1. The roots of the equation \(ax^2+bx+c=0\) will be equal in magnitude but opposite in sign if- (a) \(c=0, a≠0\) (b) \(b=0, a≠0\) (c) \(c=0, a=0\) (d) \(b=0, a=0\)

2. If the roots of the equation \( ax^2+bx+c=0 \,(a\ne 0) \) are real and equal, then (a) \(c=\cfrac{-b}{2a}\) (b) \(c=\cfrac{b}{2a}\) (c) \(c= \cfrac{-b^2}{4a}\) (d) \(c = \cfrac{b^2}{4a}\)

3. If one root of the quadratic equation \(ax^2 + abcx + bc = 0\) (\(a \ne 0\)) is the reciprocal of the other, then — (a) abc=1 (b) b=ac (c) bc=1 (d) a=bc

4. If one root of the equation \(x^2 - ax - 15 = 0\) is 3, then what is the value of \(a\)? True / False

5. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ? (a) 3 (b) -3 (c) \(\cfrac{1}{3}\) (d) -\(\cfrac{1}{3}\)

6. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?" (a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

7. If \(9x^2 - 13x + 9 = 0\), then what is the value of \(x + \cfrac{1}{x}\)? (a) \(\cfrac{9}{4}\) (b) \(\cfrac{4}{9}\) (c) \(\cfrac{13}{9}\) (d) 1

8. What is the value of \( x \) in the equation \( x^2 + x - 6 = 0 \)? (a) -3,2 (b) 2,3 (c) -2,3 (d) -2,-3

9. In the equation \(x(x+5) = 7\), the coefficient of \(x^0\) is : (a) 5 (b) -5 (c) 7 (d) -7

10. If \(x = \cfrac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}}\), then what is the value of \(bx^2 - ax + b\)? (a) 0 (b) 1 (c) 2 (d) 3

11. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be. True / False

12. The roots of the equation \(x^2 - 18x + 8 = 0\) are — (a) Real , Rational , Unequal (b) equal,Rational (c) Real , Rational , equal (d) None of the above

13. If \(a^2 + a + 1 = 0\), then write the correct relation from the following — (a) \(a^4=a\) (b) \(a^3=a\) (c) \(a^2=a+1\) (d) \(\(a^3=a+1\)

14. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)? (a) -11 (b) 11 (c) 12 (d) 14

15. Determine for which value of \(a\) the equation \((a - 2)x^2 + 3x + 5 = 0\) will not be a quadratic equation. (a) \(a=0\) (b) \(a=2\) (c) \(a=4\) (d) \(a=-2\)

16. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\). (a) 1 (b) 2 (c) 3 (d) 4

17. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), find the value of \(\cfrac{1}{α} + \cfrac{1}{β}\). (a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

18. The length of the diagonal of a rectangular field is 15 meters, and its length is 3 meters more than its width. What is the length of the rectangular field? (a) 9 meter (b) 12 meter (c) 15 meter (d) 18 meter

19. A man bought some kilograms of sugar for 80 rupees. If he had received 4 kilograms more for the same amount, the price per kilogram would have been 1 rupee less. What was the original price per kilogram of the sugar? (a) 15 rupees (b) 16 rupees (c) 5 rupees (d) 20 rupees

20. The distance between two stations is 300 km. A train traveled from the first station to the second at a constant speed. If the train's speed had been 5 km/h faster, it would have taken 2 hours less to reach the second station. What was the train's original speed? (a) 15 km/hour (b) 25 km/hour (c) 20 km/hour (d) None of the above

21. A watch seller sold a watch for 336 rupees. The percentage profit he made was equal to the cost price of the watch. What was the cost price of the watch? (a) 121 rupees (b) 120 rupees (c) 140 rupees (d) None of the above

