Quadratic Equations PYQs Set 9 (PDF)

Set 9 of 11 · Questions 241-270 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 9 (Q241-270)
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.
241. $Let S be the set of all possible integral values of λ in the interval (- 3, 7) for which the roots of the quadratic equation λ x^{2} + 13 x + 7 = 0 are all rational numbers. Then the sum of the elements in S is$ (1) $4$ (2) $2$ (3) $3$ (4) $1$ 242. $The number of real values of m so that the equation x^{2} + (2 m + 1) x + m = 0 has equal roots is$ (1) $1$ (2) $0$ (3) $2$ (4) $3$ 243. $α and β are the roots of x^{2} + 2 x + c = 0 . If α^{3} + β^{3} = 4, then the value of c is$ (1) $-2$ (2) $3$ (3) $2$ (4) $4$ 244. $The number of real roots of the equation 5 + (2^{x} - 1) = 2^{x} (2^{x} - 2) is:$ (1) $3$ (2) $2$ (3) $4$ (4) $1$ 245. $If (2 + i) is a root of the equation x^{3} - 5 x^{2} + 9 x - 5 = 0, then the other roots are$ (1) $1 and (2 - i)$ (2) $-1 and (3 + i)$ (3) $0 and 1$ (4) $-1 and (- 2 + i)$ 246. $Let a, b \in R, a \neq = 0 be such that the equation, a x^{2} - 2 bx + 5 = 0 has a repeated root α, which is also a root of the equation, x^{2} - 2 bx - 10 = 0 . If β is the other root of this equation, then α^{2} + β^{2} is equal to:$ (1) $25$ (2) $26$ (3) $24$ (4) $28$ 247. $If α and β are the roots of the quadratic equation x^{2} + x + 1 = 0, then the equation whose roots are α^{2021}, β^{2021} is given by$ (1) $x^{2} - x + 1 = 0$ (2) $x^{2} + x - 1 = 0$ (3) $x^{2} - x - 1 = 0$ (4) $x^{2} + x + 1 = 0$ 248. $The number of real solutions of the equation 3 (x^{2} + \frac{1}{x ^{2}}) - 2 (x + \frac{1}{x}) + 5 = 0 is$ (1) $4$ (2) $0$ (3) $3$ (4) $2$ 249. $The sum of the squares of the roots of \| x - 2 \|^{2} + \| x - 2 \| - 2 = 0 and the squares of the roots of x^{2} - 2 \| x - 3 \| - 5 = 0, is$ (1) $24$ (2) $26$ (3) $36$ (4) $30$ 250. $The sum of all the real roots of the equation (e^{2 x} -4) (6 e^{2 x} - 5 e^{x} + l) = 0 is$ (1) $I o g_{e} 3$ (2) $- l o g_{e} 3$ (3) $l o g_{e} 6$ (4) $- l o g_{e} 6$ 251. $The number of roots of the equation 2 + e^{cos h^{- 1} x} - e^{sin h^{- 1} x} = 0, is$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 252. $If \frac{α}{α + 1} and \frac{β}{β + 1} are the roots of the quadratic equation x^{2} + 7 x + 3 = 0, then the equation having roots α and β is$ (1) $3 x^{2} - x - 3 = 0$ (2) $11 x^{2} + 13 x + 3 = 0$ (3) $13 x^{2} + 11 x + 13 = 0$ (4) $11 x^{2} + 3 x + 13 = 0$ 253. $The number of non-real roots of the equation x^{10} - 3 x^{8} + 5 x^{6} - 5 x^{4} + 3 x^{2} - 1 = 0 is$ (1) $8$ (2) $6$ (3) $4$ (4) $2$ 254. $Solve the equation, 3^{x^{2} - x} = 25 - 4^{x^{2} - x}$ (1) $-1 only$ (2) $2 only$ (3) $Both -1 and 2$ (4) $No solution$ 255. $What is the quotient of x^{3} - 5 x^{2} + 2 x + 7 when divided with (x - 1) ?