z = 66 II. 2x + y + 3z = 42 III. 3x + 2y + 4z = 63189. I. (x + z)3 = 1728 II. 2x + 3y = 35 III. x - z = 2190. I. 4x + 5y = 37 II. x + z = 8 III. 7x + 3y = 36


(1) if x < y (2) if x y(3) if x = y or no relation can be established (4) if x > y (5) if x y

191. I. 7x + 3y = 77 II. 2x + 5y = (2601)12

192. I. 23x (6 17)x 2 17 0 II. 210y (18 5 17)y 9 17 0

289


193. I.12(289) x 324 203 II.

12(484) y 225 183

194. I. 679x2 - 168x2 = 3066 II. 3 3144y 9y 1536

Directions (Q. 195-197): In the following questions two equations numbered I and II aregiven. Solve both the equations and give answer

(1) if x < y (2) if x y (3) if x y (4) if x > y(5) if x = y or no relationship can be established

195. I. 3x + 4y = (1681)12 II. 3x + 2y = (961)

12

196. I. 23x (6 17)x 2 17 0 II. 210y (15 17) 3 17 0

197. I. x2 - 16x + 63 = 0 II. y2 - 2y - 35 = 0

290


SHORT ANSWER1. (3) 2. (4) 3. (1) 4. (2) 5. (2) 6. (1) 7. (3) 8. (4)9. (1) 10. (2) 11. (1) 12. (5) 13. (1) 14. (2) 15. (4) 16. (5)17. (2) 18. (5) 19. (2) 20. (1) 21. (4) 22. (5) 23. (3) 24. (2)25. (1) 26. (1) 27. (5) 28. (4) 29. (2) 30. (3) 31. (3) 32. (1)33. (3) 34. (2) 35. (4) 36. (1) 37. (3) 38. (2) 39. (4) 40. (5)41. (4) 42. (5) 43. (3) 44. (1) 45. (2) 46. (1) 47. (4) 48. (4)49. (1) 50. (3) 51. (1) 52. (2) 53. (4) 54. (2) 55. (3) 56. (3)57. (3) 58. (1) 59. (1) 60. (5) 61. (1) 62. (1) 63. (5) 64. (4)65. (2) 66. (4) 67. (1) 68. (2) 69. (5) 70. (3) 71. (2) 72. (2)73. (2) 74. (3) 75. (4) 76. (1) 77. (2) 78. (5) 79. (2) 80. (3)81. (3) 82. (1) 83. (4) 84. (5) 85. (1) 86. (1) 87. (5) 88. (1)89. (5) 90. (2) 91. (4) 92. (5) 93. (1) 94. (3) 95. (3) 96. (1)97. (3) 98. (4) 99. (1) 100. (5) 101. (3) 102. (1) 103. (5) 104. (4)105. (1) 106. (3) 107. (1) 108. (2) 109. (5) 110. (1) 111. (1) 112. (4)113. (3) 114. (1) 115. (4) 116. (2) 117. (1) 118. (3) 119. (3) 120. (3)121. (3) 122. (2) 123. (4) 124. (4) 125. (2) 126. (5) 127. (1) 128. (4)129. (3) 130. (2) 131. (1) 132. (2) 133. (5) 134. (2) 135. (4) 136. (3)137. (2) 138. (4) 139. (4) 140. (5) 141. (3) 142. (1) 143. (2) 144. (5)145. (2) 146. (3) 147. (4) 148. (1) 149. (5) 150. (1) 151. (2) 152. (1)153. (1) 154. (3) 155. (3) 156. (2) 157. (1) 158. (3) 159. (5) 160. (1)161. (2) 162. (4) 163. (1) 164. (1) 165. (3) 166. (5) 167. (2) 168. (4)169. (5) 170. (1) 171. (5) 172. (1) 173. (3) 174. (1) 175. (3) 176. (5)177. (2) 178. (2) 179. (1) 180. (2) 181. (2) 182. (5) 183. (2) 184. (3)185. (5) 186. (3) 187. (3) 188. (2) 189. (1) 190. (2) 191. (4) 192. (3)193. (4) 194. (1) 195. (4) 196. (5) 197. (2)

291


01. 3; I. 6x2 - 9x - 10x + 15 = 0or, 3x(2x - 3) - 5(2x - 3) = 0or, (3x - 5) (2x - 3) = 0

5 3x ,3 2

II. 10y2 - 15y - 14y + 21 = 0or, 5y(2y - 3) - 7(2y - 3) = 0or, (5y - 7) (2y - 3) = 0

7 3y ,5 2

x y2. 4; I. 12x2 + 32x - 21x - 56 = 0

or, 4x(3x + 8) - 7(3x + 8) = 0or, (4x - 7) (3x + 8) = 0

7 8x ,4 3

II. 4y2 - 8y - 7y + 14 = 0or, 4y(y - 2) - 7(y - 2) = 0or, (4y - 7) (y - 2) = 0

7y 2,4

x y3. 1; I. 3x2 + 9x + 4x + 12 = 0

or, 3x(x + 3) + 4(x + 3) = 0or, (3x + 4) (x + 3) = 0

4x , 33

II. y2 + 5y + 4y + 20 = 0or, y(y + 5) + 4(y + 5) = 0or, (y + 4) (y + 5) = 0y = - 4, - 5 x > y

4. 2; I. 8x2 - 8x - 7x + 7 = 0or, 8x(x - 1) -7(x - 1) = 0or, (8x - 7) (x - 1) = 0

7x ,18

II. 2y2 - 4y - 3y + 6 = 0or, 2y(y - 2) -3(y - 2) = 0or, (y - 2) (2y - 3) = 0

3y 2,2

5. 2; Eqn (I) × 4 + Eqn (II) × 328x - 12y = 5215x + 12y = 12043x = 172 x = 4 and y = 5

x < y6. 1; I. 2x2 - 6x - 5x + 15 = 0

or,2x(x - 3) - 5(x - 3) = 0or, (2x - 5) (x - 3) = 0

5x 3,2

II. 21y2 - 14y - 9y + 6 = 0or,7y(3y - 2) - 3 (3y - 2) = 0or,(7y - 3)(3y - 2) = 0

3 2y ,7 3

x > y

7. 3; I. 5x2 - 5x - 11x + 11 = 0or,5x(x - 1) - 11(x - 1) = 0or,(x - 1) (5x - 11) = 0

x = 1,115 x > y

II. 5y2 - 5y + 2y - 2 = 0or, 5y (y - 1) + 2(y - 1) = 0or, (5y + 2)(y - 1) = 0

y = 1, - 25 x > y

8. 4; I. x2 + 4x + 7x + 28 = 0or, x(x + 4) +7(x + 7) = 0or, (x + 4) (x + 7) = 0 x = - 4, - 7II. 2y2 + 8y + 5y + 20 = 0or, 2y(y + 4) + 5(y + 4) = 0or, (y + 4) (2y + 5) = 0

y = -4,52

x < y

9. 1; I. 6x2 + 15x + 14x + 35 = 0or, 3x(2x + 5) + 7(2x + 5) = 0or,(3x + 7) (2x + 5) = 0

7 5x ,3 2

II. 3y2 + 9y + 10y + 30 = 0or, 3y(y + 3) +10(y + 3) = 0or,(3y + 10) (y + 3) = 0

10y 3,3

x > y

10. 2; eqn (I) × 5 - eqn (II) × 210x + 25y = 3010x ± 22y = 18- - - . 3y = 12 y = 4 and x = -7 y > x

11. 1; I. 1225x 4900 0

DETAIL - EXPLANATIONS

292


or, 35x + 70 = 0 or, x =70 235

II. 3y + 7 = 0 or y = 73

:. x > y

12. 5; I. 2 2

18 6x 12 8x x

or, x = 13 = .333

II. y2 = 16.95 - 9.68 - 5.64 = 1.63:. y = ±1.277

13. 1; I. 3 32 1331 1363x6 6

II. 5y3 - 4y3 = 5894

or y3 =5894 :. x > y

14. 2; I. 2x2 - llx + 12 = 0

or, x = 4, 32

II. 4y2 - 8y + 3 = 0

3 1y ,2 2

:. x > y

15. 4; I. 7 35 5x 9 169 x

7 35 5or, x x 169 9

7 35or, x 1521

or, x2 = 1521

x = ± 39

II. 11 124 4 273y y y

7

or, 1 114 24y 39

or, y = 39x < y

16. 5; I. x = 4 2401 x = 7

II. 2y2 - 16y + 7y - 56 = 0 2y(y - 8) + 7(y - 8) = 0 (2y + 7) (y - 8) = 0

7y 8,2

Hence, no relation exists between x and y.17. 2; I. 5x2 + 10x - 7x - 14 = 0

or, 5x(x + 2) - 7(x + 2) = 0or, (x + 2) (5x - 7) = 0

x = - 2, 75

II. 2y2 - 4y - 5y + 10 = 0or, 2y(y - 2) - 5(y - 2) = 0

or, (2y - 5)(y - 2) = 0

or, y = 2, 52

x < y18. 5; I. 8x2 + 24x + 7x + 21 = 0

or, 8x(x + 3) + 7(x + 3) = 0or, (x + 3) (8x + 7) = 0

x = - 3, 78

II. 5y2 + 20y - 9y - 36 = 0or, 5y(y + 4) - 9(y + 4) = 0or, (y + 4) (5y - 9) = 0

y = -4, 95

Hence, no relation exists between x and y.