22. The coefficient of \(x\) in the quadratic equation \(x+\cfrac{1}{x}=6\). (a) 6 (b) -6 (c) 0 (d) 1

23. If the roots of the equation ( ???? + 2 ) ???? 2 − ( ???? − 3 ) ???? + 3 ???? − 1 = 0 are equal in magnitude but opposite in sign, what is the value of ???? ? (a) -3 (b) 1 (c) 3 (d) None of the above

24. If the speed of the stream is 2 km/h, and Ratan Majhi takes 10 hours to go 21 km downstream and return the same distance upstream, what is the speed of the boat in still water? (a) 7 km/h (b) 5 km/h (c) 8 km/h (d) 14 km/h

25. If ???? = 3 + 2 and ???? = 3 − 2 , then what is the value of 8 ???? ???? ( ???? 2 + ???? 2 ) ? (a) 24 (b) 80 (c) 16 (d) 8

26. It takes Mahim 3 hours more than Majid to clean our house garden. Together, they can finish the work in 2 hours. How long would it take Majid to complete the work alone? (a) 3 hours (b) 4 hours (c) 5 hours (d) 6 hours

27. A rectangular playground is 45 meters long and 40 meters wide. Around it, there is a road of uniform width along all four sides. The area of the road is 450 square meters. What is the width of the road? (a) 5 meter (b) 2.5 meter (c) 1.5 meter (d) 4.5 meter

28. What are the roots of the equation \(x^2 - 4x + 4 = 0\)? (a) \(2,2\) (b) \(2,-2\) (c) \(\cfrac{1}{2},\cfrac{1}{2}\) (d) \(\cfrac{1}{2},-\cfrac{1}{2}\)

29. If the roots of the quadratic equation \(ax^2+bx+c=0\) are real and unequal, the value of \(b^2-4ac\) will be: (a) >0 (b) <0 (c) 0 (d) None of these

30. If the equation \((x+2)^3 = x(x-1)^2\) is expressed in the form of the quadratic equation \(ax^2 + bx + c = 0\) \((a ≠ 0)\), the coefficient of \(x^0\) (the constant term) will be. (a) -8 (b) -1 (c) 3 (d) 8

31. The roots of the equation \(ax^2+bx+c=0\) will be real and equal when – (a) \(b^2>4ac \) (b) \(b^2=4ac \) (c) \(b^2≠ 4ac \) (d) \(b^2<4ac\)

32. The roots of the equation \(\cfrac{x^2}{x} = 6\) are (a) 0 (b) 6 (c) 0 and 6 (d) -6

33. If the equation \(ax^2 + bx + c = 0 (a ≠ 0)\) has equal roots. (a) \(c =-\cfrac{b}{2a}\) (b) \(c =\cfrac{b}{2a}\) (c) \(c =-\cfrac{b^2}{4a}\) (d) \(c =\cfrac{b^2}{4a}\)

34. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is \(4\), the value of \(b\) will be – (a) \(\cfrac{5}{3}\) (b) \(\cfrac{3}{5}\) (c) 12 (d) -12

35. The expression \(x^4 - 2x^2 + k\) will be a perfect square when the value of \(k\) is: (a) \(\cfrac{1}{4}\) (b) 0 (c) \(\cfrac{1}{2}\) (d) 1

36. If the equation \(3x^2 - 6x + p = 0\) has real and equal roots, then the value of \(p\) is – (a) \(\cfrac{5}{3}\) (b) -\(\cfrac{1}{3}\) (c) -3 (d) 3

37. If \(25x^2 - 20ax + b\) is a perfect square, which of the following relations is true? (a) \(b=2a^2\) (b) \(b=4a^2\) (c) \(b=a^2\) (d) \(b=3a^2\)

38. The roots of the equation \(x^2 = 6x\) (a) 0 (b) 6 (c) 0 and 6 (d) 0 and -6

39. What is the number of solutions of the equation \(x^2 = x\)? (a) 1 (b) 2 (c) 0 (d) 3

40. The sum of the roots of the equation \(x^2 - 6x + 2 = 0\) will be – (a) 2 (b) -2 (c) 6 (d) -6

41. A root of the equation \(ax^2 + bx + c = 0\) being zero requires the condition— (a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of these

