Quadratic Equations PYQs Set 1 (PDF)

Set 1 of 11 · Questions 1-30 of 316

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 1 (Q1-30)
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1. $If α, β are the roots of x^{2} - x + 1 = 0, then the quadratic equation whose roots are α^{2015}, β^{2015} is$ (1) $x^{2} - x + 1 = 0$ (2) $x^{2} + x + 1 = 0$ (3) $x^{2} + x - 1 = 0$ (4) $x^{2} - x - 1 = 0$ 2. $The number of all possible positive integral values of α for which the roots of the quadratic equation, 6 x^{2} - 11x + α = 0 are rational numbers is:$ (1) $3$ (2) $4$ (3) $5$ (4) $2$ 3. $If α, β, γ are the roots of the equation 4 x^{3} + 12 x^{2} - 7 x + 165 = 0 and α + 5, β + 5, γ + 5 are the roots of the equation a x^{3} + b x^{2} + cx + d = 0 then the product of the roots of the second equation is$ (1) $27$ (2) $0$ (3) $-3$ (4) $35 + 4$ 4. $A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in:$ (1) $30 days$ (2) $36 days$ (3) $24 days$ (4) $42 days$ 5. $If α, β, γ are the roots of the equation x^{3} - 5 x^{2} - 2 x + 24 = 0 then \frac{βγ}{α} + \frac{γα}{β} + \frac{αβ}{γ} =$ (1) $244$ (2) $\frac{- 1}{6}$ (3) $61$ (4) $\frac{- 61}{6}$ 6. $If the numbers appeared on the two throws of a fair six faced die are α and β, then the probability that x^{2} + αx + β > 0, for all x \in R, is:$ (1) $\frac{17}{36}$ (2) $\frac{4}{9}$ (3) $\frac{1}{2}$ (4) $\frac{19}{36}$ 7. $The number of solutions of the equation (\frac{9}{x} - \frac{9}{x} + 2) (\frac{2}{x} - \frac{7}{x} + 3) = 0 is:$ (1) $2$ (2) $3$ (3) $1$ (4) $4$ 8. $The integer ’ k ’, for which the inequality x^{2} - 2 (3 k - 1) x + 8 k^{2} - 7 > 0 is valid for every x in R is:$ (1) $3$ (2) $2$ (3) $4$ (4) $0$ 9. $The sum of all the real numbers satisfying the equation x^{2} + \| x - 3 \| = 4 is$ (1) $0$ (2) $1$ (3) $2$ (4) $-1$ 10. $Let α be a root of the equation 1 + x^{2} + x^{4} = 0. Then the value of α^{1011} + α^{2022} - α^{3033} is equal to:$ (1) $1$ (2) $α$ (3) $l + α$ (4) $l + 2 α$ 11. $If a, b and c are the roots of x^{3} + qx + r = 0, then (a - b)^{2} + (b - c)^{2} + (c - a)^{2} =$ (1) $- 6 q$ (2) $- 4 q$ (3) $6 q$ (4) $4 q$ 12. $If 4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2 x - 1} then the value of x is$ (1) $\frac{7}{2}$ (2) $\frac{5}{2}$ (3) $\frac{1}{2}$ (4) $\frac{3}{2}$ 13. $If α, β, γ are the roots of f (x) = x^{3} - 9 x^{2} + 26 x - 24, then \frac{1}{α}, \frac{1}{β}, \frac{1}{γ} are the roots of$ (1) $24 x^{3} + 26 x^{2} + 9 x - 1$ (2) $24 x^{3} - 26 x^{2} + 9 x - 1$ (3) $24 x^{3} + 26 x^{2} - 9 x - 1$ (4) $24 x^{3} - 26 x^{2} + 9 x + 1$ 14. $If x^{2} + 2 px - 2 p + 8 > 0 for all real values of x, then the set of all possible values of p is$ (1) $(2, 4)$ (2) $(- \infty, - 4)$ (3) $(2, \infty)$ (4) $(- 4, 2)$ 15. $If f (x) = 2 x^{3} + m x^{2} - 13 x + n and 2,3 are the roots of the