19. 2; I. y 1089or, y = 33

II. 12 y 12 33 45x 15

3 3 3

x < y20. 1; I. 15x2 + 68x + 77 = 0

or, 15x2 + 35x + 33x + 77 = 0or, 5x(3x + 7) + 11(3x + 7) = 0or, (5x + 11) (3x + 7) = 0

7 11x ,3 5

II. 3y2 + 29y + 68 = 0or, 3y2 + 12y + 17y + 68 = 0or, 3y(y + 4) + 17(y + 4) = 0or, (y + 4) (3y + 17) = 0

y = -4, 173

x > y

21. 4; I. 2x2 + 2x - x - 1 = 0or, 2x(x + 1) - 1(x + 1) = 0or, (x + 1) (2x - 1) = 0

x = - l, 12

II. 6y2 - 3y - 10y + 5 = 0or, 3y(2y - 1) - 5(2y - 1) = 0or, (3y - 5)(2y - 1) = 0

y = - 3, 113

x y22. 5; I. 21x2 - 45x - 77x + 165 = 0

or, 3x(7x - 15) - 11 (7x - 15) = 0or, (3x - 11) (7x - 15) = 0

293


11 15x ,3 7

II. 3y2 + 9y - 11y - 33 = 0or,3y(y + 3) - 11(y + 3) = 0or,(3y - 11) (y + 3) = 0

y = - 3, 113

Hence, no relation can be established between xand y.

23. 3; I. 5x2 - 20x - 9x + 36 = 0or, 5x(x - 4) - 9(x - 4) = 0or,(x - 4) (5x - 9) = 0

x = 4, 95

II. 10y2 + 15y - 18y - 27 = 0or, 5y(2y + 3) - 9(2y + 3) = 0or, (2y + 3) (5y - 9) = 0

y = 9 3,5 2

x y24. 2; eqn (I) × 3 - eqn (II) × 4

21x + 12y = 920x + 12y = 12- - - . x = - 3and y = 6 x < y

25. 1; I. 7x2 - 21x - 33x + 99 = 0or, 7x(x - 3) - 33(x - 3) = 0or, (x - 3) (7x - 33) = 0

x = 3, 337

II. 4y2 - 6y - 10y + 15 = 0or, 2y(2y - 3) - 5(2y - 3) = 0or, (2y - 3)(2y - 5) = 0

3 5 y= ,2 2

26. 1; I. 5x2 - 45x - 42x + 378 = 0or, 5x(x - 9) - 42(x - 9) = 0or, (5x - 42) (x - 9) = 0

x = 9, 425

II. 3y2 - 24y - 25y + 200 = 0or, 3y(y - 8) - 25(y - 8) = 0or, (y - 8) (3y - 25) = 0

25y 8,3

x > y27. 5; I. 10x2 - 16x + 15x - 24 = 0

or, 2x(5x - 8) + 3(5x - 8) = 0or, (2x + 3) (5x - 8) = 0

3 8x ,2 5

II. y2 - 2y = 0or, y(y - 2) = 0

y = 0, 2ie no relationship exists between x and y.

28. 4; I. x2 - 2x - 3x + 6 = 0or, x(x - 2) - 3(x - 2) = 0or, (x - 2) (x - 3) = 0

x = 2, 3II. 2y2 - 6y - 9y + 27 = 0or, 2y(y - 3) - 9(y - 3) = 0or, (y - 3) (2y - 9) = 0

y = 3, 92

x y29. 2; I. eqn (I) × 5 + eqn (II) × 2

15x + 10y = 150514x - 10y = 14829x = 1653

x = 1653

29 = 57

and y = 65 x < y30. 3; I. 14x2 - 37x + 24 = 0

or, 14x2 - 21x - 16x + 24 = 0or, 7x(2x - 3) - 8(2x - 3) = 0or, (2x - 3) (7x - 8) = 0

x =3 8,2 7

II. 28y2 - 53y + 24 = 0or, 28y2 - 21y - 32y + 24 = 0or, 7y(4y - 3) - 8(4y - 3) = 0or, (7y - 8) (4y - 3) = 0

8 3y ,7 4

x y31. 3; eqn (I) × 7

eqn (II) × 11 77x + 35y = 819 - 77x ± 143y = 1683 - 108y = - 864 y = 8, x = 7 ie x < y

32. 1; I. 6x2 + 21x + 30x + 105 = 0or, 3x(2x + 7) + 15(2x + 7) = 0or, (3x + 15) (2x + 7) = 0

x = -5, 72

II. 2y2 + 12y + 13y + 78 = 0or, 2y(y + 6) + 13(y + 6) = 0or, (2y + 13) (y + 6) = 0

294


y = 13, 62

x < y33. 3; eqn (I) × 4

eqn (II) × 7 24x + 28y = 208- 98x ± 28y = 245 - 74x = - 37

x = 12 , y = 7

x < y34. 2; I. x2 + 5x + 6x + 30 = 0

or, x(x + 5) + 6(x + 5) = 0or, (x + 5) (x + 6) = 0 x = - 5, - 6II. y2 + 12y + 36 = 0or, (y + 6)2 = 0or, y + 6 = 0 y = - 6ie x y

35. 4; I. 2x2 + 2x - x - 1 = 0or, 2x(x + 1) - 1(x + 1) = 0or, (2x - 1) (x + 1) = 0

1x , 12

II. 2y2 - 2y - y + 1 = 0or, 2y(y - 1) - 1(y - 1) = 0or, (2y - 1)(y - 1) = 0

1y ,12

ie x y36. 1; 7x + 6y + 4z = 122 ... (i)

4x + 5y + 3z = 88 ... (ii)9x + 2y + z = 78 ... (iii)From (i) and (ii)

5x - 2y = 14... (iv)From (ii) and (iii)

23x + y = 146 ... (v)From (iv) and (v),

x = 6, y = 8Putting the value of x and y in eqn (i), we get

z = 8:. x < y = z

37. 3; 7x + 6y = 110 ... (i)4x + 3y = 59 ... (ii)x + z = 15 ... (iii)From eqn (i) and (ii), x = 8, y = 9Put the value of x in eqn (iii).Then, z = 7x < y > z

38. 2; 2 1/2 4 1/4(6 ) (6 )x

6 6 6 ... (i)

2y + 3z = 33 ... (ii)6y + 5z = 71 ... (iii)From eqn (ii) and (iii),

y = 6 and z = 7x = y , z

39. 4; 8x + 7y = 135 ... (i)5x + 6y = 99 ... (ii)9y + 8z = 121 ... (iii)From eqn (i) and (ii),

x = 9, and y = 9Putting the value of y in eqn (iii),

z = 5:. x = y > z

40. 5; (x + y)3 = 1331or, x + y = 11 ... (i)(x + y)2 = 121(x - y)2 + 4xy = 121x - y = 3... (ii)

[value of xy from eqn (iii)]From eqn (i) and (ii), x = 7, y = 4Put the value x and y in the eqn

x - y + z = 07 - y + z = 03 + z = 0z = -3

41. 4; I. 7x2 - 7x - 2x + 2 = 0or, 7x(x - 1) - 2(x - 1) = 0 (7x - 2) (x - 1) = 0

or, x = 2,17

II. y2 - y - 3y + 3 = 0or, y(y - 1) - 3(y - 1) = 0or, (y - 3) (y - 1) = 0 y = 1, 3 x y

42. 5; I. x2 = 64 x = ±8II. 2y2 + 9y + 16y + 72 = 0or, y(2y + 9) + 8(2y + 9) = 0or, (y + 8) (2y + 9) = 0