42. If \(\alpha, \beta\) are the roots of the equation \(3x^2+8x+2=0\), then the value of \( \cfrac{1}{\alpha}+\cfrac{1}{\beta}\) is – (a) \(-\cfrac{3}{8}\) (b) 4 (c) \(\cfrac{2}{3}\) (d) -4

43. What is the value of \(x\) in the equation \(x^2 + x - 6 = 0\)? (a) -3,2 (b) 2,3 (c) -2,3 (d) -2,-3

44. Under what condition will one root of the quadratic equation \(ax^2 + bx + c = 0\) be zero? (a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of these

45. The product of the roots of the equation \(x^2−7x+3=0\) is—? (a) 7 (b) -7 (c) 3 (d) -3

46. The product of the roots of the equation \(x^2−7x+3=0\) is—? (a) 7 (b) -7 (c) 3 (d) -3

47. The two roots of the equation \(x^2 = 100\) are \(\pm 10\). True / False

48. The equation \(x^6+5x^3+6=0\) is quadratic in terms of \(x^3\). True / False

49. Solve: \[ 3x + 2 + \frac{3}{3x + 2} = -4 \]

50. If 9 is added to three times a positive number, the sum equals twice the square of that number. Find the number.

51. What value of \(P\) will make the equation \((P - 3)x^2 - 5x + 10 = 0\) not a quadratic equation? \(P =\) _____

52. If \(b^2 = 4ac\) for the quadratic equation \(ax^2 + bx + c = 0\), then the roots are real and ——.

53. Solve: \[ \frac{x - 2}{x + 2} + 6\left(\frac{x - 2}{x - 6}\right) = 1 \]

54. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.

55. Solve: \[ \frac{x}{x+1} + \frac{x+1}{x} = 2\frac{1}{6} \]

56. Solve: \[ \frac{x - 3}{x + 3} - \frac{x + 3}{x - 3} + 6\frac{6}{7} = 0 \quad \text{where } x \ne -3, 3 \]

57. সমাধান করো:
\(\cfrac{x+1}{2}+\cfrac{2}{x+1}=\cfrac{x+1}{3}+\cfrac{3}{x+1}-\cfrac{5}{6}\) - translate in english

58. Solve: \[ \left(\frac{x - a}{x + a}\right)^2 - 5\left(\frac{x - a}{x + a}\right) + 6 = 0 \]

59. A positive integer, when its positive square root is subtracted from it, gives 110. Find the positive integer.

60. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).

61. If \( \alpha \) and \( \beta \) are the roots of the equation \(x^2 - 22x + 105 = 0\), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).

62. Without solving, determine all values of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) has real and equal roots.

63. If one root of the equation \(ax^2+bx+c=0 (a≠0)\) is twice the other, show that \(2b^2=9ac\).

64. A's speed is 1 meter/second more than B's speed. In a 180-meter race, A finishes 2 seconds earlier than B. What is B's speed in meters per second?

65. Solve: \[ \frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x} \] where \(x \ne 0\) and \(x \ne -(a + b)\).

66. If five times a positive integer is 3 less than twice its square, then what is the value of that number?

67. Solve:
\(\cfrac{a}{ax-1}+\cfrac{b}{bx-1}=a+b, \)
\([x\ne \cfrac{1}{a}, \cfrac{1}{b}]\)

68. Write the polynomial \(x^2 - 7x + 2\) as a quadratic polynomial.

69. "Write the polynomial \(7x^5 - x(x + 2)\) in the form of a quadratic polynomial."

70. Solve: \[ \frac{1}{x} - \frac{1}{3} = \frac{1}{x + 2} - \frac{1}{5} \]

71. "Write the polynomial \(2x(x + 5) + 5\) in the form of a quadratic polynomial."

