Quadratic Equations PYQs Set 8 (PDF)

Set 8 of 11 · Questions 211-240 of 316

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 8 (Q211-240)
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211. $Number of roots common to the equations x^{3} + x^{2} - 2 x - 2 = 0 and x^{3} - x^{2} - 2 x + 2 = 0 is$ (1) $1$ (2) $2$ (3) $3$ (4) $0$ 212. $If x, y \in R and x^{2} + y + 4 i and - 3 + x^{2} yi are conjugates to each other, then (\| x \| + \| y \|)^{2} =$ (1) $17$ (2) $16$ (3) $25$ (4) $9$ 213. $The least value of α \in R for which 4 α x^{2} + \frac{1}{x} \geq 1, for all x > 0, is (JEE Advanced 2016)$ (1) $\frac{1}{64}$ (2) $\frac{1}{32}$ (3) $\frac{1}{27}$ (4) $\frac{1}{25}$ 214. $The sum of the roots of the equation e^{4 t} - 10 e^{3 t} + 29 e^{2 t} - 22 e^{t} + 4 = 0 is$ (1) $log_{e} 10$ (2) $2 log_{e} 2$ (3) $log_{2} 29$ (4) $2 log_{10} 2$ 215. $The number of real roots of the equation x\|x\| - 5\|x + 2\| + 6 = 0, is$ (1) $5$ (2) $4$ (3) $6$ (4) $3$ 216. $Let S be the set of positive integral values of a for which \frac{a x ^{2} + 2 ( a + 1 ) x + 9 a + 4}{x ^{2} - 8 x + 32} < 0, \forall x \in R . Then, the number of elements in S is:$ (1) $1$ (2) $0$ (3) $\infty$ (4) $3$ 217. $If α and β are the roots of the equation, 7 x^{2} - 3 x - 2 = 0 then the value of \frac{α}{1 - α ^{2}} + \frac{β}{1 - β ^{2}} equal to:$ (1) $\frac{1}{24}$ (2) $\frac{27}{32}$ (3) $\frac{3}{8}$ (4) $\frac{27}{16}$ 218. $The sum of the fourth powers of the roots of the equation 16 x^{2} - 10 x + 1 = 0 is$ (1) $\frac{257}{4096}$ (2) $\frac{257}{2048}$ (3) $\frac{257}{1024}$ (4) $\frac{257}{512}$ 219. $Let α and β be the roots of x^{2} - 3 x + p = 0 and γ and δ be the roots of x^{2} - 6 x + q = 0 . If α, β, γ, δ form a geometric progression. Then ratio (2 q + p) : (2 q - p) is:$ (1) $3 : 1$ (2) $5 : 3$ (3) $9 : 7$ (4) $33 : 31$ 220. $If α, β, γ are the roots of the equation x^{3} - 3 x^{2} + 3 x + 1 = 0 , then α^{2} β^{2} + β^{2} γ^{2} + γ^{2} α^{2} =$ (1) $9$ (2) $15$ (3) $8$ (4) $20$ 221. $If the sum of the roots of the quadratic equations is 1 and sum of the squares of the roots is 13 , then find that equation.$ (1) $x^{2} + x - 6 = 0$ (2) $x^{2} - x + 6 = 0$ (3) $x^{2} - x - 6 = 0$ (4) $x^{2} + x + 6 = 0$ 222. $If α and β are the roots of the equation 2 x (2 x + 1) = 1 then β is equal to:$ (1) $2 α^{2}$ (2) $- 2 α (α + 1)$ (3) $2 α (α - 1)$ (4) $2 α (α + 1)$ 223. $Suppose α, β, γ are roots of x^{3} + x^{2} + 2 x + 3 = 0 . If f (x) = 0 is a cubic polynomial equation whose roots are α + β, β + γ, γ + α, then f (x) is equal to$ (1) $x^{3} + 2 x^{2} - 3 x - 1$ (2) $x^{3} + 2 x^{2} - 3 x + 1$ (3) $x^{3} + 2 x^{2} + 3 x - 1$ (4) $x^{3} + 2 x^{2} + 3 x + 1$ 224. $Let a, b \in R and the roots α, β of the equation z^{2} + az + b = 0 be complex. If the origin, α and β represent the vertices of an equilateral triangle on the Argand plane, then$ (1) $a = b$ (2) $a^{2} = 3 b$ (3) $a^{2} = 4 b$ (4) $a = 3 b$ 225. $If x \in R and 1 \leq \frac{3 x ^{2} - 7 x + 8}{x ^{2} + 1} \leq 2, then the minimum and maximum values of x are respectively.