Quadratic Equations PYQs Set 4 (PDF)

Set 4 of 11 · Questions 91-120 of 316

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 4 (Q91-120)
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91. $Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots (4, 3). Rahul made a mistake in writing down coefficient of x to get roots (3, 2). The correct roots of equation are:$ (1) $6, 1$ (2) $4, 3$ (3) $-6 , -1$ (4) $-4 , -3$ 92. $The solution set of the inequation 3^{x} + 3^{1 - x} - 4 < 0, is$ (1) $(0, 1)$ (2) $(0, 2)$ (3) $(1, 2)$ (4) $(1, 3)$ 93. $The number of solutions of the equations x + y + z = 12; x^{2} + y^{2} + z^{2} = 50; x^{3} + y^{3} + z^{3} = 216 is$ (1) $6$ (2) $24$ (3) $3$ (4) $9$ 94. $Let α_{θ} and β_{θ} be the distinct roots of 2 x^{2} + (cosθ) x - 1 = 0, θ \in (0, 2 π) . If m and M are the minimum and the maximum values of α_{θ}^{4} + β_{θ}^{4}, then 16 (M + m) equals:$ (1) $24$ (2) $25$ (3) $17$ (4) $27$ 95. $The number of all possible positive integral values of α for which the roots of the quadratic equation, 6 x^{2} - 11 x + α = 0 are rational numbers is:$ (1) $3$ (2) $2$ (3) $4$ (4) $5$ 96. $Let p, q \in R . If 2 - 3 is a root of the quadratic equation, x^{2} + px + q = 0, then:$ (1) $p^{2} - 4 q + 12 = 0$ (2) $q^{2} - 4 p - 16 = 0$ (3) $q^{2} + 4 p + 14 = 0$ (4) $p^{2} - 4 q - 12 = 0$ 97. $If α and β are non-real roots of x^{3} - x^{2} - x - 2 = 0, then α^{2020} + β^{2020} + α^{2020} \cdot β^{2020} =$ (1) $1$ (2) $2020$ (3) $1 + α + β$ (4) $-1$ 98. $Given that the roots of x^{3} + 3 p x^{2} + 3 qx + r = 0 are in harmonic progression. Then,$ (1) $2 q^{3} = r (3 pq - r)$ (2) $q^{3} = r (3 pq - r)$ (3) $q^{3} = - r (3 pq - r)$ (4) $q^{3} = r (r + 3 pq)$ 99. $If the roots of the equation 16 x^{3} - 44 x^{2} + 36 x - 9 = 0 are in harmonic progression, then its greatest root is$ (1) $\frac{3}{4}$ (2) $\frac{3}{2}$ (3) $\frac{1}{2}$ (4) $- \frac{1}{2}$ 100. $Let S = (x^{3} + a x^{2} + bx + c : a, b, c \in N) and (a, b, c \leq 20) be a set of polynomials. Then the number of polynomials in S, which are divisible by x^{2} + 2, is$ (1) $10$ (2) $20$ (3) $6$ (4) $120$ 101. $If x^{2} + px + 1 is a factor of a x^{3} + bx + c, then$ (1) $a^{2} + c^{2} = ab + 3$ (2) $a^{2} - c^{2} = ab$ (3) $a^{2} - c^{2} = - ab$ (4) $a^{2} + c^{2} = ab$ 102. $The common roots of the equations z^{3} + 2 z^{2} + 2 z + 1 = 0 and z^{2018} + z^{2017} + 1 = 0 satisfy the equation$ (1) $z^{2} - z + 1 = 0$ (2) $z^{4} + z^{2} + 1 = 0$ (3) $z^{8} + z^{3} + 1 = 0$ (4) $z^{12} + z^{6} - 1 = 0$ 103. $When b = 17, it is found that the roots of the equation x^{2} + bx + c = 0 are -2 and -15. If α, β are the roots of the same equation when b = 13 then \| α - β \| =$ (1) $7$ (2) $13$ (3) $17$ (4) $30$ 104. $The number of real roots of the equation x^{2} - 4 x + 3 + x^{2} - 9 = 4 x^{2} - 14 x + 6 is$ (1) $0$ (2) $1$ (3) $3$ (4) $2$ 105. $If the roots of x^{5} - a x^{4} + b x^{3} - c x^{2} + dx - 1 = 0 are all positive such that their arithmetic mean and geometric mean are equal, then a + b + c + d =$ (1) $10$ (2) $15$ (3) $20$ (4) $30$ 106. $If α, β are the roots of the equation x^{2} - 23 x + 4 = 0 then α^{6} + β^{6} =$ (1) $128$ (2) $-64$ (3) $64$ (4) $-128$ 107. $The sum of all the solutions of the equation (8)^{2 x} - 16 \cdot (8)^{x} + 48 = 0 is :$ (1) $1 + log_{8} (6)$ (2) $1 + log_{6} (8)$ (3) $log_{8} (6)$ (4) $log_{8} (4)$ 108. $The coefficients a, b and c of the quadratic equation, a x^{2} + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is:$ (1) $\frac{1}{54}$ (2) $\frac{1}{72}$ (3) $\frac{1}{36}$ (4) $\frac{5}{216}$ 109. $If cot xcoty = a and x + y = \frac{π}{6}, then the quadratic equation satisfying cotx and coty is$ (1) $t^{2} + (1 - a) 3 t + a = 0$ (2) $3 t^{2} + (1 - a) t + a 3 = 0$ (3) $3 t^{2} + (a - 1) t + a 3 = 0$ (4) $t^{2} + (a - 1) 3 t + a = 0$ 110. $Let f (x) be a quadratic polynomial such that f (- 1) + f (2) = 0 If one of the roots of f (x) = 0 is 3, then its other root lies in:$ (1) $(- 1, 0)$ (2) $(- 3, - 1)$ (3) $(0, 1)$ (4) $(1, 3)$ 111. $The number of solutions, of the equation e^{sin x} - 2 e^{- sin x} = 2 is$ (1) $2$ (2) $more than 2$ (3) $1$ (4) $0$ 112. $For what values of a \in Z, the quadratic expression (x + a) (x + 1991) + 1 can be factorised as (x + b) (x + c), where b, c \in Z ?$ (1) $1990$ (2) $1989$ (3) $1991$ (4) $1992$ 113. $The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has$ (1) $only purely imaginary roots$ (2) $all real roots$ (3) $two real and two purely imaginary roots$ (4) $neither real nor purely imaginary roots$ 114. $Let α, β be two roots of the equation x^{2} + (20)^{1/4} x + (5)^{1/2} = 0 . Then α^{8} + β^{8} is equal to$ (1) $10$ (2) $100$ (3) $50$ (4) $160$ 115. $If α, β, γ, δ are the roots of the equation x^{4} - 8 x^{3} + 11 x^{2} + 32 x - 60 = 0 and α < β < γ < δ, then 4 α + 3 β + 2 γ + δ =$ (1) $0$ (2) $1$ (3) $9$ (4) $10$ 116. $The sum of the solutions of the equation ∣ x - 2∣ + x (x - 4) + 2 = 0, (x > 0) is equal to:$ (1) $9$ (2) $12$ (3) $4$ (4) $10$ 117. $Let α and β be two real numbers such that α + β = 1 and α β = - 1 . Let P_{n} = α^{n} + β^{n}, P_{n - 1} = 11 and P_{n + 1} = 29 for some integer n \geq 1 . Then, the value of P_{n}^{2} is$ 118. $Let p and q be two real numbers such that p + q = 3 and p^{4} + q^{4} = 369. Then (\frac{1}{p} + \frac{1}{q})^{- 2} is equal to _________ .$ 119. $If for some p, q, r \in R, not all have same sign, one of the roots of the equation (p^{2} + q^{2}) x^{2} - 2 q (p + r) x + q^{2} + r^{2} = 0 is also a root of the equation x^{2} + 2 x - 8 = 0, then \frac{q ^{2} + r ^{2}}{p ^{2}} is equal to-$ 120. $The sum of the cubes of all the roots of the equation x^{4} - 3 x^{3} - 2 x^{2} + 3x + 1 = 10 is______.$