9y 8,2

ie, no relation between x and y.43. 3; I. x2 + x - 20 = 0

or, x2 + 5x - 4x - 20 = 0or, x(x + 5) - 4(x + 5) = 0or, (x - 4) (x + 5) = 0 x = 4, - 5II. 2y2 - 10y - 9y + 45 = 0or, 2y(y - 5) - 9(y - 5) = 0or, (y - 5) (2y - 9) = 0

9y 5,2

x < y

295


44. 1; Eqn (I) × 2Eqn (II) × 714x + 6y = 5214x + 119y = - 287 - - + . - 113y = 339 y = - 3 and x = 5, ie x > y

45. 2; I. 3x2 - 9x - 11x + 33 = 0or, 3x(x - 3) - 11(x - 3) = 0or, (3x - 11) (x - 3) = 0

x = 3, 113

II. 2y2 - 6y - 5y + 15 = 0or, 2y(y - 3) - 5(y - 3) = 0or, (y - 3) (2y - 5) = 0

5y 3,2

x y46. 1; I. 4x2 - 28x - 15x + 105 = 0

or, 4x(x - 7) - 15(x - 7) = 0or, (x - 7) (4x - 15) = 0

x = 7, 154

II. 7y2 - 14y - 15y + 30 = 0or, 7y(y - 2) - 15(y - 2) = 0or, (y - 2)(7y - 15) = 0

y = 2, 157

x > y47. 4; I. x2 + 8x + 5x + 40 = 0

or, x(x + 8) + 5(x + 8) = 0or, (x + 5) (x + 8) = 0 x = - 5, - 8II. y2 + 2y + 5y + 10 = 0or, y(y + 2) + 5(y + 2) = 0or, (y + 2)(y + 5) = 0 y = - 2, - 5 x y

48. 4; I. x = 3 2197 x = 13II. 2y2 - 28y - 26y + 364 = 0or, 2y(y - 14) - 26(y - 14) = 0or, (2y - 26) (y - 14) = 0 y = 14, 13 x y

49. 1; I. 5x2 - 15x - 12x + 36 = 0or, 5x(x - 3) - 12(x - 3) = 0or, (5x - 12) (x - 3) = 0

12x , 35

II. y2 - y - 2y + 2 = 0or, y(y - 1) - 2(y - 1) = 0or, (y - 1)(y - 2) = 0 y = 1, 2 x > y

50. 3; eqn (I) × 5 + eqn (II) × 8 65x - 40y + 405 = 0120x + 40y + 520 = 0185x + 0 + 925 = 0

925x 5185

13x 81y8

65 81 16 28 8

x < y51. 1; I. 15x2 - 10x - 9x + 6 = 0

or, 5x(3x - 2) -3(3x - 2) = 0or, (5x - 3) (3x - 2) = 0

3 2x ,5 3

II. 6y2 - 3y - 2y + 1 = 0or, 3y(2y - 1) -1(2y - 1) = 0or, (3y - 1)(2y - 1) = 0

1 1y ,3 2

x > y

52. 2; I. x 172

x = 13.11II. y2 - 14y - 15y + 210 = 0or, y(y - 14) - 15(y - 14) = 0or, (y - 14) (y - 15) = 0 y = 14, 15 x < y

53. 4; I. 3x2 -12x - 8x + 32 = 0or, 3x(x - 4) - 8(x - 4) = 0or, (x - 4) (3x - 8) = 0

x = 4, 83

II. 2y2 - 8y - 11y + 44 = 0or, 2y(y - 4) -11(y - 4) = 0or, (y - 4) (2y - 11) = 0

11y 4,2

x y54. 2; 4 × eqn (I) - 3 × eqn (II),

12x + 32y = -812x + 54y = 3

- - - .

296


-22y = -11

y = 12 and x = -2

x < y55. 3; I. 2x2 - 8x - 7x + 28 = 0

or, 2x(x - 4) - 7(x - 4) = 0or, (x - 4) (2x - 7) = 0

x = 4, 72

II. 10y2 - 35y + 34y - 119 = 0or, 5y(2y - 7) + 17(2y - 7) = 0or, (2y - 7)(5y + 17)

7 17y ,2 5

x y56. 3; I. 676x2 - 1 = 0

or x2 =1 1x

676 26

II. 3

1 1y y2413824

ie, x < y57. 3; On solving these two equations, we get

x = -2, y = 6ie, x < y

58. 1; I. 7x2 - 28x + 30x - 120 = 0or, 7x(x - 4) + 30(x - 4) = 0or, (x - 4) (7x + 30) = 0

x = 4, 307

II. y2 + 6y + 5y + 30 = 0or, y(y + 6) + 5(y + 6) = 0or, (y + 5) (y + 6) = 0y = -5, - 6ie, x > y

59. 1; I. x2 = 7xor, x2 - 7x = 0or, x(x - 7) = 0 x = 0, 7II. (y + 7)2 = 0or, (y + 7) = 0 y = -7ie, x > y

60. 5; I. 2x2 - 6x + 11x - 33 = 0or, 2x(x - 3) + 11(x - 3) = 0or, (2x + 11) (x - 3) = 0

x = 3, 112

II. y2 - 3y + 2y - 6 = 0or, y(y - 3) + 2(y - 3) = 0or, (y + 2)(y - 3) = 0 y = - 2, 3 i.e no relation exists between x and y

61. 1; I. x2 + 12x + 36 = 0

or, (x + 6)2 = 0or, x + 6 = 0or, x = - 6II. y2 + 15y + 56 = 0or, y2 + 7y + 8y + 56 = 0or, y(y + 7) + 8(y + 7) = 0or, (y + 7) (y + 8) = 0 y = -7, -8 x > y

62. 1; I. x2 = 35

x = ± 35II. y2 + 13y + 42 = 0or, y2 + 6y + 7y + 42 = 0or, y(y + 6) + 7(y + 6) = 0or, (y + 6) (y + 7) = 0 y = -6, - 7 x > y

63. 5; I. 2x2 - 3x - 35 = 0or, 2x2 - 10x + 7x - 35 = 0or, 2x(x - 5) + 7(x - 5) = 0or, (2x + 7) (x - 5) = 0

x =7 ,2

5

II. y2 - 7y + 6 = 0or, y2 - y - 6y + 6 = 0or, y(y - 1) - 6(y - 1)or, (y - 1)(y - 6) = 0 y = 1, 6No relation can be established between x and y.

64. 4; I. 6x2 - 29x + 35 = 0or, 6x2- 15x - 14x + 35 = 0or, 3x(2x - 5) -7(2x - 5) = 0or, (3x - 7) (2x - 5) = 0

7 5x ,3 2

II. 2y2 - 19y + 35 = 0or, 2y2 - 14y - 5y + 35 = 0or, 2y(y - 7) -5 (y - 7) = 0or, (2y - 5)(y - 7) = 0

y = 5 , 72

x y65. 2; I. 12x2 - 47x + 40 = 0

or, 12x2 - 32x - 15x + 40 = 0or, 4x(3x - 8) -5(3x - 8) = 0or, (4x - 5) (3x - 8) = 0

x = 5 8,4 3

II. 4y2 + 3y - 10 = 0

297


or, 4y2 + 8y - 5y - 10 = 0or, 4y(y + 2) -5(y + 2) = 0or, (4y - 5) (y + 2) = 0

y = 5 , 24

x y66. 4; I. x2 + 7x - 4x - 28 = 0

or, x(x + 7) - 4 (x + 7) = 0or, (x - 4)(x + 7) = 0 x = 4, - 7II. y2 - 11y + 28 = 0or, y2 - 7y - 4y + 28 = 0or, y (y - 7) -4(y - 7) = 0or, (y - 4) (y - 7) = 0 y = 4, 7x y

67. 1; I. 6x2 - 17x + 12 = 0or, 6x2 - 9x - 8x + 12 = 0or, 3x (2x - 3) - 4 (2x - 3) = 0or, (3x - 4) (2x - 3) = 0

4 3x ,3 2

II. 6y2 - 3y - 4y + 2 = 0or, 3y (2y - 1) - 2 (2y - 1) =0or, (3y - 2) (2y - 1) = 0

2 1y ,3 2

x > y

68. 2; I. 256x576

16 2x24 3

II. 3y2 + y - 2 = 0or, 3y2 + 3y - 2y - 2 = 0or, 3y (y + 1) - 2(y + 1) = 0or, (3y - 2) (y + 1) = 0

y = 2 , 13

x y69. 5; I. x2 = 64

x = ± 8II. y2 = 9yor, y2 - 9y = 0or, y (y - 9) = 0 y = 0, 9 no relationship can be established between xand y.