72. In a garden, saplings have been planted in rows. The number of rows is 5 more than the number of saplings in each row. If a total of 336 saplings have been planted, how many saplings were planted in each row?

73. "Write the polynomial \(2x - 1\) in the form of a quadratic polynomial."

74. If the product of two consecutive odd numbers is 783, find the two numbers.

75. Write whether the equation \(x - 1 + \frac{1}{x} = 6,\ (x ≠ 0)\) can be expressed in the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a ≠ 0\).

76. Write whether the equation \( x + \frac{3}{x} = x^2 \), where \( x \ne 0 \), can be expressed in the form \( ax^2 + bx + c = 0 \), where \( a, b, c \) are real numbers and \( a \ne 0 \).

77. Write whether the equation \( x^2 - 6\sqrt{x} + 2 = 0 \), where \( a, b, c \) are real numbers and \( a \ne 0 \), can be expressed in the form \( ax^2 + bx + c = 0 \).

78. Write whether the equation \( (x - 2)^2 = x^2 - 4x + 4 \), where \( a, b, c \) are real numbers and \( a \ne 0 \), can be expressed in the form \( ax^2 + bx + c = 0 \).

79. Determine with respect to which power of the variable the equation \(x^6 - x^3 - 2 = 0\) becomes a quadratic equation.

80. If the sum of the squares of the roots of the equation \(6x^2 + x + k = 0\) is \(\frac{25}{36}\), then the value of \(k\) will be \(12\).

81. If \(x + \frac{9}{x} = 6\), find the numerical value of \(x^2\).

82. Let’s translate that into English: Express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).

83. Translate: Express \((x + 2)^3 = x(x^2 - 1)\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\) (i.e., the constant term).

84. What are the two consecutive positive odd numbers whose product is 143?

85. What are the two consecutive numbers whose squares add up to 313?

86. The equation \((a - 2)x^2 + 3x + 5 = 0\) will not be a quadratic equation for the value of \(a =\) ______.

87. In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the two digits is 12 less than the number itself. What could the number be?

88. Check whether 1 and -1 are roots of the quadratic equation \(x^2 + x + 1 = 0\).

89. Check whether 0 and -2 are roots of the quadratic equation \(8x^2 + 7x = 0\).

90. Check whether \(\cfrac{5}{6}\) and \(\cfrac{4}{3}\) are roots of the equation \(x + \cfrac{1}{x} = \cfrac{13}{6}\).

91. If the roots of a quadratic equation are 2 and -3, write the equation.

92. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).

93. Solve: \((x – 7)(x – 9) = 195\)

94. Solve: \(\left(\cfrac{x+4}{x−4}\right)^2 − 5\left(\cfrac{x+4}{x−4}\right) + 6 = 0\), where \(x ≠ 4\)

95. In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the two digits is 12 less than the number itself. What are the possible values of the unit digit?

96. The area of a rectangular park is 600 square meters and its perimeter is 100 meters. Find the length and breadth of the park.

97. Translate to English: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is double the other, show that \(2b^2 = 9ac\).

98. For what value of \(k\) will the equation \(2x^2 + 3x + k = 0\) have real and equal roots?

99. Solve: \(\frac{x}{x+1} + \frac{x+1}{x} = 2\frac{1}{12}\)

100. Solve: \((2x - 1) + \frac{3}{2x - 1} = 4\)

101. Due to an increase of ₹1 in the price per kilogram of rice, Mr. Nutubabu was able to buy 1 kg less rice for ₹600 than before. Find the original price per kilogram of rice.

