Quadratic Equations PYQs Set 10 (PDF)

Set 10 of 11 · Questions 271-300 of 316

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 10 (Q271-300)
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271. $Let P_{n} = α^{n} + β^{n}, n \in N . If P_{10} = 123, P_{9} = 76, P_{8} = 47 and P_{1} = 1 then the quadratic equation having roots \frac{1}{α} and \frac{1}{β} is:$ (1) $x^{2} + x - 1 = 0$ (2) $x^{2} - x + 1 = 0$ (3) $x^{2} + x + 1 = 0$ (4) $x^{2} - x - 1 = 0$ 272. $If the quadratic equations 3 x^{2} - 7 x + 2 = 0 and k x^{2} + 7 x - 3 = 0 have a common root then the positive value of k is$ (1) $6$ (2) $11/4$ (3) $4$ (4) $7/2$ 273. $Let α and β be the roots of the equation x^{2} + 2 ax + (3 a + 10) = 0 such that α < 1 < β . Then the set of all possible values of a is :$ (1) $(- \infty, \frac{- 11}{5})$ (2) $(- \infty, - 2) \cup (5, \infty)$ (3) $(- \infty, - 3)$ (4) $(- \infty, \frac{- 11}{5}) \cup (5, \infty)$ 274. $If 2 + 4 i is one of the roots of x^{2} + bx + c = 0 with b, c \in R then (b, c) =$ (1) $(4, - 20)$ (2) $(4, 20)$ (3) $(- 4, - 20)$ (4) $(- 4, 20)$ 275. $The number of solutions of the equations x + y + z = 1; x^{2} + y^{2} + z^{2} = 1; x^{3} + y^{3} + z^{3} = 1 is$ (1) $6$ (2) $3$ (3) $9$ (4) $12$ 276. $If 3 x^{2} - 7 x + 2 = 0 and 15 x^{2} - 11 x + a = 0 have a common root and a is a positive real number, then the sum of the roots of the equation 15 x^{2} - ax + 7 = 0, is$ (1) $\frac{76}{15}$ (2) $\frac{38}{15}$ (3) $\frac{2}{15}$ (4) $\frac{36}{15}$ 277. $The number of positive real roots of the equation 3^{x + 1} + 3^{- x + 1} = 10 is$ (1) $3$ (2) $2$ (3) $1$ (4) $Infinitely many$ 278. $Let α, β, γ be the roots of the equation x^{3} + px + q = 0 and f (x) = 3 p^{2} x^{2} + p^{2} x + 3 q . Then \sum α^{2} β + \sum α^{4} =$ (1) $f (1)$ (2) $f (- 1)$ (3) $f (0)$ (4) $f (2)$ 279. $If f (x) = x^{4} - 2 x^{3} + 3 x^{2} - ax + b is divided by x - 1 and x +1 , the remainders are 5 and 19 , respectively. If f (x) is divided by x - 2 . The remainder is$ (1) $8$ (2) $5$ (3) $10$ (4) $12$ 280. $The product of all the rational roots of the equation (x^{2} - 9 x + 11)^{2} - (x - 4) (x - 5) = 3, is equal to$ (1) $14$ (2) $21$ (3) $28$ (4) $7$ 281. $Let α, β, γ be the three roots of the equation x^{3} + bx + c = 0 if βγ = 1 = - α then b^{3} + 2 c^{3} - 3 α^{3} - 6 β^{3} - 8 γ^{3} is equal to$ (1) $\frac{155}{8}$ (2) $21$ (3) $\frac{169}{8}$ (4) $19$ 282. $Suppose a, b denote the distinct real roots of the quadratic polynomial x^{2} + 20 x - 2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x^{2} - 20 x + 2020 . Then the value of ac (a - c) + ad (a - d) + bc (b - c) + bd (b - d) is (JEE Advanced 2020)$ (1) $0$ (2) $8000$ (3) $8080$ (4) $16000$ 283. $If c and d are the roots of x^{2} + ax + b = 0, then a root of x^{2} + (4 c + a) x + (b + 2 ac + 4 c^{2}) = 0 is$ (1) $d + 2 c$ (2) $d + c$ (3) $d - c$ (4) $d - 2 c$ 284. $The product of real roots of the equation 4 x^{4} - 24 x^{3} + 57 x^{2} + 18 x - 45 = 0 if one of the root is 3 + i 6 is$ (1) $- 5/16$ (2) $5/16$ (3) $3/4$ (4) $- 3/4$ 285. $If α, β, γ are the roots of the equation x^{3} + x^{2} + x + r = 0 and α^{3} + β^{3} + γ^{3} = 5, then r =$ (1) $\frac{- 1}{2}$ (2) $1$ (3) $-1$ (4) $\frac{1}{2}$ 286. $The equation of lowest degree with rational coefficients having roots 3 + 2 i and 3 - 2 is$ (1) $(x^{4} - 2 x^{2} + 25) (x^{4} - 10 x^{2} + 1) = 0$ (2) $(x^{2} - 23 x + 5) (x^{2} - 23 x + 1) = 0$ (3) $(x^{4} - 2 x^{2} + 25) (x^{4} + 10 