$ (1) $1,2$ (2) $5,12$ (3) $6,10$ (4) $1,6$ 226. $Let p and q be real numbers such that p \neq = 0, p^{3} \neq = q and p^{3} \neq = - q . If α and β are nonzero complex numbers satisfying α + β = - p and α^{3} + β^{3} = q, then a quadratic equation having \frac{α}{β} and \frac{β}{α} as its roots is$ (1) $(p^{3} + q) x^{2} - (p^{3} + 2 q) x + (p^{3} + q) = 0$ (2) $(p^{3} + q) x^{2} - (p^{3} - 2 q) x + (p^{3} + q) = 0$ (3) $(p^{3} - q) x^{2} - (5 p^{3} - 2 q) x + (p^{3} - q) = 0$ (4) $(p^{3} - q) x^{2} - (5 p^{3} + 2 q) x + (p^{3} - q) = 0$ 227. $C osec 1 8^{0} is a root of the equation:$ (1) $x^{2} + 2x - 4 = 0$ (2) $4 x^{2} + 2x - 1 = 0$ (3) $x^{2} - 2x + 4 = 0$ (4) $x^{2} - 2x - 4 = 0$ 228. $Let α, β be the roots of the quadratic equation 12 x^{2} - 20 x + 3 λ = 0, λ \in Z . If \frac{1}{2} ⩽ \| β - α \| ⩽ \frac{3}{2}, then the sum of all possible values of λ is:$ (1) $1$ (2) $6$ (3) $3$ (4) $4$ 229. $If the roots of the quadratic equation a x^{2} + bx + c = 0 are imaginary, then for all real values of x, the minimum value of the expression 3 a^{2} x^{2} + 6 abx + 2 b^{2} is$ (1) $< 4 ab$ (2) $> 4 ac$ (3) $> - 4 ac$ (4) $< - 4 ab$ 230. $If α, β, γ are the roots of the equation 3 x^{3} - 26 x^{2} + 52 x - 24 = 0 such that α, β, γ are in geometric progression and α < β < γ, then 3 α + 2 β + γ =$ (1) $\frac{68}{3}$ (2) $\frac{56}{3}$ (3) $12$ (4) $24$ 231. $If ' x' is real, then maximum value of \frac{3 x ^{2} + 9 x + 17}{3 x ^{2} + 9 x + 7} is -$ (1) $41$ (2) $1$ (3) $\frac{17}{7}$ (4) $\frac{1}{4}$ 232. $If α, β and γ are the roots of x^{3} - x + 1 = 0, then \frac{1 + α}{1 - α} + \frac{1 + β}{1 - β} + \frac{1 + γ}{1 - γ} is equal to$ (1) $1$ (2) $0$ (3) $2$ (4) $-2$ 233. $If α, β and γ are the roots of the equation x^{3} - a x^{2} + bx - c = 0 then, α^{- 2} + β^{- 2} + γ^{- 2} =$ (1) $\frac{b ^{2} - 3 ac}{c ^{2}}$ (2) $\frac{b ^{2} - ac}{c ^{2}}$ (3) $\frac{b ^{2} - 2 ac}{c ^{2}}$ (4) $\frac{b ^{2} - 4 ac}{c ^{2}}$ 234. $If α, β, γ are the roots of x^{3} - 6 x^{2} + 11 x - 6 = 0, then the equation having the roots α^{2} + β^{2}, β^{2} + γ^{2} and γ^{2} + α^{2} is$ (1) $x^{3} - 28 x^{2} + 245 x - 650 = 0$ (2) $x^{3} - 28 x^{2} + 245 x + 650 = 0$ (3) $x^{3} + 28 x^{2} - 245 x - 650 = 0$ (4) $x^{3} + 28 x^{2} + 245 x - 650 = 0$ 235. $If α, β, γ are the roots of x^{3} + p x^{2} + qx + r = 0, then the value of (1 + α^{2}) (1 + β^{2}) (1 + γ^{2}) is$ (1) $(r - p)^{2} + (r - q)^{2}$ (2) $(1 + p)^{2} + (1 + q)^{2}$ (3) $(r + p)^{2} + (q + 1)^{2}$ (4) $(r - p)^{2} + (q - 1)^{2}$ 236. $If the sum of all the roots of the equation e^{2 x} - 11 e^{x} - 45 e^{- x} + \frac{81}{2} = 0 is l o g_{e} P, then p is equal to_______.$ 237. $The product of all positive real values of x satisfying the equation x^{(16 (log_{5} x)^{3} - 68 log_{5} x)} = 5^{- 16} is____[2022 Adv.]$ 238. $The least positive value of a for which the equation, 2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a has real roots is$ 239. $If f (x) and g (x) are two polynomials such that the polynomial P (x) = f (x^{3}) + xg (x^{3}) is divisible by x^{2} + x + 1, then P(1) is equal to ___$ 240. $The number of the real roots of the equation (x + 1)^{2} + \| x - 5 \| = \frac{27}{4} is$