equation f (x) = 0, then the values of m and n are$ (1) $- 5, - 30$ (2) $- 5, 30$ (3) $5,30$ (4) $5, - 30$ 16. $If λ be the ratio of the roots of the quadratic equation in x, 3 m^{2} x^{2} + m (m - 4) x + 2 = 0, then the least value of m for which λ + \frac{1}{λ} = 1, is:$ (1) $2 - 3$ (2) $4 - 32$ (3) $- 2 + 2$ (4) $4 - 23$ 17. $If α, β are the irrational roots of the equation 3 p^{2} x^{3} + p x^{2} + qx + 3 = 0 when p = 1 and q = - 7 then \| α - β \| =$ (1) $\frac{3 13}{2}$ (2) $\frac{3}{2}$ (3) $\frac{2 13}{3}$ (4) $4$ 18. $If x^{2} - 3 ax + 14 = 0 and x^{2} + 2 ax - 16 = 0 have a common root, then a^{4} + a^{2} .$ (1) $2$ (2) $90$ (3) $6$ (4) $20$ 19. $The number of integer solutions of the equation \| 1 - i \|^{x} = 2^{x} is$ (1) $1$ (2) $0$ (3) $2$ (4) $3$ 20. $Let α and β be the roots of x^{2} - 6 x - 2 = 0, with α > β . If a_{n} = α^{n} - β^{n} for n \geq 1, then the value of \frac{a _{10} - 2 a _{8}}{2 a _{9}} is (IIT-JEE, 2011)$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 21. $The sum of the complex roots of the equation x^{4} - 2 x^{3} + x - 380 = 0 is$ (1) $- 3 i + 3$ (2) $3 i - 3$ (3) $-1$ (4) $1$ 22. $The minimum value of the sum of the squares of the roots of x^{2} + (3 - a)x + 1 = 2a is:$ (1) $4$ (2) $5$ (3) $6$ (4) $8$ 23. $If the set of all a \in R, for which the equation 2 x^{2} + (a - 5) x + 15 = 3 a has no real root, is the interval (α, β), and X = {x \in Z : α < x < β}, then \sum_{x \in X} x^{2} is equal to :$ (1) $2109$ (2) $2129$ (3) $2119$ (4) $2139$ 24. $If a, b, c are distinct and the roots of (b - c) x^{2} + (c - a) x + (a - b) = 0 are equal, then a, b and c are in$ (1) $arithmetic progression$ (2) $geometric progression$ (3) $harmonic progression$ (4) $arithmetico-geometric progression$ 25. $If α and β be two roots of the equation x^{2} - 64 x + 256 = 0 . Then the value of (\frac{α ^{3}}{β ^{5}})^{\frac{1}{8}} + (\frac{β ^{3}}{α ^{5}})^{\frac{1}{8}} is$ (1) $3$ (2) $2$ (3) $4$ (4) $1$ 26. $A value of b for which the equations x^{2} + bx - 1 = 0, x^{2} + x + b = 0 have one root in common is$ (1) $- 2$ (2) $- i 3$ (3) $i 5$ (4) $2$ 27. $Let α, β be the roots of the quadratic equation x^{2} + 6 x + 3 = 0 . Then \frac{α ^{23} + β ^{23} + α ^{14} + β ^{14}}{α ^{15} + β ^{15} + α ^{10} + β ^{10}}$ (1) $81$ (2) $9$ (3) $72$ (4) $729$ 28. $Let S = {α : log_{2} (9^{2 a - 4} + 13) - log_{2} (\frac{5}{2} . 3^{2 a - 4} + 1) = 2} Then the maximum value of β for which the equation x^{2} - 2 (\sum α)^{2} x + \sum (α + 1)^{2} β = 0 has real roots, is ______$ 29. $Let α, β be roots of x^{2} + 2 x - 8 = 0 . If U_{n} = α^{n} + β^{n}, then \frac{U _{10} + 2 U _{9}}{2 U _{8}} is equal to_______$ 30. $If the value of real number a > 0 for which x^{2} - 5 ax + 1 = 0 and x^{2} - ax - 5 = 0 have a common real roots is \frac{3}{2 β} then β is equal to .$