Quadratic Equations PYQs Set 3 (PDF)

Set 3 of 11 · Questions 61-90 of 316

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 3 (Q61-90)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
61. $The largest interval containing x for which x^{12} - x^{9} + x^{4} - x + 1 > 0 is$ (1) $0 < x < 1$ (2) $- 4 < x < 2$ (3) $- \infty < x < \infty$ (4) $- 2^{10} < x < 2^{10}$ 62. $If the difference between the roots of the equation x^{2} + ax + 1 = 0 is less than 5, then the set of possible values of ' a' is$ (1) $(- 3, 3)$ (2) $(- 3, \infty)$ (3) $(3, \infty)$ (4) $(- \infty, - 3)$ 63. $If e^{ix} is a solution of the equation z^{n} + p_{1} z^{n - 1} + p_{2} z^{n - 2} + \dots . + p_{n} = 0, where p_{i} are real (i = 1, 2, 3, \dots ., n), then p_{n} sinnx + p_{n - 1} sin (n - 1) x + \dots + p_{1} sinx + 1 =$ (1) $cos (n + 1) x$ (2) $sin (n (n + 1)) x$ (3) $1$ (4) $0$ 64. $If k \in R, then roots of (x - 2) (x - 3) = k^{2} are always$ (1) $real and distinct$ (2) $real and equal$ (3) $complex number$ (4) $rational numbers$ 65. $Let λ \neq = 0 be a real number. Let α, β be the roots of the equation 14 x^{2} - 31 x + 3 λ = 0 and α, γ be the roots of the equation 35 x^{2} - 53 x + 4 λ = 0 . Then 3 α / β and 4 α / λ are the roots of the equation$ (1) $7 x^{2} + 245 x - 250 = 0$ (2) $7 x^{2} - 245 x + 250 = 0$ (3) $49 x^{2} - 245 x + 250 = 0$ (4) $49 x^{2} + 245 x + 250 = 0$ 66. $The set of all values of ' a' for which the expression \frac{a x ^{2} - 2 x + 3}{2 x - 3 x ^{2} + a} assumes all real values for real values of x, is$ (1) $(2, 3)$ (2) $R - (2, 3)$ (3) $ϕ$ (4) $[1, 5]$ 67. $Let - \frac{π}{6} < θ < - \frac{π}{12} . Suppose α_{1} and β_{1} are the roots of the equation x^{2} - 2 xsecθ + 1 = 0 and α_{2} and β_{2} are the roots of the equation x^{2} + 2 xtanθ - 1 = 0 . If α_{1} > β_{1} and α_{2} > β_{2}, then α_{1} + β_{2} equals$ (1) $2 (secθ - tanθ)$ (2) $2 secθ$ (3) $- 2 tanθ$ (4) $0$ 68. $If α, β are the roots of the equation, x^{2} - x - 1 = 0 and S_{n} = 2023 α^{n} + 2024 β^{n}, then$ (1) $2 S_{12} = S_{11} + S_{10}$ (2) $S_{12} = S_{11} + S_{10}$ (3) $2 S_{11} = S_{12} + S_{10}$ (4) $S_{11} = S_{10} + S_{12}$ 69. $The sum, of the squares of all the roots of the equation x^{2} + \| 2 x - 3 \| - 4 = 0, is$ (1) $3 (3 - 2)$ (2) $6 (3 - 2)$ (3) $6 (2 - 2)$ (4) $3 (2 - 2)$ 70. $Let α, β be the roots of the equation x^{2} + 22 x - 1 = 0 . The quadratic equation, whose roots are α^{4} + β^{4} and \frac{1}{10} (α^{6} + β^{6}), is:$ (1) $x^{2} - 190 x + 9466 = 0$ (2) $x^{2} - 180 x + 9506 = 0$ (3) $x^{2} - 195 x + 9506 = 0$ (4) $x^{2} - 195 x + 9466 = 0$ 71. $The value of λ such that sum of the squares of the roots of the quadratic equation, x^{2} + (3 - λ) x + 2 = λ has the least value is:$ (1) $\frac{15}{8}$ (2) $1$ (3) $\frac{4}{9}$ (4) $2$ 72. $Let α, β be the distinct roots of the equation x^{2} - (t^{2} - 5 t + 6) x + 1 = 0, t \in R and a_{n} = α^{n} + β^{n} . Then the minimum value of \frac{a _{2023} + a _{2025}}{a _{2024}} is$ (1) $- \frac{1}{8}$ (2) $- 1/4$ (3) $- 1/2$ (4) $1/4$ 73. $If a and b are the roots of the equation y^{2} + y + 1 = 0, then the value of a^{4} + b^{4} + a^{- 1} b^{- 1} is$ (1) $1$ (2) $0$ (3) $5$ (4) $2$ 74. $The smallest value of the constant m > 0 for which