70. 3; I. x2 + 6x - 7 = 0

or, x2 + 7x - x - 7 = 0or, x(x + 7) -1 (x + 7)= 0or, (x - 1) (x + 7) = 0 x = 1, -7II. 41y + 17 = 140or, 41y = 140 - 17 = 123

y = 123 341

x < y71. 2; I. x2 + 3x - 28 = 0

or, x2 + 7x - 4x - 28 = 0or, x (x + 7) - 4 (x + 7) = 0or, (x - 4) (x + 7) = 0or, x = 4, -7II. y2 + 9y + 7y + 63 = 0or, y(y + 9) + 7(y + 9) = 0or, (y + 7)(y + 9) = 0or, y = -7, -9 x y

72. 2; I. x = 3 2197 x = 13II. y2 = 169 y = ±13 x y

73. 2; I. 8x2 - 40x - 9x + 45 = 0or, 8x (x - 5) -9 (x - 5) = 0or, (8x - 9) (x - 5) = 0or, x = 5, 9/8II. 8y2 + 8y - 9y -9 = 0or, 8y (y + 1) -9 (y + 1) = 0or, (8y - 9) (y + 1) = 0

9y , 18

x y74. 3; 42x - 17y = -67

42x + 72y = -156 eqn (II) × 6 - - + .

-89y = 89

89y 1 and x 289

x < y75. 4; I. x2 - 8x + 15 = 0

or, x2 - 3x - 5x + 15 = 0or, x (x - 3) -5 (x - 3) = 0or, (x - 3) (x - 5) = 0or, x = 3, 5II. 2y2 - 10y + 55 = 0or, 2y (y - 5) -11 (y - 5) = 0or, (y - 5)(2y - 11) = 0

298


or, y = 5, 112

x y76. 1; I. 2.3p - 20.01 = 0

20.01p 8.72.3

II. 2.9q - p = 0

or, p = 2.9q8.7

q 8.72.9

ie p > q

77. 2; I. p = 1764 p = 42II. q2 = 1764 q = + 42ie p > q

78. 5; I. p2 - 26p + 168 = 0 p2 - 12p - 14p + 168 = 0 p(p - 12) - 14(p - 12) = 0 (p - 12) (p - 14) = 0 p = 12, 14II. q2 - 25q + 156 = 0 q2 - 13q - 12q + 156 = 0q(q - 13) - 12(q - 13) = 0 (q - 12) (q - 13) = 0q = 12, 13Hence, no relation can be established between pand q

79. 2; I. p2 - l3q + 42 = 0p2 - 6p - 7p + 42 = 0p(p - 6) - 7(p - 6) = 0(p - 6) (p - 7) = 0 p = 6, 7II. q2 + q - 42 = 0q2 + 7q - 6p - 42 = 0q(q + 7) - 6(q + 7) = 0(q - 6)(q + 7) = 0 q = 6, - 7 ie p q

80. 3; eqn(I) × 3 18p - 15q = -141eqn (II) × 5 25p + 15q = 55

43p = -86

86p 243

5p + 3q = 113q = 11 - 5p 3q = 11 + 103q = 21 q = 7 ie p < q

81. 3; I. 2x2 + 13x - 7 = 0or 2x2 + 14x - x - 7 = 0or 2x (x + 7) - 1 (x + 7) = 0or (2x - 1) (x + 7) = 0

1x , 72

II. 2y2 - 5y + 3 = 0or 2y2 - 2y - 3y + 3 = 0

or 2y(y - 1) - 3(y - 1) = 0or (2y - 3) (y - 1) = 0

3y 1, 2

Hence x < y82. 1; I. 2x2 - 8x - 7x + 28 = 0

or 2x (x - 4) - 7(x - 4) = 0or (2x - 7) (x - 4) = 0

7x 4, 2

II. 4y2 - 16y + 15 = 0or 4y2 - 6y - 10y + 15 = 0or 2y (2y - 3) - 5(2y - 3) = 0or (2y - 5) (2y - 3) = 0

5 3y ,2 2

Hence x > y83. 4; I. x2 + 8x + 16 = 0

or (x + 4)2 = 0or x + 4 = 0 x = -4II. y2 = 16 y = ±4Hence, x y

84. 5; I. x2 - 2x - 24 = 0or x2 + 4x - 6x - 24 = 0or x(x + 4) - 6(x + 4) = 0or (x - 6) (x + 4) = 0 x = 6, - 4II. y2 + 8y = 0or y(y + 8) = 0 y = 0, - 8 ie No relation can be establishedbetween x and y.

85. 1; I. x2 + 4x = 0or x(x + 4) = 0 x = 0, - 4II. y2 + 10y + 25 = 0or (y + 5)2 = 0or y + 5 = 0 y = - 5 x > y

86. 1; I. 2x2 + 2x – x – 1 = 0or 2x(x + 1) – 1(x + 1) = 0or (2x – 1) (x + 1) = 0

1x 1,2

II. 2y2 + 3y + 10y + 15 = 0or y(2y + 3) + 5(2y + 3) = 0or (y + 5) (2y + 3) = 0

3y 5,2

x > y87. 5; I. x2 + 4x + 8x + 32 = 0

or x(x + 4) + 8(x + 4) = 0or (x + 4) (x + 8) = 0 x = – 4, – 8II. 2y2 + 6y + 9y + 27 = 0or 2y(y + 3) + 9(y + 3) = 0or (2y + 9) (y + 3) = 0

299


9y , 32

No relation can be established between x and y.

88. 1; I. 6x2 – 9x – 8x + 12 = 0or 3x(2x – 3) – 4(2x – 3) = 0or (2x – 3) (3x – 4) = 0

3 4x ,2 3

II. 7y2 – 7y – 6y + 6 = 0or 7y(y – 1) – 6(y – 1) = 0or (7y – 6) (y – 1) = 0

6y 1,7

x > yNo relation between ‘x’ and ‘y’.

89. 5; I. x2 – 11x – 71x + 781 =0or x(x – 11) – 71(x – 11) = 0or(x – 11)(x – 71) = 0 x = 11, 71II. y2 = 5041 y = ± 71

90. 2; I. 6x2 – 15x – 32x + 80 = 0or 3x(2x – 5) – 16(2x – 5) = 0or (3x – 16) (2x – 5) = 0

16 5x ,3 2

II. 2y2 – 4y – 5y + 10 = 0or 2y(y – 2) – 5(y – 2) = 0or (y – 2) (2y – 5) = 0

5y 2,2

x y

91. 4; I. 3x2 - 12x + 5x - 20 = 0or 3x(x - 4) + 5(x - 4) = 0or (3x + 5) (x - 4) = 0

5x ,43

II. y2 - 8y + 16 = 0or (y - 4)2 = 0 (y - 4) = 0or y = 4 x y

92. 5; I. x2 - 72 = 0or x2 = 72 x = + 8.485II. y2 - y - 8y + 8 = 0or y(y - 1) - 8(y - 1) = 0or (y - 1) (y - 8) = 0y = 1, 8

93. 1; I. 9x2- l14x + 361 =0or (3x - 19)2 = 03x - 19 = 0

x = 193 = 6.33

II. y2 = 36 y = ±6 x > y

94. 3; I. 13x + 17y = 107 eqn (II) × 1313x ± 143y = ± 533 160y = 640

640y 4 and x=11y-41 160

x = 44 - 41 = 3 x < y

95. 3; I. 9x2 + 18x + 9 = 0or x2 + 2x + 1 = 0or (x + l)2 = 0 x + 1 = 0, or x = -1II. y2 - y - 2y + 2 = 0or y(y - 1) -2(y - 1) = 0or (y - 1) (y - 2) = 0

x < y

96. 1; eqn (I) ×3 - eqn (II) × 412x + 21y = 12612x - 44y = -4 - + + . 65y = 130 y = 2and x = 7

97. 3; I. 9x2 - 18x - 1 lx + 22 = 0or 9x(x - 2)- 11(x - 2) = 0or (x - 2)(9x - 11) = 0

11x 2,9

II. y2 - 3y - 4y + 12 - 0or y(y - 3) - 4(y - 3) = 0or (y - 3) (y - 4) = 0 y = 3, 4 x < y

98. 4; I. 3x2 - 4x - 32 = 0or 3x2 - 12x + 8x - 32 = 0or 3x(x - 4) + 8(x - 4) = 0or (3x + 8) (x - 4) = 0

8x 4,3

II. 2y2 - 8y - 9y + 36 = 0or 2y(y - 4) - 9(y - 4) = 0or (2y - 9) (y - 4) = 0or (2y - 9) (y - 4) = 0

9y 4,2

x y99. 1; I. 3x2 - 21x + 2x - 14 = 0

or 3x(x - 7) + 2(x - 7) = 0or (3x + 2) (x - 7) = 0

2x 7,3

II. 2y2 + 5y + 3 = 0or 2y2 + 2y + 3y + 3 = 0or 2y(y + 1) + 3(y + 1) = 0or (2y + 3) (y + 1) = 0

300


3y , 12

x > y100. 5; I. x2 + 14x + 49 = 0

or (x + 7)2 = 0 x + 7 = 0or, x = -7II. y2 + 9y = 0or y(y + 9) = 0 y = 0, -9 ie no relation between x and y.

