Quadratic Equations PYQs Set 2 (PDF)

Set 2 of 11 · Questions 31-60 of 316

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 2 (Q31-60)
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31. $If α, β, γ are the roots of the equation x^{3} + p x^{2} + qx + r = 0, then (α + β) (β + γ) (γ + α) =$ (1) $p - qr$ (2) $r - pq$ (3) $q - rp$ (4) $r + pq$ 32. $The sum of all distinct roots of the equation x^{5} - 3 x^{4} + 5 x^{3} - 5 x^{2} + 3 x - 1 = 0 is$ (1) $1$ (2) $2$ (3) $3$ (4) $23$ 33. $Let f(x) be a quadratic polynomial such that f(-2) + f(3) = 0. If one of the roots of f(x) = 0 is -1, then the sum of the roots of f(x) = 0 is equal to:$ (1) $\frac{11}{3}$ (2) $\frac{7}{3}$ (3) $\frac{13}{3}$ (4) $\frac{14}{3}$ 34. $The product of the roots of the equation 9 x^{2} - 18 \| x \| + 5 = 0 is:$ (1) $\frac{25}{9}$ (2) $\frac{25}{81}$ (3) $\frac{5}{9}$ (4) $\frac{5}{27}$ 35. $The numbers of pairs (a, b) of real numbers, such that whenever α is a root of the equation x^{2} + ax+ b = 0, α 2 - 2 is also a root of this equation, is:$ (1) $6$ (2) $2$ (3) $4$ (4) $8$ 36. $If the quadratic equations x^{2} - 7 x + 3 c = 0 and x^{2} + x - 5 c = 0 have a common root, then for non-zero real value of c the sign of the expression x^{2} - 3 x + c is$ (1) $negative for all x \in R$ (2) $positive for all x \in (1, 3)$ (3) $negative for all x \in (1, 3)$ (4) $positive for all x \in R$ 37. $If α_{1}, α_{2}, \dots .., α_{n} are roots of x^{n} + px + q = 0, then (α_{n} - α_{1}) (α_{n} - α_{2}) \dots .. (α_{n} - α_{n - 1}) =$ (1) $n α_{a}^{n - 1} + q$ (2) $α_{1}^{2} + α_{2}^{2} + \dots + α_{n - 1}^{2}$ (3) $α_{n}^{n - 1} + p$ (4) $n α_{n}^{n - 1} + p$ 38. $If sin 2 θ and cos 2 θ are solutions of x^{2} + bx - c = 0, then$ (1) $b^{2} + 2 c + 1 = 0$ (2) $b^{2} + 2 c - 1 = 0$ (3) $b^{2} - 2 c + 1 = 0$ (4) $b^{2} - 2 c - 1 = 0$ 39. $If \frac{5}{2} is the sum of two roots of the equation 6 x^{6} - 25 x^{5} + 31 x^{4} - 31 x^{2} + 25 x - 6 = 0 then the sum of all non-real roots of the equation is$ (1) $does not exist$ (2) $0$ (3) $\frac{5}{3}$ (4) $\frac{2}{5}$ 40. $The value of 4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + \dots \infty}}}}$ (1) $2 + \frac{2}{5} 30$ (2) $2 + \frac{4}{5} 30$ (3) $4 + \frac{4}{5} 30$ (4) $5 + \frac{2}{5} 30$ 41. $If the sum of the squares of the reciprocals of the roots α and β of the equation 3 x^{2} + λx - 1 = 0 is 15, then 6 (α^{3} + β^{3})^{2} is equal to:$ (1) $18$ (2) $24$ (3) $36$ (4) $96$ 42. $If α, β are the roots of a x^{2} + bx + c = 0, then the quadratic equation whose roots are 5 α, 5 