f (x) = 9 mx - 1 + \frac{1}{x} \geq 0 for all x > 0, is$ (1) $\frac{1}{9}$ (2) $\frac{1}{16}$ (3) $\frac{1}{36}$ (4) $\frac{1}{81}$ 75. $α, β are the roots of x^{2} - 10 x - 8 = 0 with α > β . If a_{n} = α^{n} - β^{n} for n \in N, then the value of \frac{a _{10} - 8 a _{8}}{5 a _{9}} is$ (1) $-3$ (2) $3$ (3) $-2$ (4) $2$ 76. $If the difference between the roots of x^{2} + ax + b = 0 and that of the roots of x^{2} + bx + a = 0 is same and a \neq = b, then$ (1) $a - b - 4 = 0$ (2) $a - b + 4 = 0$ (3) $a + b + 4 = 0$ (4) $a + b - 4 = 0$ 77. $If α and β are the roots of the equation x^{2} + px + 2 = 0 and \frac{1}{α} and \frac{1}{β} are the roots of the equation 2 x^{2} + 2 qx + 1 = 0 then (α - \frac{1}{α}) (β - \frac{1}{β}) (α + \frac{1}{β}) (β + \frac{1}{α}) is equal to$ (1) $\frac{9}{4} (9 - q^{2})$ (2) $\frac{9}{4} (9 + p^{2})$ (3) $\frac{9}{4} (9 + q^{2})$ (4) $\frac{9}{4} (9 - p^{2})$ 78. $The quadratic equations x^{2} - 6 x + a = 0 and x^{2} - cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3 . Then the common root is$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 79. $Statement (I): The set of solutions of \| x \|^{2} - 4 \| x \| + 3 < 0 is the interval (- 3, 3) . Statement (II): If x < 3 or x > 5 then x^{2} - 8 x + 15 > 0 . Which of the above statements is (are) true?$ (1) $Statement I is true, but Statement II is false$ (2) $Statement II is true, but Statement I is false$ (3) $Both Statement I and Statement II are true$ (4) $Both Statement I and Statement II are false$ 80. $Let x, y > 0. If x^{3} y^{2} = 2^{15}, then the least value of 3x + 2y is$ (1) $30$ (2) $32$ (3) $36$ (4) $40$ 81. $If α and β are the roots of the equation a x^{2} + bx + c = 0 and the equation having roots \frac{1 - α}{α} and \frac{1 - β}{β} is p x^{2} + qx + r = 0, then r =$ (1) $a + 2 b$ (2) $ab + bc + ca$ (3) $a + b + c$ (4) $abc$ 82. $The equation e^{sinx} - e^{- sinx} - 4 = 0 has:$ (1) $infinite number of real roots$ (2) $no real roots$ (3) $exactly one real root$ (4) $exactly four real roots$ 83. $Let f (x) = x^{3} + a x^{2} + bx + c be polynomial with integer coefficients. If the roots of f (x) are integer and are in Arithmetic Progression, then a cannot take the value$ (1) $-642$ (2) $1214$ (3) $1323$ (4) $1626$ 84. $Let α, β be the roots of the equation x^{2} - 2 x + 2 = 0 Then α^{14} + β^{14} is equal to$ (1) $-64$ (2) $- 642$ (3) $-128$ (4) $- 1282$ 85. $The number of points, where the curve f(x) = e^{8 x} - e^{6 x} - 3 e^{4 x} - e^{2 x} + 1, x \in R cuts x-axis, is equal to............$ 86. $The smallest value of k, for which both the roots of the equation x^{2} - 8 kx + 16 (k^{2} - k + 1) = 0 are real, distinct and have values at least 4, is$ 87. $Let a ϵ R and let, β be the roots of the equation x^{2} + 60^{\frac{1}{4}} x + a = 0 . If a^{4} + β^{4} = - 30, then the product of all possible values of a is _______$ 88. $The number of real solutions of the equation x \| x + 5 \| + 2 \| x + 7 \| - 2 = 0 is______$ 89. $Let λ ϵ R and let the equation E be \| x \|^{2} + 2 \| x \| + \| λ - 3 \| = 0 then the largest element in the set S= {X + λ : x is an integer solution of E} is$ 90. $The number of real solutions of the equation e^{4 x} + 4 e^{3 x} - 58 e^{2 x} + 4 e^{x} + 1 = 0 is$