101. 3; I. 9x2 = 1

2 1x9

1x3

II. 4y2 + 11y - 3 = 0or, 4y2 + 12y - y - 3 = 0or, 4y(y + 3) - 1(y + 3) = 0

1y , 34

Hence, there is no relation between x and y.102. 1; I. 3x2 + 5x - 2 = 0

or, 3x2 + 6x - x - 2 = 0or, 3x(x + 2) - 1(x + 2) = 0or, (3x - 1) (x + 2) = 0

1x 2,3

II. 2y2 - 2y - 5y + 5 = 0or, 2y(y - 1) - 5(y - 1) = 0

5y 1,2

Hence, x < y103. 5; I. 6x2 + 13x + 5 = 0

or, 6x2 + 3x + 10x + 5 = 0or, 3x(2x + 1) + 5(2x + 1) = 0or, (3x + 5) (2x + 1) = 0

5 1x ,3 2

II. 3y2 + 11y + 10 = 0or, 3y2 + 6y + 5y + 10 = 0or, 3y(y + 2) + 5(y + 2) = 0or, (3y + 5) (y + 2) = 0

5y , 23

Hence, x y104. 4; I. 7x - 4y = 29

II. 5x + 3y = 50(I) × 3 + (II) × 4

21x - 12y = 8720x + 12y = 200 41x = 287 x = 7Putting the value of x in (I), we gety = 5Hence, x > y

105. 1; I. x2 = 5

x 5 2.236

II. 4y2 - 24y + 35 = 0or, 4y2 - 14y - 10y + 35 = 0or, 2y(2y - 7) - 5(2y - 7) = 0or, (2y - 5) (2y - 7) = 0

5 7y , 2.5, 3.52 2

Hence, x < y106. 3; I. 35x2 - 28x - 25x + 20 = 0

or 7x(5x - 4) - 5(5x - 4) = 0or (7x - 5) (5x - 4) = 0

5 4x ,7 5

II. 56y2 - 48y - 49y + 42 = 0or 8y(7y - 6) - 7(7y - 6) = 0or (8y - 7) (7y - 6) = 0

7 6y ,8 7

x < y

107. l ; I. x = 3 4913 x = 17II. 13y = 246 - 3xor 13y = 246 - 51 = 195 y = 15 x > y

108. 2; I. x2 - 7x + 2x - 14 = 0or x(x - 7) + 2(x - 7) = 0(x + 2) (x - 7) = 0 x = -2, 7II. y2 + 5y + 2y + 10 = 0or y(y + 5) + 2(y + 5) = 0or (y + 2) (y + 5) = 0 y = -2, -5x y

109. 5; I. x2 = 3481 x = ± 59

II. 3y2 = 3 216000 3y2 = 60

y = ± 20No relation

110. 1; I. 5x2 + 5x - 3x - 3 = 0or 5x (x + 1) - 3(x + 1) = 0or (5x - 3) (x + 1) = 0

3x , 15

II. 2y2 + 4x + 3y + 6 = 0or 2y(y + 2) + 3(y + 2) = 0

301


or (2y + 3) (y + 2) = 0

–3y , 22

ie x > y111. 1; I. 20x2 - 35x - 32x + 56 = 0

or 5x(4x - 7) - 8(4x - 7) = 0or (5x - 8) (4x - 7) = 0

x = 8 7,5 4

II. 56y2 - 32y - 35y + 20 = 0or 8y(7y - 4) - 5(7y - 4) = 0or (8y - 5) (7y - 4) = 0

5 4y ,8 7

x > y

112. 4; I. x4 = 65536 x = +16

II. y = 3 4096 y = 16 x y

113. 3; I. 2x2 + 16x - 5x - 40 = 0or 2x(x + 8) - 5(x + 8) = 0or (2x - 5) (x + 8) = 0

x = 5 , 82

II. 4y2 - 16y - 11y + 44 = 0or 4y(y - 4) - 11(y - 4) = 0or(4y - 11) (y - 4) = 0

y = 4, 114 x < y

114. 1; I. 7x = 4y + 85or 7x = 4 × 26 + 85 (Put y = 26)

x = 189

7 = 27

II. y = 3 17576 y = 26 x > y

115. 4; I. x2 = 14641 x = ±121

II. y = 14641 y = 121 x y

116. 2; I. x2 + 42 = 13xor x2 - 13x + 42 = 0or x2 - 7x - 6x + 42 = 0or x(x - 7) - 6(x - 7) = 0or (x - 6) (x - 7) = 0 x = 6, 7

II. 4y 1296

y = 6 x y

117. 1; I. x2 + x - 2 = 0or x2 + 2x - x - 2 = 0or x(x + 2) - 1(x + 2) = 0or (x - 1) (x + 2) = 0 x = 1, - 2II. y2 + 7y + 12 = 0or y2 + 3y + 4y + 12 = 0or y(y + 3) + 4(y + 3) = 0

or (y + 3) (y + 4) = 0y = -3, -4 x > y

118. 3; I. 3x2 - 23x + 40 = 0or 3x2 - 15x - 8x + 40 = 0or 3x(x - 5) - 8(x - 5) = 0or (3x - 8) (x - 5) = 0

8x 5, 3

II. 2y2 - 23y + 66 = 0or 2y2 - 12y - 11y + 66 = 0or 2y (y - 6) -11 (y - 6) = 0or (y - 6)(2y - 11) = 0

11y 6, 2

x < y119. 3; I. 15x2 - 25x - 21x + 35 = 0

or 5x(3x - 5) - 7(3x - 5) = 0or (5x - 7) (3x - 5) = 0

57x ,5 3

II. 4y2 - 8y - 7y + 14 = 0or 4y(y - 2) - 7(y - 2) = 0or (4y - 7) (y - 2) = 0

y = 2, 74

x < y120. 3; I. x2 - x + 6x -6 = 0

or x(x - 1) + 6(x - 1) = 0or (x - 1) (x + 6) = 0 x = 1, -6II. 2y2 - 6y - 5y + 15 = 0or 2y(y - 3) - 5(y - 3) = 0or (y - 3) (2y - 5) = 0

y = 3, 52

x < y121. 3; I. 2x2 - 21x + 54 = 0

or 2x2 - 12x - 9x + 54 = 0or 2x(x - 6) - 9(x - 6) = 0or (x - 6) (2x - 9) = 0

9x 6,2

II. y2 - 14y + 49 = 0or (y - 7)2 = 0or y - 7 = 0 y = 7 Hence x < y

122. 2; I. x2 - 19x + 70 = 0or x2 - 5x - 14x + 70 = 0or x(x - 5) - 14(x - 5) = 0or (x - 5) (x - 14) = 0 x = 5, 14II. 2y2 - 10y - 7y + 35 = 0or 2y(y - 5) - 7(y - 5) = 0or (y - 5) (2y - 7) = 0

y = 5, 72 Hence x y

123. 4; I. 3x2 + 5x - 8 = 0or 3x2 - 3x + 8x - 8 = 0or 3x(x - 1) + 8(x - 1) = 0

302


or (x - 1) (3x + 8) = 0

8x 1,3

II. y2 - 4y + 3 = 0or y2 - y - 3y + 3 = 0or y(y - 1) - 3(y - 1) = 0or(y - l)(y - 3) = 0 y = 1, 3 Hence, x y

124. 4; I. 12x2 - 16x + 5 = 0or 12x2 - 6x - 10x + 5 = 0or 6x(2x - 1) - 5(2x - 1) = 0or (6x - 5) (2x - 1) = 0