102. If one root of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) is double the other, show that \(2b^2 = 9ac\).

103. If the roots of the equation are -4 and 3, then find the corresponding quadratic equation.

104. Solve: \[ \frac{x+3}{x-3} + \frac{x-3}{x+3} = 2\frac{1}{2} \quad \text{where } x \ne 3, -3 \]

105. If the sum and product of the roots of the equation \(x^2 - x = k(2x - 1)\) are equal, what is the value of \(k\)?

106. What is the ratio of the sum and product of the roots of the equation \[ 7x^2 - 66x + 27 = 0? \]

107. Solve using the quadratic formula: \(3x^2 - 11x + 8 = 0\)

108. Find the equation whose roots are the squares of the roots of the equation \(x^2 + x + 1 = 0\).

109. A club had ₹195 in its fund. After each member of the club contributed an amount equal to the number of members, the total money in the club was equally divided among all members, and each received ₹28. Find the number of members in the club.

110. If the price per dozen pens is reduced by ₹6, then 3 more pens can be bought for ₹30. Find the original price per dozen pens before the reduction.

111. If \(\cfrac{1}{3}\) is one root of the quadratic equation \(3x^2-10x+3=0\), determine the other root.

112. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

113. What should be the value of \(a\) so that \(36x^2 – 24x + a\) becomes a perfect square?

114. Solve:
\(\cfrac{a}{ax - 1} + \cfrac{b}{bx - 1} = a + b\),
\([x \ne \cfrac{1}{a}, \cfrac{1}{b}]\)

115. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is 2, find the value of \(k\).

116. If the roots of the quadratic equation \((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + a^2) = 0\) are equal, prove that \(\cfrac{a}{b} = \cfrac{c}{d}\).

117. If \(x=\cfrac{\sqrt{a+2b}+\sqrt{a-2b}}{\sqrt{a+2b}-\sqrt{a-2b}}\), then show that \(bx^2 - ax + b = 0\).

118. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .

119. Five times a positive integer is 3 less than twice the square of that integer. Form the required quadratic equation to find the integer, and then solve the equation to determine its value.

120. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).

121. If \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

122. Solve: \(\frac{x}{3} + \frac{3}{x} = 4\frac{1}{4}\)

123. The length of a rectangular field is \(\frac{3}{2}\) times its breadth. If the length is reduced by 1200 cm and the breadth is increased by 1200 cm, the field becomes a square. Find the area of the field.

124. Are the equations \(x^2 = x\) and \(\frac{x^2}{x} = 1\) identical? Justify your answer.

125. Solve: \(\frac{1}{x−3}−\frac{1}{x+5}=\frac{1}{6}\)

126. If the product of two consecutive positive odd numbers is 143, form the equation and find the two numbers using Sridhar Acharya's formula (the quadratic formula).

127. Solve: \[ \frac{1}{x^2} - \frac{1}{x} = 0 \]

128. The difference between a proper fraction and its reciprocal is \(\frac{9}{20}\). Find the fraction.

129. Solve: \(\cfrac{x + 5}{2 - x} + 2 \cdot \cfrac{2 - x}{x + 5} = 3\)

130. Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.

131. Solve: \[ 6\left(\cfrac{x-2}{x-6}\right)+\cfrac{x-2}{x+2}=1 \quad [x\ne -2,6] \]

132. Solve: \[ \frac{1}{x - a + b} = \frac{1}{x} - \frac{1}{a} + \frac{1}{b} \]

133. In the equation \( ax^2 + bx + c = 0 \), if one root is twice the other, prove that \( 2b^2 = 9ac \).

134. Solve: \[ 3x - \frac{5}{3x + 2} = 2 \]

135. A rectangular field has an area of 2000 square meters and a perimeter of 180 meters. Find the length and breadth of the field.

136. The sum of the squares of two consecutive positive even numbers is 340. Find the two numbers.

137. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

138. The distance between two places is 200 km. The time taken to travel from one place to the other by a motor car is 2 hours more than the time taken by a jeep. The speed of the jeep is 5 km/h more than that of the motor car. Find the speed of the motor car.

139. Divide 50 into two parts such that the sum of their reciprocals is \(\frac{1}{12}\).

140. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]