Quadratic Equations PYQs Set 7 (PDF)

Set 7 of 11 · Questions 181-210 of 316

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 7 (Q181-210)
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181. $Find α^{4} + β^{4} if α, β are the roots of equation x^{2} + x + 1 = 0 .$ (1) $1 αβ$ (2) $2 αβ$ (3) $αβ$ (4) $- αβ$ 182. $The sum of the real roots of the equation \| x - 2 \|^{2} + \| x - 2 \| - 2 = 0 is$ (1) $4$ (2) $-4$ (3) $2$ (4) $-2$ 183. $Let α, β; α > β, be the roots of the equation x^{2} - 2 x - 3 = 0 . Let P_{n} = α^{n} - β^{n}, n \in N . Then (113 - 102) P_{10} + (112 + 10) P_{11} - 11 P_{12} is equal to$ (1) $103 P_{9}$ (2) $113 P_{9}$ (3) $102 P_{9}$ (4) $112 P_{9}$ 184. $\sum_{k = 1}^{6} sin (\frac{2 πk}{7}) - icos (\frac{2 πk}{7}) =$ (1) $1$ (2) $- i$ (3) $i$ (4) $-1$ 185. $If α, β, γ are the roots of the equation x^{3} - 9 x^{2} + 23 x - 15 = 0, then α^{3} + β^{3} + γ^{3} =$ (1) $36$ (2) $92$ (3) $153$ (4) $244$ 186. $The sum of all the roots of the equation (x - 1)^{2} - 5 \| x - 1 \| + 6 = 0, is:$ (1) $1$ (2) $5$ (3) $3$ (4) $4$ 187. $The smallest positive integral value of a, for which all the roots of x^{4} - a x^{2} + 9 = 0 are real and distinct, is equal to$ (1) $4$ (2) $9$ (3) $3$ (4) $7$ 188. $Suppose, α is minimum value of x^{2} + bx + 5 and β is maximum value of - x^{2} + ax + 5 . If [α, β] is the interval of maximum length for x in which x^{2} - 10 x + 24 \leq 0, then a^{2} b^{2} is equal to$ (1) $25$ (2) $16$ (3) $4$ (4) $18$ 189. $The sum of all the roots of the equation \| x^{2} - 8x + 15\| - 2x + 7 = 0 is$ (1) $9 - 3$ (2) $9 + 3$ (3) $11 - 3$ (4) $11 + 3$ 190. $If a_{k} = cos α_{k} + isin α_{k}, k = 1, 2, 3 and a_{1}, a_{2}, a_{3} are the roots of the equation x^{3} + bx + c = 0, then the real part of b =$ (1) $-1$ (2) $1$ (3) $0$ (4) $\frac{2}{3}$ 191. $Let λ \neq = 0 be in R . If α and β are the roots of the equation, x^{2} - x + 2 λ = 0 and α and γ are the roots of the equation, 3 x^{2} - 10 x + 27 λ = 0 then \frac{βγ}{λ} is equal to:$ (1) $18$ (2) $9$ (3) $27$ (4) $36$ 192. $If α, β, γ are the roots of x^{3} + p x^{2} + qx + r = 0, then α^{3} + β^{3} + γ^{3} =$ (1) $p^{3} - 3 pq + r$ (2) $p^{2} - 2 pq + r$ (3) $3 pq - 3 r - p^{3}$ (4) $3 pq + 3 r + p^{3}$ 193. $Let α and β be the roots of the equation px^{2} + qx - r = 0, where p \neq = 0 . If p, q and r be the consecutive terms of a non-constant G.P and \frac{1}{α} + \frac{1}{β} = \frac{3}{4}, then the value of (α - β)^{2} is:$ (1) $\frac{80}{9}$ (2) $9$ (3) $\frac{20}{3}$ (4) $8$ 194. $The equation e^{4 x} + 9 e^{3 x} + 13 e^{2 x} - 8 e^{x} + 1 = 0, x ϵ R has$ (1) $Two solutions and both are negative$ (2) $No solution$ (3) $Four solutions two of which are negative$ (4) $Two solutions and only one of them is negative$ 195. $If α, β, γ, δ are the roots of the equation x^{4} + x^{2} + 1 = 0 such that α + β = - 1, γ + δ = 1, α^{2} = β and γ^{2} = - δ, then α^{2023} + β^{2023} + γ^{2022} + δ^{2022} =$ (1) $1$ (2) $0$ (3) $1 + 3 ω$ (4) $ω - 2 ω^{2}$ 196. $If the sum of two roots of the equation x^{3} - 7 p x^{2} + 5 qx - 6 r = 0 is zero, then$ (1) $5 p = \frac{6 q}{7 r}$ (2) $5 q = \frac{6 r}{7 p}$ (3) $5 r = \frac{6 p}{7 q}$ (4) $pqr = 35$ 197. $If \frac{k}{kx + 3} + \frac{3}{3 x - k} = \frac{12 x + 5}{( kx + 3 ) ( 3 x - k )} \forall x \in R - (\frac{3}{k}, \frac{k}{3}), then both the roots of the equation kx^{2} - 7 x + 3 = 0 are$ (1) $Rational numbers$ (2) $Irrational numbers$ (3) $Complex numbers$ (4) $Integers$ 198. $If the equation x^{4} + a x^{3} + b x^{2} + cx + d = 0 has three equal roots, then that root is$ (1) $\frac{6 c - ab}{8 b - 3 a ^{2}}$ (2) $\frac{ab - 6 c}{8 b + 3 a ^{2}}$ (3) $\frac{6 c - ab}{3 a ^{2} - 4 b}$ (4) $\frac{6 c - ab}{3 a ^{2} - 8 b}$ 199. $Consider the two sets: A = {m \in R : both the roots of x^{2} - (m + 1) x + m + 4 = 0 are real} and B = [- 3, 5) . Which of the following is not true?$ (1) $A \cap B = {- 3}$ (2) $B - A = (- 3, 5)$ (3) $A \cup B = R$ (4) $A - B = (- \infty, - 3) \cup (5, \infty)$ 200. $If the roots of the equation 6 x^{3} - 11 x^{2} + 6 x - 1 = 0 are in harmonic progression, then the roots of x^{3} - 6 x^{2} + 11 x - 6 = 0 will be in$ (1) $Geometric Progression$ (2) $Arithmetic Progression$ (3) $Harmonic Progression$ (4) $Arithmetico-Geometric Progression$ 201. $Two dice are thrown independently. Let A be the event that the number appeared on the 1^{s t} die is less than the number appeared on the 2^{n d} die. B be the event that the number appeared on the 1^{s t} die is even and that on the second die is odd, and C be the event that the number appeared on the 1^{s t} die is odd and that on the 2^{n d} is even. Then$ (1) $The number of favourable cases of the event (A \cup B) \cap C is 6$ (2) $A and B are mutually exclusive$ (3) $The number of favourable cases of the events a, B and C are 15, 6 and 6 respectively.$ (4) $B and C are independent$ 202. $If x = - 5 + 2 - 4, then the value of x^{4} + 9 x^{3} + 35 x^{2} - x + 4 is$ (1) $80$ (2) $160$ (3) $-160$ (4) $-80$ 203. $If α, β \in R are such that 1 - 2 i (here i^{2} = - 1) is a root of z^{2} + αz + β = 0, then (α - β) is equal to:$ (1) $7$ (2) $- 3$ (3) $3$ (4) $- 7$ 204. $If α and β (α < β) are the roots of the equation (- 2 + 3) (\| x - 3 \|) + (x - 6 x) + (9 - 23) = 0, x ⩾ 0, then \frac{β}{α} + αβ is equal to :$ (1) $8$ (2) $11$ (3) $9$ (4) $10$ 205. $Let S = (x \in R : (3 + 2)^{x} + (3 - 2)^{x} = 10) .Then the number of elements in S is:$ (1) $4$ (2) $0$ (3) $2$ (4) $1$ 206. $The sum of all the real solutions of the equation log_{(x + 3)} (6 x^{2} + 28 x + 30) = 5 - 2 log_{(6 x + 10)} (x^{2} + 6 x + 9) is equal to$ (1) $2$ (2) $1$ (3) $0$ (4) $4$ 207. $The minimum value of the sum of real numbers a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8}, and a^{10} with a > 0 is____. (IIT-JEE 2011)$ 208. $Let a, b, c be the length of three sides of a triangle satisfying the condition (a^{2} + b^{2}) x^{2} - 2 b (a + c) . x + (b^{2} + c^{2}) = 0 . If the set of all possible values of x is the interval (α, β), then 12 (α^{2} + β^{2}) is equal to______$ 209. $The sum of all integral values of k(k \neq 0) for which the equation \frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k} in x has no real roots, is .$ 210. $If α, β are roots of the equation x^{2} + 5 (2) x + 10 = 0, α > β and P_{n} = α^{n} - β^{n} for each positive integer n, then the value of (\frac{P _{17} P _{20} + 5 2 P _{17} P _{19}}{P _{18} P _{19} + 5 2 P _{18}^{2}}) is equal to$