1 5x ,2 6

II. 18y2 - 45y + 25 = 0or 18y2 - 30y - 15y + 25 = 0or 6y(3y - 5) - 5(3y - 5) = 0or (3y - 5) (6y - 5) = 0

5 5y , Hence, x y3 6

125. 2; I. 3x2 + 11x + 8 = 0or 3x2 + 3x + 8x + 8 = 0or 3x(x+ 1) + 8(x + 1) = 0or (x + 1) (3x + 8) = 0

8x 1,3

II. 3y2 + 20y + 32 = 0or 3y2 + 12y + 8y + 32 = 0or 3y(y + 4) + 8(y + 4) = 0or (3y + 8) (y + 4) = 0

8y 4,3

Hence, x y

126. 5; I. x = 3 357911 x = 71

II. y = 5041 y = 71 x = y

127. 1; Eqn(l) × 9 - Eqn (II) × 545x + 63y = -38745x - 85y = 205

- + - . 148y = -592 y = -4 and x = -3 x > y

128. 4; I. x2 + 11x + 30 = 0or x(x + 5) + 6(x + 5) = 0 or (x + 5) (x + 6) = 0 x = -5, -6II. y2 + 4y + 5y + 20 = 0or y(y + 4) + 5(y + 4) = 0or (y + 4) (y + 5) = 0 y = -4,-5 x y

129. 3; I. 4x2 + 4x - x - l = 0or 4x(x+ 1)- l(x + 1) = 0or (4x - 1) (x + 1) = 0

x = -l, 14

II. 6y2 - 3y - 2y + 1 = 0or 3y(2y - 1) - l(2y - 1) = 0

or (3y - 1) (2y - 1) = 0

1 1y ,2 3

x < y130. 2; I. 3x2 + 9x + 6x + 18 = 0

or 3x(x + 3) + 6(x + 3) = 0or (x + 3)(3x + 6) = 0 x = -3, -2II. 2y2 + 6y + 9y+ 27 = 0or 2y(y + 3) + 9(y + 3) = 0or (2y + 9)(y + 3) = 0

9y 3,2

x y131. 1; 4x + 3y = 40 .........(i) ×6

6x - 5y = 22 .........(ii) ×4 24x +18y = 240 24x - 20y = 88

- + - . 38y = 152

152y 438

Putting the value of y in equation (i), we have4x + 3 x 4 = 40or, 4x = 40 12 = 28 x = 7Hence, x > y.

132. 2; 22x 4x 13x 2 13 0 ...(i)

or, 2x x 2 13 x 2 0

or, x 2 2x 13 0

13x 2,2

210y 18y 5 13y 9 13 0 ...(ii)

or, 2y 5y 9 13(5y 9) 0

or, (2y 13)(5y 9) 0

9 13y ,5 2

Hence, x y.133. 5; 6x2 + 17 - 3x2 - 20 = 0 ... (i)

or, 3x2 = 3 x ± l5y2 - 12 - 9y2 + 16 = 0 .... (ii)or, 4y2 = 4 y ± 1

Hence x = y.134. 2; 13x + 17 = 134 .... (i) .

117x 9.13

(36l)1/2y2 - 270 = 1269or, 19y2 = 1629 + 270 = 1539

2 1539y 8119

y ± 9

303


Hence, x y.135. 4; 64x2 = 256 .... (i)

or, x2 = 4 x = ± 214y3 - 12y3 = 16 .... (ii)or, 2y3 = 16 y3 = 8 y = 2Hence x y.

136. 3; 15x - 21y = -7291x + 21y = 602106x = 530

x = 5, y = 7 x < y

137. 2; I. x2 - 13x + 40 = 0or x2 - 5x - 8x + 40 = 0or x(x - 5) -8(x - 5) = 0or (x - 5)(x - 8) = 0 x = 5, 8II. y2 + 3y - 40 = 0or y2 - 5y + 8y - 40 = 0or y(y - 5) + 8(y - 5) = 0or (y - 5) (y + 8) = 0 y = 5, -8Hence, x y

138. 4; I. 8x2 -26x + 15 = 0or 8x2 - 20x - 6x + 15 = 0or 4x(2x - 5) - 3(2x - 5) = 0or (4x - 3) (2x - 5) = 0

x = 3 5,4 2

II. 2y2-17y + 30 = 0or 2y2 - 12y - 5y + 30 - 0or 2y(y - 6) - 5(y - 6) = 0or (2y - 5) (y - 6) = 0

y = 5 , 62

x y139. 4; I. x2 = 484

x = + 22II. y2 - 45y + 506 = 0or y2 - 22y - 23y + 506 = 0or y(y - 22) - 23(y - 22) = 0or (y - 22) (y - 23) = 0 y = 22, 23 x y

140. 5; I. 13x -21 = 200- 4xor 13x + 4x = 200 + 21

x 221 1317

II. y = 3 2197 y = 13 x = y

141. 3; I. (p + q)2 = 3136 p + q = +56II. q + 2513 = 2569or, q = 2569 - 2513 = 56Putting the value of q in (I) we have,p = 0. -112 p < q

142. 1; I. 4p2 - 16p + 15 = 0or, 4p2 - 10p - 6p + 15 = 0

or, 2p(2p - 5) - 3(2p - 5) = 0or, (2p - 3) (2p - 5) = 0

p = 3 5,2 2

II. 2q2 + 5q - 7 = 0or, 2q2 + 7q - 2q - 7 = 0or, q(2q + 7) - 1(2q + 7) = 0or, (q - 1) (2q + 7) = 0

q = 71,2

143. 2; I. p2 = 49 P = ±7II. q2 + 15q + 56 = 0or, q2 + 8q + 7q + 56 = 0or, q(q + 8) + 7(q + 8) = 0or, (q + 7) (q + 8) = 0 q = -7, -8 p q

144. 5; I. 2p2 + 5p - 12 = 0or, 2p2 + 8p - 3p - 12 = 0or, 2p(p + 4) - 3(p + 4) = 0or, (2p - 3) (p + 4) = 0

p = 3 , 42

II. 2q2 - q - 1 = 0or, 2q2 - 2q + q - 1 = 0or, 2q(q - 1) + 1(q - 1) = 0or, (2q + 1) (q - 1) = 0

q = 1, 12

No reation between ‘p’ and ‘q’.145. 2; I. p2 - 12p + 35 = 0

or p2 - 5p - 7p + 35 = 0or p(p - 5) - 7(p - 5) = 0or (p - 7) (p - 5) = 0 p = 5, 7II. q2 - 25 = 0or, q2 = 25 q = +5 p q

146. 3; I. 3x2 + 6x + x + 2 = 0or 3x(x + 2) + 1(x + 2) = 0or (x + 2) (3x + 1) = 0

x = –2,13

II. 2y2 + 4y + 5y + 10 = 0or 2y(y + 2) + 5(y + 2) = 0or (2y + 5) (y + 2) = 0

y = –2, 52

x y147. 4; I. x2 + 2x – x – 2 = 0

or x(x + 2)– 1(x + 2) = 0or (x – 1) (x + 2) = 0 x = 1, –2II. y2 – y – 2y + 2 = 0or y(y – 1) – 2(y – 1) = 0or (y – 1)(y – 2) y = l, 2x y

304


148. 1; I. 20x2 – 15x – 36x + 27 = 0or 5x(4x – 3) – 9(4x – 3) = 0or (5x – 9) (4x – 3) = 0

x = 9 3,5 4

II. 5y2 – 10y – 6y + 4 = 0or 5y(3y – 2) – 2(3y – 2) = 0or (5y – 2) (3y – 2)

2 2y ,5 3

x > y149. 5; I. 7x2 + 21x – 5x – 15 = 0

or 7x(x + 3) – 5(x + 3) = 0or (x + 3) (7x – 5) = 0

5x 3,7

II. y2 – 7y + y – 7 = 0or y(y – 7) + 1(y – 7) = 0or (y + 1) (y – 7) y =–1, 7 no relation between ‘x’ and ‘y’.

150. 1; I. x2 = 729 x = ±27II. y2 + 58y + 840 = 0or y2 + 28y + 30y + 840 = 0or y(y + 28) + 30(y + 28) = 0or (y + 30) (y + 28) = 0 y = –30, –28 x > y

151. 2; I. 1215 9 (x)

x x

1215 9or, x x

x

or, x = 4 x = 4II. y10 - (36)5 = 0or, y10 = (36)5

or y =15210(36) 36

y = 36 6 x y

152. 1; 5x + 2y = 96 ... (i)21x + 15y = 489 ... (ii)Now, eqn (i) × 15 and eqn (ii) × 2

