Quadratic Equations PYQs Set 5 (PDF)

Set 5 of 11 · Questions 121-150 of 316

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 5 (Q121-150)
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121. $For 0 < c < b < a, let (a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0 and α \neq = 1 be one of its root. Then, among the two statements (I) If α \in (- 1, 0), then b cannot be the geometric mean of a and c (II) If α \in (0, 1), then b may be the geometric mean of a and c$ (1) $Both (I) and (II) are true$ (2) $Neither (I) nor (II) is true$ (3) $Only (II) is true$ (4) $Only (I) is true$ 122. $If α satisfies the equation \frac{x}{2 x + 1} + \frac{2 x + 1}{x} = 2, then the roots of the equation α^{2} x^{2} + 4 αx + 3 = 0 are$ (1) $1,3$ (2) $- 1, 1$ (3) $2, - 3$ (4) $3,4$ 123. $Let a, b \in R be such that the equation a x^{2} - 2bx + 15 = 0 has a repeated root α . If α and β are the roots of the equation x^{2} - 2bx + 21 = 0, then α^{2} + β^{2} is equal to:$ (1) $37$ (2) $58$ (3) $68$ (4) $92$ 124. $The quadratic equation p (x) = 0 with real coefficients has purely imaginary roots. Then the equation p (p (x)) = 0 has (JEE Advanced 2014)$ (1) $only purely imaginary roots$ (2) $all real roots$ (3) $two real and two purely imaginary roots$ (4) $neither real nor purely imaginary roots$ 125. $If α is a multiple root of the equation x^{5} - 6 x^{4} + 11 x^{3} - 2 x^{2} - 12 x + 8 = 0 then 3 α^{2} - 2 α + 1 =$ (1) $-2$ (2) $1$ (3) $0$ (4) $9$ 126. $If α and β are the roots of x^{2} + 7 x + 3 = 0 and \frac{2 α}{3 - 4 α}, \frac{2 β}{3 - 4 β} are the roots of a x^{2} + bx + c = 0 and GCD of a, b, c is 1 , then a + b + c =$ (1) $11$ (2) $0$ (3) $243$ (4) $81$ 127. $If one root of the equation 4 x^{2} - 2 x + k - 4 = 0 is the reciprocal of the other, then the value of k is$ (1) $-8$ (2) $8$ (3) $-4$ (4) $4$ 128. $Let α and β be the roots of equation p x^{2} + qx + r = 0, p \neq = 0 . If p, q, r are in the A.P. and \frac{1}{α} + \frac{1}{β} = 4, then the value of \| α - β \| is:$ (1) $\frac{34}{9}$ (2) $\frac{2 13}{9}$ (3) $\frac{61}{9}$ (4) $\frac{2 17}{9}$ 129. $If (x^{2} + 5 x + 5)^{x + 5} = 1, then the number of integers satisfying this equation is$ (1) $2$ (2) $3$ (3) $4$ (4) $5$ 130. $If α, β, 2 β are the real roots of the equation x^{3} - 9 x^{2} + k = 0 and k \in R - 0, then 14 β =$ (1) $28$ (2) $36$ (3) $18$ (4) $54$ 131. $If the minimum value of the quadratic expression x^{2} + 5 x - 2 is M and it exists at a, then \frac{M}{a} is equal to$ (1) $3.3$ (2) $\frac{33}{5}$ (3) $2.5$ (4) $-0.25$ 132. $If a \in R and the equation - 3 (x - [x])^{2} + 2 (x - [x]) + a^{2} = 0 (where [x] denotes the greatest integer \leq x) has no interal solution, then all possible values of a lie in the interval:$ (1) $(- 2, - 1)$ (2) $(- \infty, - 2) \cup (2, \infty)$ (3) $(- 1, 0) \cup (0, 1)$ (4) $(1, 2)$ 133. $If α, β, where α < β, are the roots of the equation λ x^{2} - (λ + 3) x + 3 = 0 such that \frac{1}{α} - \frac{1}{β} = \frac{1}{3}, then the sum of all possible values of λ is$ (1) $8$ (2) $6$ (3) $2$ (4) $4$ 134. $If 2, 3, 6 are the roots of the polynomial f (x) = x^{3} + a x^{2} + bx + c, where a, b, c \in C . Then the value of a - c is$ (1) $-11$ (2) $36$ (3) $25$ (4) $11$ 135. $Which among the following equations has roots which are negative of the roots of the equation x^{3} - x^{2} + x - 4 = 0 ?