$ (1) $x^{2} + 4 x - 2$ (2) $x^{2} - 4 x + 2$ (3) $x^{2} + 4 x + 2$ (4) $x^{2} - 4 x - 2$ 256. $The number of integral value of m for which the equation (1 + m^{2}) x^{2} - 2 (1 + 3 m) x + (1 + 8 m) = 0 has no real root is:$ (1) $1$ (2) $2$ (3) $infinitely many$ (4) $3$ 257. $Let f : R \to R be defined by f (x) = 2 x + 3 . If α, β are the roots of the equation f (x^{2}) - 2 f (\frac{x}{2}) - 1 = 0 then α^{2} + β^{2} =$ (1) $13$ (2) $25$ (3) $5$ (4) $18$ 258. $If α, β, γ are the roots of the equation 5 x^{3} - 2 x - 4 = 0, then α^{3} + β^{3} + γ^{3} =$ (1) $\frac{12}{5}$ (2) $\frac{18}{29}$ (3) $4$ (4) $-4$ 259. $The value of 3 + \frac{1}{4 + \frac{1}{3 + \frac{1}{4 + \frac{1}{3 + \dots \infty}}}}$ (1) $1.5 + 3$ (2) $2 + 3$ (3) $3 + 2 + 3$ (4) $4 + + 3$ 260. $If (x - iy)^{1/3} = a - ib, then the value of \frac{x}{2 a} + \frac{y}{2 b} is$ (1) $2 (a^{2} - b^{2})$ (2) $4 (a^{2} - b^{2})$ (3) $a^{2} - b^{2}$ (4) $12 a^{2} - b^{2}$ 261. $Let α, β be the roots of the equation x^{2} - \| a \| x - \| b \| = 0 such that \| α \| < \| β \| . If \| a \| < β - 1, then the positive root of log_{\| α \|} (\frac{x ^{2}}{β ^{2}}) - 1 = 0, is$ (1) $< \| α \|$ (2) $< α$ (3) $< β$ (4) $> β$ 262. $Let α and β be the roots of the equation, 5 x^{2} + 6 x - 2 = 0 If S_{n} = α^{n} + β^{n}, n = 1, 2, 3, \dots then:$ (1) $5 S_{6} + 6 S_{5} = 2 S_{4}$ (2) $6 S_{6} + 5 S_{5} + 2 S_{4} = 0$ (3) $6 S_{6} + 5 S_{5} = 2 S_{4}$ (4) $5 S_{6} + 6 S_{5} + 2 S_{4} = 0$ 263. $The quadratic equation whose sum of the roots is 11 and sum of squares of the roots is 61 is$ (1) $x^{2} + 11 x - 30 = 0$ (2) $x^{2} + 11 x + 30 = 0$ (3) $x^{2} - 11 x - 30 = 0$ (4) $x^{2} - 11 x + 30 = 0$ 264. $Let α, β (α > β) be the roots of the quadratic equation x^{2} - x - 4 = 0 . If P_{n} = α^{n} - β^{n}, n \in N, then \frac{P _{15} P _{16} - P _{14} P _{16} - P _{15}^{2} + P _{14} P _{15}}{P _{13} P _{14}} is equal to____.$ 265. $The sum of all real values of x for which \frac{3 x ^{2} - 9 x + 17}{x ^{2} + 3 x + 10} = \frac{5 x ^{2} - 7 x + 19}{3 x ^{2} + 5 x + 12} is equal to$ 266. $Let λ \neq 0 be in R. If α and β are the roots of the equation x^{2} - x + 2λ = 0, and α and γ are the roots of equation 3 x^{2} - 10 x + 27 λ = 0, then \frac{βγ}{λ} is equal to.$ 267. $The number of real roots of the equation e^{4 x} - e^{3 x} - 4 e^{2 x} - e^{x} + 1 = 0 is equal to$ 268. $If the equation a (b - c) x^{2} + b (c - a) x + c (a - b) = 0 has equal roots, where a + c = 15 and b = \frac{36}{5}, then a^{2} + c^{2} is equal to$ 269. $Let a, b be two non-zero real numbers. If p and r are the roots of the equation x^{2} - 8 ax + 2 a = 0 and q and s are the roots of the equation x^{2} + 12 bx + 6 b = 0, such that \frac{1}{p}, \frac{1}{q}, \frac{1}{r}, \frac{1}{s} are in A.P., then a^{- 1} - b^{- 1} is equal to____.$ 270. $If a and b are the roots of the equation x^{2} - 7x-1 = 0, then the value of \frac{a ^{21} + b ^{21} + a ^{17} + b ^{17}}{a ^{19} + b ^{19}} is equal to$