75x + 30y = 144042x + 30y = 978- - - .33x = 462 x = 14

Putting the value of x in eqn (i), we get5 × 14 + 2y = 96 or, 2y = 96 - 70 = 26

or, y =262 = 13

x > y

153. 1; I. 1

2 22(441) x 111 (15)

or, 12 22(21) x 225 111 336

or, 21x2 = 336

x = 336 1621

II. 2 3121y 6 260 or, 11y2 + 63 = 260or, 11y2 = 260 - 216 = 44or, y2 = 4 y = +2 x > y

154. 3; I. 17x = 169+ 14 + 25 + 4xor, 13x = 208

x = 208 1613

II. 9y - 4y = 345 - 260 = 85or, 5y = 85 y = 17 x < y

155. 3; I. 3x2 - 13x + 14 = 0or, 3x2 - 7x - 6x + 14 = 0or,3x(x - 2) -7(x - 2) = 0or, (3x - 7) (x - 2) = 0

x = 7 ,23

II. y2 - 7y + 12 = 0or, y2 - 4y - 3y + 12 = 0or, y(y - 4) -3(y - 4) = 0or,(y -3)(y - 4) = 0 y = 4, 3 x < y

156. 2; I. 2x2 - 8x - 7x + 28 = 0or 2x(x - 4) - 7(x - 4) = 0or (2x - 7) (x - 4) = 0

7x 4,2

II. 2y2 + 10y - 7y - 35 = 0or 2y(y + 5) - 7(y + 5) = 0or (2y - 7)(y + 5) = 0

7y , 52

x y157. 1; 28x - 20y = 96

28x + 21y = 301- - - . -41y = -205 y = 5 and x = 7 x > y

158. 3; I. x = 3 2744 14

II. y = 487 = 22

484 32

x < y159. 5; I. x2 - x - 8x + 8 = 0

or x(x - 1) - 8(x - 1) = 0or (x - 1) (x - 8) = 0 x = 1, 8II. 2y2 - y - 10y + 5 = 0or y(2y - 1) - 5(2y - 1) = 0or (y - 5) (2y - 1) = 0

y = 5, 12

305


160. 1; I. 2x2 + 2x + x + 1 = 0or 2x(x + 1) + 1(x + 1) = 0or (x + 1)(2x + 1) = 0

1x 1, 0.52

II. 6y2 + 9y + 8y + 12 = 0or 3y(2y + 3) + 4(2y + 3) = 0or (3y + 4) (2y + 3) = 0

4 3y , 1.33, 1.53 2

x > y161. 2; I. 3x2 - 29x + 56 = 0

or 3x2 - 21x - 8x + 56 = 0or 3x(x - 7) - 8(x - 7) = 0or (3x - 8) (x - 7) = 0

x = 8 ,73

II. 3y2 - 5y - 8 = 0or 3y2 + 3y - 8y - 8 = 0or 3y(y + 1) - 8(y + 1) = 0or (3y - 8) (y + 1) = 0or (3y - 8) (y + 1) = 0

y = 81,3

x y162. 4; I. 5x2 + 26x - 24 = 0

or 5x2 + 30x - 4x - 24 = 0or 5x(x + 6) - 4(x + 6) = 0or (5x - 4) (x + 6) = 0

x = 4 , 65

II. 5y2 - 30y - 4y + 24 = 0or 5y(y - 6) - 4(y - 6) = 0or (5y - 4) (y - 6) = 0 4

y = 4 ,65

x y163. 1; I. x2 - 7x = 0

or x (x - 7) = 0 x = 0, 7II. 2y2 + 5y + 3 = 0or 2y2 + 2y + 3y + 3 = 0or 2y(y + 1) + 3(y + 1) = 0or (2y + 3) (y + 1) = 0

y = 31,2

x > y164. l ; 7x - 4y = 40 ...(i)

and 8x + 8y = 8or x + y = 1 ...(ii)Solving (i) and (ii), we have x = 4, y = -3 x > y

165. 3; I. 15x2 - 4!x + 14 = 0or 15x2 - 6x - 35x + 14 = 0or 3x(5x - 2) - 7(5x - 2) = 0or (3x - 7)(5x - 2) = 0

x = 7 2,3 5

II. 2y2 - 13y + 20 = 0or 2y2 - 8y - 5y + 20 = 0or 2y(y - 4) - 5(y - 4) = 0or (2y - 5) (y - 4) = 0

y = 54,2

x < y

166. 5; I. 2x 8 3x 45 0

2or, x 5 3x 3 3x 45 0

or,x(x 5 3) 3 3(x 5 3) 0

or,(x 3 3)(x 5 3) 0

x 3 3, 5 3

II. 2y 2y 24 0

2or y 4 2y 3 2y 24 0

or (y 4 2y)(y 3 2) 0

or (y 3 2)(y 4 2)

y 3 2,4 2 Hence relation cannot be established between xand y.

167. 2; I. x 7 2x 24 0

or x 4 2x 3 2x 24 0

or x( x 4 2) 3 2( x 4 2) 0

or ( x 3 2)( x 4 2) 0

Now, if x 3 2 0

then x 3 2

x = 9 × 2 = 18

If x 4 2 0

then x 4 2

x = 16 × 2 = 32

II. y 5 2y 12 0

or y 3 2y 2 2y 12 0

or y( y 3 2) 2 2( y 3 2) 0

or ( y 2 2)( y 3 2) 0

If ( y 2 2) 0

then y 2 2

y = 4 × 2 = 8

if y 3 2 0

then y 3 2

y = 9 × 2= 18 x y

168. 4; I. 12x2 - 17x + 6 = 0or 12x2 - 9x - 8x + 6 = 0or 3x(4x - 3) - 2(4x - 3) = 0or (3x - 2) (4x - 3) = 0

306


If 3x - 2 = 0then 3x = 2

2x3

If 4x - 3 = 0

then x = 34

II. 20y2 - 31y + 12 = 0or 20y2 - 16y - 15y + 12 = 0or 4y(5y - 4) - 3(5y - 4) = 0or (4y - 3) (5y - 4) = 0

3 4y ,4 5

Hence x y169. 5; I. 3x2 - 8x + 4 = 0

or 3x2 - 6x - 2x + 4 = 0or (3x - 2) (x - 2) = 0

2x 2,3

II. 4y2 - 15y + 9 = 0or 4y2 - 12y - 3y + 9 = 0or 4y(y - 3) - 3(y - 3) = 0or (4y - 3) (y - 3) = 0

3y , 34

Relation cannot be established between x and y.170. 1; I. x2 - 16x + 63 = 0

or x2 - 9x - 7x + 63 = 0or x(x - 9) - 7(x - 9) = 0or (x - 7) (x - 9) = 0 x = 7, 9II. y2 - 2y - 35 = 0or y2 - 17y + 5y - 35 = 0or y(y - 7) + 5(y - 7) = 0or (y + 5) (y - 7) = 0 y = -5, 7Hence, x y

171. 5; I. 63x 94 x 35 0

or, or,63x 49 x 45 x 35 0

or, (9 x 7)(7 x 5) 0

49 25x ,81 49

II. 32y 52 y 21 0

or, 32y 28 y – 24 y 21 0

or,(4 y 3).(8 y 7) 0

9 49y ,

16 64Therefore relation can’t be established between xand y.

172. 1; I. 2x 7 3x 35 15 5 5x

2or, x 5 5x 7 3x 35 15 0

or, (x 7 3)(x 5 5) 0

x 7 3, 5 5

II. 2y 5 5y 30 0

2or, y 3 5y 2 5y 30 0

or, (y 3 5), (y 2 5) 0

y 3 5, 2 5173. 3; I. 14x2 + 11x - 15 = 0

or (7x - 5) (2x + 3) = 0

5 3x ,7 2

II. 20y2 - 31y + 12 = 0or (4y - 3), (5y - 4) = 0

3 4y ,4 5

x < y174. 1; I. 5x + 4y = 41 ... (i)

II. 4x + 5y = 40 ... (ii)On solving both equations, we havex = 5 and y = 4 x > y

175. 3; I.

152

2

(18)x 0x

5 152 2or x (18)

x = (18)3

II.

92(19)y 0

y

3 92 2or y (19)

y = (19)3

x < y

176. 5; I. 63x 194 x 143 0

or63x 117 x 77 x 143 0

or (7 x 13)(9 x 11) 0

169 121x ,49 81

II. 99y 225 y 150 0

or 99y 90 y 165 y 150 0

or (11 y 10)(9 y 15) 0

100 225y ,121 81

Therefore relation cannot be established betweenx and y.