x^{2} + 1) = 0$ (4) $(x^{4} - 10 x^{2} + 1) (x^{4} + 2 x^{2} + 25) = 0$ 287. $If both the roots of the quadratic equation x^{2} - mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval:$ (1) $(- 5, - 4)$ (2) $(4, 5)$ (3) $(5, 6)$ (4) $(3, 4)$ 288. $Consider the equation x^{2} + 4 x - n = 0, where n \in [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to$ (1) $6$ (2) $5$ (3) $8$ (4) $7$ 289. $If 2,1 and 1 are roots of the equation x^{3} - 4 x^{2} + 5 x - 2 = 0 , then the roots of (x + \frac{1}{3})^{3} - 4 (x + \frac{1}{3})^{2} + 5 (x + \frac{1}{3}) - 2 = 0$ (1) $\frac{7}{3}, \frac{4}{3}, \frac{4}{3}$ (2) $\frac{5}{3}, \frac{2}{3}, \frac{2}{3}$ (3) $\frac{- 5}{3}, \frac{- 2}{3}, \frac{- 2}{3}$ (4) $\frac{- 7}{3}, \frac{- 4}{3}, \frac{- 4}{3}$ 290. $Let α and β be the roots of x^{2} - 6 x - 2 = 0 . If a_{n} = α^{n} - β^{n} for n \geq 1, then the value of \frac{a _{10} - 2 a _{8}}{3 a _{9}} is:$ (1) $4$ (2) $1$ (3) $2$ (4) $3$ 291. $If α, β are the roots of x^{2} + bx + c = 0, γ, δ are the roots of x^{2} + b_{1} x + c_{1} = 0 and γ < α < δ < β, then (c - c_{1})^{2} <$ (1) $(b_{1} - b) (b c_{1} - b_{1} c)$ (2) $1$ (3) $(b - b_{1})^{2}$ (4) $(c - c_{1}) (b_{1} c - b_{1} c_{1})$ 292. $If α and β are the distinct roots of the equation x^{2} + (3)^{1/4} x + 3^{1/2} = 0, then the value of α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1) is equal to:$ (1) $56 \times 3^{25}$ (2) $56 \times 3^{24}$ (3) $52 \times 3^{24}$ (4) $28 \times 3^{25}$ 293. $Let S be the set of all non-zero real numbers α such that the quadratic equation α x^{2} - x + α = 0 has two distinct real roots x_{1} and x_{2} satisfying the inequality \| x_{1} - x_{2} \| < 1. Which of the following intervals is(are) a subset(s) of S?$ (1) $(- \frac{1}{2}, - \frac{1}{5})$ (2) $(- \frac{1}{5}, 0)$ (3) $(0, \frac{1}{5})$ (4) $(\frac{1}{5}, \frac{1}{2})$ 294. $All the values of ' m' for which both roots of the equation x^{2} - 2 mx + m^{2} - 1 = 0 are greater than -2 but less than 4 lie in the interval$ (1) $m > 3$ (2) $-1< m < 3$ (3) $1 < m < 4$ (4) $- 2 < m < 0$ 295. $Let the abscissae of the two points P and Q on a circle be the roots of x^{2} - 4x - 6 = 0 and the ordinates of P and Q be the roots of y^{2} + 2y - 7 = 0. If PQ is a diameter of the circle x^{2} + y^{2} + 2ax + 2by + c = 0, then the value of (a + b - c) is$ (1) $12$ (2) $13$ (3) $14$ (4) $16$ 296. $Let S be the set of all non-zero real numbers such that the quadratic equation α x^{2} - x + α = 0 has two distinct real roots x_{1} and x_{2} satisfying the inequality (x_{1} - x_{2}) < 1 . Which of the following intervals is(are) a subset(s) of S ? (JEE Advanced 2015)$ (1) $(- \frac{1}{2}, - \frac{1}{5})$ (2) $(- \frac{1}{5}, 0)$ (3) $(0, \frac{1}{5})$ (4) $(\frac{1}{5}, \frac{1}{2})$ 297. $The set of all values of k > -1, for which the equation (3x2 + 4x + 3) 2- (k + 1) (3x2 + 4x + 3) (3x2 + 4x + 2) + k (3 x^{2} + 4x + 2)2 = 0 has real roots is:$ (1) $(1, \frac{5}{2})$ (2) $[2, 3)$ (3) $(- \frac{1}{2}, 1)$ (4) $(\frac{1}{2}, \frac{3}{2}) - (1)$ 298. $The number of distinct real roots of the equation \| x + 1 \| \| x + 3 \| - 4 \| x + 2 \| + 5 = 0, is$ 299. $The number of distinct real roots of the equation is 3 x^{4} + 4 x^{3} - 12 x^{2} + 4 = 0.$ 300. $Let m and n be the numbers of real roots of the quadratic equations x^{2} -12x + [x] + 31 = 0 and x^{2} -5\|x+ 2\|-4 = 0 respectively, where [x] denotes the greatest integer \leq x. Then m^{2} + mn + n^{2} is equal to$