Quadratic Equations PYQs Set 6 (PDF)

Set 6 of 11 · Questions 151-180 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 6 (Q151-180)
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.
151. $Let f (x) = a x^{2} + b x + c be such that f (1) = 3, f (- 2) = λ and f (3) = 4 . If f (0) + f (1) + f (- 2) + f (3) = 14, then λ is equal to$ (1) $-4$ (2) $\frac{13}{2}$ (3) $\frac{23}{2}$ (4) $4$ 152. $If m is chosen in the quadratic equation (m^{2} + 1) x^{2} - 3 x + (m^{2} + 1^{2}) = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:$ (1) $105$ (2) $83$ (3) $85$ (4) $43$ 153. $If tan α and tan β are the roots of the equation x^{2} + px + q = 0, then the value of sin^{2} (α + β) + pcos (α + β) sin (α + β) + q cos^{2} (α + β) is$ (1) $p + q$ (2) $p$ (3) $q$ (4) $p p + q$ 154. $If one real root of the quadratic equation 81 x^{2} + kx + 256 = 0 is cube of the other root, then a value of k is:$ (1) $- 81$ (2) $100$ (3) $144$ (4) $- 300$ 155. $The set of all real values of λ for which the quadratic equations, (λ^{2} + 1) x^{2} - 4 λx + 2 = 0 always have exactly one root in the interval (0, 1) is:$ (1) $(- 3, - 1)$ (2) $(2, 4]$ (3) $(0, 2)$ (4) $(1, 3]$ 156. $The number of distinct solutions of the equations x^{11} - x^{7} + x^{4} - 1 = 0 is$ (1) $9$ (2) $11$ (3) $10$ (4) $8$ 157. $If the equations x^{2} + 2 x + 3 = 0 and a x^{2} + bx + c = 0, a, b, c \in R, have a common root, then a : b : c is$ (1) $1 : 2 : 3$ (2) $3 : 2 : 1$ (3) $1 : 3 : 2$ (4) $3 : 1 : 2$ 158. $Let α, β be the roots of the equation x^{2} - 2 x + 6 = 0 and \frac{1}{α ^{2}} + 1, \frac{1}{β ^{2}} + 1 be the roots of the equation x^{2} + ax + b = 0 . Then the roots of the equation x^{2} - (a + b - 2) x + (a + b + 2) = 0 are:$ (1) $non-real complex numbers$ (2) $real and both negative$ (3) $real and both positive$ (4) $real and exactly one of them is positive$ 159. $Let f (x) = \frac{6 x ^{2} - 18 x + 21}{6 x ^{2} - 18 x + 17} . If m is the maximum value of f (x) and f (x) > n \forall x \in R . Then 14 m - 7 n =$ (1) $-1$ (2) $23$ (3) $35$ (4) $42$ 160. $If α, β and γ are the roots of the equation x^{3} - 6 x^{2} + 11 x + 6 = 0, then Σ α^{2} β + Σ α β^{2} is equal to$ (1) $80$ (2) $84$ (3) $90$ (4) $-84$ 161. $If 1 + 2 and 2 - i are the roots of the equation x^{4} + b x^{3} + c x^{2} + dx + e = 0 where b, c, d, e are rational numbers, then the roots of the equation b x^{2} + cx + d = 0 are$ (1) $real and different$ (2) $real and equal$ (3) $purely imaginary$ (4) $complex conjugate$ 162. $Let a_{n}, a_{n - 1}, \dots, a_{1}, a_{0} \in C and f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0} is a polynomial. If the polynomial f (x) is monic then$ (1) $a_{n} \neq = 0$ (2) $a_{n} = 1$ (3) $a_{n} > 0$ (4) $a_{n} < 0$ 163. $If α, β, γ are roots of x^{3} - 5 x + 4 = 0, then (α^{3} + β^{3} + γ^{3})^{2} is equal to$ (1) $12$ (2) $13$ (3) $169$ (4) $144$ 164. $Let α and β be the roots of x^{2} - 6 x - 2 = 0 . If a_{n} = α^{n} - β^{n} for n \geq 1, then the value of \frac{a _{10} - 2 a _{8}}{2 a _{9}} is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 165. $Let - \frac{π}{6} < θ < - \frac{π}{12} . Suppose α_{1} and β_{1} are the roots of the equation x^{2} - 2 xsecθ + 1 = 0 and α_{2} and β_{2} are the roots of the equation x^{2} + 2 xtanθ - 1 = 0 . If α_{1} > β_{1} and α_{2} > β_{2}, then α_{1} + β_{2} equals (JEE Advanced 2016)$ (1) $2 (secθ - tanθ)$ (2) $2 secθ$ (3) $- 2 tanθ$ (4) $0$ 166. $Consider the quadratic equation (c - 5) x^{2} - 2 cx + (c - 4) = 0, c \neq = 5 . Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is:$ (1) $18$ (2) $12$ (3) $10$ (4) $11$ 167. $Let the transformed equation of 2 x^{4} - 8 x^{3} + 3 x^{2} - 1 = 0 so that the term containing the cubic power of x is absent be 2 x^{4} + b x^{2} + cx + d = 0 . Then b =$ (1) $-18$ (2) $-15$ (3) $-9$ (4) $-16$ 168. $If 2 and 6 are the roots of the equation a x^{2} + bx + 1 = 0, then the quadratic equation, whose roots are \frac{1}{2 a + b} and \frac{1}{6 a + b}, is :$ (1) $2 x^{2} + 11 x + 12 = 0$ (2) $x^{2} + 8 x + 12 = 0$ (3) $4 x^{2} + 14 x + 12 = 0$ (4) $x^{2} + 10 x + 16 = 0$ 169. $Let α and β be two roots of the equation x^{2} + 2 x + 2 = 0, then α^{15} + β^{15} is equal to:$ (1) $- 256$ (2) $512$ (3) $- 512$ (4) $256$ 170. $If one root of the equation i x^{2} - 2 (i + 1) x + (2 - i) = 0 is (2 - i), then the other root is$ (1) $- i$ (2) $2 + i$ (3) $i$ (4) $2 - i$ 171. $If f (x) \in Q [x] be a non-zero polynomial such that all its roots are irrational, then the degree of f (x) is$ (1) $an even number$ (2) $an odd number$ (3) $0$ (4) $can't determine$ 172. $If α, β and γ are roots of the equation x^{3} + 4 x - 19 = 0 . Then, the value of \frac{α ^{3}}{19 - 4 α} + \frac{β ^{3}}{19 - 4 β} + \frac{γ ^{3}}{19 - 4 γ} is equal to$ (1) $0$ (2) $3$ (3) $-3$ (4) $2$ 173. $Let a, b and c be the sides of a scalane triangle. If λ is a real number such that the roots of the equation x^{2} + 2 (a + b + c) x + 3 λ (ab + bc + ca) = 0 are real, then the interval in which λ lies is$ (1) $(- \infty, \frac{4}{3})$ (2) $(\frac{5}{3}, \infty)$ (3) $(\frac{1}{3}, \frac{5}{3})$ (4) $(\frac{4}{3}, \infty)$ 174. $If α and β are the roots of the equation x^{2} + 2 x + 2 = 0, then α^{15} + β^{15} =$ (1) $-512$ (2) $-256$ (3) $256$ (4) $512$ 175. $The number of real roots of the equation x \| x - 2 \| + 3 \| x - 3 \| + 1 = 0 is:$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 176. $The number of distinct real roots of x^{4} - 4 x^{3} + 12 x^{2} + x - 1 = 0 is____.(IIT-JEE, 2011)$ 177. $For x \in R, the number of real roots of the equation 3 x^{2} - 4∣ x^{2} - 1∣ + x - 1 = 0 is ____ . (JEE Advanced 2021)$ 178. $The number of distinct real roots of the equation x^{5} (x^{3} - x^{2} - x + 1) + x (3 x^{3} - 4 x^{2} - 2x + 4) - 1 = 0 is$ 179. $Let α, β be the roots of the equation x^{2} - 4 λx + 5 = 0 and α, γ be the roots of the equation x^{2} - (32 + 23) x + 7 + 3 λ 3 = 0 . If, β + γ = 32 then (α + 2 β + γ)^{2} is equal to:$ 180. $The number of solutions of the equation log_{4} (x - 1) = log_{2} (x - 3) is$