177. 2; I. 16x2 - 40x - 39 = 0or 16x2 - 52x + 12x - 39 = 0or (4x- 13) (4x + 3)

13 3x ,4 4

307


II. 12y2 - 113y + 255 = 0or 12y2 - 45y - 68y + 255 = 0or (4y - 15) (3y - 17) = 0

15 17y ,4 3

Therefore y > xor, x < y

178. 2; I. x 7 3x 36 0

or x 7 3. x 36 0

or x 3 3. x 4 3. x 36 0

or ( x 3 3)( x 4 3) 0

x = 27, 48

II. y 5 2y 7 2y 70 0

or y 5 2. y 7 2. y 70 0

or ( y 5 2)( y 7 2) 0

y = 50, 98 x < y

179. 1; I. 2x 7 7x 84 0

or (x 4 7)(x 3 7) 0

x 4 7,3 7

II. 2y 5 5y 30 0

or (y 2 5)(y 3 5) 0

y 2 5, 3 5

x > y180. 2; I. 10x + 6y = 13

II. 45x + 24y = 56

On solving both eqns, x =4 5, y5 6

x < y181. 2; I. x2 - 2x - 15 = 0

or,x2 - 5x + 3x - 15 = 0or, x(x - 5) + 3(x - 5) = 0or,(x - 5) (x + 3) = 0x = 5, -3II. y2 + 5y + 6 = 0or, y2 + 3y + 2y + 6 = 0or, y(y + 3) + 2(y + 3) = 0or,(y + 3)(y + 2) = 0y = -3, -2

x y

182. 5; I. x2 - x - 12 = 0or, x2 - 4x + 3x - 12 = 0or, x(x - 4) + 3(x - 4) = 0or, (x - 4) (x + 3) = 0 x = 4, -3II. y2-3y + 2 = 0or, y2 - 2y - y + 2 = 0

or, y(y - 2) - 1 (y - 2) = 0or, (y - 2)(y - 1) = 0 y = 2, lHence, no relation can be established.

183. 2; I. x 169 0

or,x 169

x 13

II. y2 - 169 = 0or, y2 = 169

or, y = 169

y = ±13

Hence, x y

184. 3; I. x2 - 32 = 112or, x2= 112 + 32 = 144

or, x = 144

x = ±12

II. y - 256 = 0

or, y = 256

y = 16Hence, x < y

185. 5; I. x2 - 25 = 0or, x2 = 25

or, x = 25

x = ±5II. y2 - 9y + 20 = 0or, y2 - 5y - 4y + 20 = 0or, y(y - 5) - 4(y - 5) = 0or, (y - 5) (y - 4) = 0 y = 5, 4Hence, no relation can be established.

186. 3; 3x + 5y = 69 ... (i)9x + 4y = 108 ... (ii)x + z = 12 ... (iii)Now, from (i) and (ii), we have3x + 5y = 69 ... (i) × 49x + 4y = 108 ... (ii) × 512x + 20y = 27645x + 20y = 540

- 33x = - 264On subtracting, we getor, 33x = 264

x = 26433

= 8

Putting the value of x in equation (i), we get 3 × 8+ 5y = 69or, 5y = 69 - 24 = 45

y = 455 = 9

Again, putting the value of x in equation (iii),we get

308


x + z = 12or, z = 12 - 8 = 4Hence, x < y > z

187. 3; I. 1 13 43 4y 9 9 9 9 9

...(I)

II. 2x + 5z = 54 .. (ii)III. 6x + 4z = 74or, 3x + 2z = 37 ... (iii)From equation (ii) × 2 - (iii) × 5, we get

4x + 10z = 10815x + 10z = 185

- - - . - 11x = - 77or, 11x = 77 x = 7Putting the value of x in equation (ii), we get2 × 7 + 5z = 54or, 5z = 40 z = 8Hence, x < y > z

188. 2; I. 2x + 3y + 4z = 66 ... (i)II. 2x + y + 3z = 42 ... (ii)III.3x + 2y + 4z = 63 ... (iii)From (iii) and (i),x - y = - 3 ...(iv)From equation (i) × 3 - equation (ii) × 46x + 9y + 12z = 1988x + 4y + 12z = 168- - - - .2x + 5y = 30 ... (v)Solving equation (iv) and (v), we get

x = 5, y = 8Now, on putting the value of x and y in equation(i),

10 + 24 + 4z = 66or, 4z = 32

32z 84

Hence, x < y = z189. 1; I. (x + z)3 = 1728 = 123

or, x + z = 12 ...(i)II. 2x + 3y = 35 ... (ii)III. x - z = 2 ...(iii)Now, equation (i) and (ii),x = 7, z = 5Putting the value x in question (ii) we have,2 × 7 + 3y = 35or, 3y = 35 - 14 = 21

or, y = 213

= 7

Hence, x = y > z190. 2; 4x + 5y = 37 ... (i)

x + z = 8 ... (ii)7x + 3y = 36 ... (iii)From equation (i) and (iii),

4x + 5y = 37 ... (i) × 37x + 3y = 36 ... (ii) × 5

or, 12x + 15y = 11135x + 15y = 180- - - .-23x = - 69 x = 3

Putting the value of x in equation (i)4 × 3 + 5y = 37

or, y =25 55

Now, putting the value of x in equation (ii)z = 5. Hence, x < y = z

191. 4; I. 7x + 3y = 77 ... (i)

II. 2x + 5y = (2601)12 = 51 ...(ii)

Now, 7x + 3y = 77 ... (i) × 52x + 5y = 51 ... (ii) × 3

or, 35x + 15y = 385 6x + 15y = 153- - - .29x = 232

232x 829

Putting the value of x in equation (i), we have7 × 8 + 3y = 77or, 3y = 77 - 56 = 21

or, y = 213

= 7

Hence, x > y

192. 3; I. 3x2 - 6x - 17x 2 17 = 0

or, 3x(x - 2) - 17 (x - 2) = 0

or, (3x 17)(x 2) 0

x = 2, 173

II. 10y2 - 18y - 5 17y 9 17y 0

or, 2y(5y - 9) - 17 (5y - 9) = 0

or, (2y - 17 ) (5y - 9) = 0

17 9or, y ,2 5

193. 4; I. 12(289) x 324 203

or, 17x - 18 = 203or, 17x = 221

x = 22117 = 13

II. 12(484) y 225 183

or, 22y - 15 = 183

309


or, 22y = 198

198y 922

Hence, x > y194. 1; I. 511x2 = 3066

2 3066or, x 6511

x 6 II. 12y3 - 9y3 = 1536or, 3y3 = 1536

or, y3 =1536

3 = 512 = 83

y = 8 Hence, x < y

195. 4, I. 3x + 4y = (4681)12 = 41

or, 3x + 4y = 41 ... (i)

II. 3x + 2y = (961)12

3x + 2y = 31 ...(ii)Solving (i) and (ii), we get

3x + 4y = 41 ...(i) × 23x + 2y = 31 ...(ii) × 46x + 8y = 8212x + 8y = 124- - - . - 6x = - 42

42x 76

Putting the value of x in equation (i), we get 3 × 7

+ 4y = 41or, 4y = 41 - 21 = 20

or, y = 204 = 5

Hence, x > y

196. 5; I. 3x2 - 6x - 17x 2 17 0 ...(i)

or, 3x(x - 2) - 17 (x - 2) = 0

or, (3x - 17 )(x - 2) = 0

or, x = 2, 173

II. 10y2 - 15y + 17y 3 17 0

or, 5y(y - 3) + 17 (y - 3) = 0

or, (5y + 17 ) (y - 3) = 0

17y 3,5

197. 2; I. x2 - 16x + 63 = 0or, x2 - 9x - 7x + 63 = 0or, x(x - 9) - 7(x - 9) = 0or, (x - 7) (x - 9) = 0 x = 7, 9II. y2 - 2y - 35 = 0or, y2 - 7y + 5y - 35 = 0or, y(y - 7) + 5(y - 7) = 0or, (y - 7) (y + 5) = 0 y = 7, - 5Hence, x y

281 EQUATION - 197notesbag.com/wp-content/uploads/2018/01/Quadractic...33. I. 6x + 7y = 52 II. 14x +...

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Transcript of 281 EQUATION - 197notesbag.com/wp-content/uploads/2018/01/Quadractic...33. I. 6x + 7y = 52 II. 14x +...