$ (1) $x^{3} - x^{2} + x - 4 = 0$ (2) $x^{3} + x^{2} + x + 4 = 0$ (3) $x^{3} + x^{2} - x - 4 = 0$ (4) $x^{3} - x^{2} - x - 4 = 0$ 136. $If the equation having the roots as the values obtained by diminishing each root of the equation x^{3} - 3 x^{2} + 2 x - 1 = 0 by K is x^{3} - x - 1 = 0, then K =$ (1) $2$ (2) $-1$ (3) $1$ (4) $-2$ 137. $The roots of the equation ∣ x^{2} - x - 6∣ = x + 2 are$ (1) $- 2, 1, 4$ (2) $0, 2, 4$ (3) $0, 1, 4$ (4) $- 2, 2, 4$ 138. $If both the roots of the equation x^{2} - 4 ax + 1 - 3 a + 4 a^{2} = 0 exceed 1, then a lies in the interval$ (1) $(- \infty, \frac{7 - 17}{8})$ (2) $(\frac{7 + 17}{8}, \infty)$ (3) $(\frac{7 - 17}{8}, \frac{1}{2})$ (4) $(\frac{1}{2}, \frac{7 + 17}{8})$ 139. $Let the set of all values of p \in R, for which both the roots of the equation x^{2} - (p + 2) x + (2 p + 9) = 0 are negative real numbers, be the interval (α, β] . Then β - 2 α is equal to$ (1) $5$ (2) $0$ (3) $20$ (4) $9$ 140. $If α, β, γ are the roots of the equation x^{3} + a x^{2} - bx + c = 0, then Σ β^{2} (γ + α) =$ (1) $\frac{a ^{2} + b - c}{3 ab}$ (2) $ac + b^{3}$ (3) $\frac{bc + a ^{2}}{3 ab}$ (4) $ab + 3 c$ 141. $Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (\frac{10}{3}, \frac{7}{3}) . If α, β are the roots of the equation a x^{2} + bx + 1 = 0, then the value of α^{2} + β^{2} - αβ is:$ (1) $\frac{71}{256}$ (2) $- \frac{69}{256}$ (3) $\frac{69}{256}$ (4) $- \frac{71}{256}$ 142. $The sum of the complex roots of the equations (x - 1)^{3} + 64 = 0 is$ (1) $6$ (2) $3$ (3) $6 i$ (4) $3 i$ 143. $If α and β be the roots of the equation x^{2} - 2 x + 2 = 0, then the least value of n for which (\frac{α}{β})^{n} = 1 is:$ (1) $2$ (2) $5$ (3) $4$ (4) $3$ 144. $The number of real roots of the equation \frac{x}{1 - x} + \frac{1 - x}{x} = \frac{13}{6} is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 145. $The sum of all the real values of x satisfying the equation (x^{2} - 7 x + 11)^{x^{2} - 6 x - 7} = 1 is$ (1) $14$ (2) $20$ (3) $13$ (4) $16$ 146. $A value of b for whichthe equations x^{2} + bx - 1 = 0, x^{2} + x + b = 0 have one root in common is (IIT-JEE, 2011)$ (1) $- 2$ (2) $- i 3$ (3) $2$ (4) $3$ 147. $The product and sum of the roots of the equation (x^{2}) - 5 \| x \| - 24 = 0 are respectively$ (1) $- 64, 0$ (2) $- 24, 5$ (3) $5, - 24$ (4) $0,72$ 148. $If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a^{4} + b^{4} + c^{4} is equal to ___$ 149. $The number of real solutions of the equation x (x^{2} + 3 \| x \| + 5 \| x + 1 \| + 6 \| x - 2 \|) = 0 is______$ 150. $Let x_{1}, x_{2}, x_{3}, x_{4} be the solution of the equation 4 x^{4} + 8 x^{3} - 17 x^{2} - 12 x + 9 = 0 and (4 + x_{1}^{2}) (4 + x_{2}^{2}) (4 + x_{3}^{2}) (4 + x_{4}^{2}) = \frac{125}{16} m . Then the value of m is$