β is$ (1) $a x^{2} + 5 bx + 5 c = 0$ (2) $a x^{2} + 5 bx + 5 c = 0$ (3) $a x^{2} + 5 bx + 5 c = 0$ (4) $a x^{2} + 5 bx + 5 c = 0$ 43. $Let the equation x (x + 2) (12 - k) = 2 have equal roots. Then the distance of the point (k, \frac{k}{2}) from the line 3 x + 4 y + 5 = 0 is$ (1) $15$ (2) $12$ (3) $53$ (4) $155$ 44. $If the sum of two roots of the equation x^{3} - 2 p x^{2} + 3 qx - 4 r = 0 is zero, then the value of r is$ (1) $\frac{3 pq}{2}$ (2) $\frac{3 pq}{4}$ (3) $pq$ (4) $2 pq$ 45. $If α, β are the roots of the equation x^{2} - (5 + 3^{l o g_{3} 5} - 5^{l o g_{5} 3}) x + 3 (3^{(l o g_{3} 5)^{1/3}} - 5^{(l o g_{5} 3)^{2/3}} - 1) = 0 then the equation whose roots are α + \frac{1}{β} and β + \frac{1}{α} is$ (1) $3 x^{2} - 20 x - 12 = 0$ (2) $3 x^{2} - 10 x - 4 = 0$ (3) $3 x^{2} - 10 x + 2 = 0$ (4) $3 x^{2} - 20 x + 16 = 0$ 46. $If 2 α = - 1 - i 3 and 2 β = - 1 + i 3, then 5 α^{4} + 5 β^{4} + 7 α^{- 1} β^{- 1} is equal to$ (1) $-1$ (2) $-2$ (3) $0$ (4) $2$ 47. $If the cubic equation x^{3} - a x^{2} + ax - 1 = 0 is identical with the cubic equation whose roots are the squares of the roots of the given cubic equation, then the non-zero real value of ' a' is$ (1) $\frac{1}{2}$ (2) $2$ (3) $3$ (4) $\frac{7}{2}$ 48. $If tan 15^{\circ} and tan 30^{\circ} are the roots of the equation x^{2} + px + q = 0, then pq =$ (1) $\frac{6 3 + 10}{3}$ (2) $\frac{10 - 6 3}{3}$ (3) $\frac{10 + 6 3}{3}$ (4) $\frac{10 - 6 3}{3}$ 49. $All the roots of the equation x^{5} + 15 x^{4} + 94 x^{3} + 305 x^{2} + 507 x + 353 = 0 are increased by some real number k in order to eliminate the 4th degree term from the equation. Now, the coefficient of x in the transformed equation is$ (1) $2$ (2) $1$ (3) $6$ (4) $0$ 50. $The polynomial equation of degree 4 having real coefficients with three of its roots as 2 \pm 3 and 1 + 2 i, is$ (1) $x^{4} - 6 x^{3} - 14 x^{2} + 22 x + 5 = 0$ (2) $x^{4} - 6 x^{3} - 19 x + 22 x - 5 = 0$ (3) $x^{4} - 6 x^{3} + 19 x - 22 x + 5 = 0$ (4) $x^{4} - 6 x^{3} + 14 x^{2} - 22 x + 5 = 0$ 51. $For real number x, if the minimum value of f (x) = x^{2} + 2 bx + 2 c^{2} is greater than the maximum value of g (x) = - x^{2} - 2 cx + b^{2}, then$ (1) $c^{2} > 2 b^{2}$ (2) $c^{2} < 2 b^{2}$ (3) $b^{2} = 2 c^{2}$ (4) $c^{2} = 2 b^{2}$ 52. $2 cot^{2} θ - cotθ - 3 =$ (1) $(2 cotθ - 3) (cotθ + 1)$ (2) $(2 cotθ - 1) (cotθ + 3)$ (3) $(2 cotθ + 3) (cotθ - 1)$ (4) $(2 cotθ + 1) (cotθ - 3)$ 53. $If α, β are the roots of quadratic equation x^{2} + bx + c = 0 such that α^{2} + β^{2} = 5 and α^{3} + β^{3} = 9, then b + c =$ (1) $-5$ (2) $-1$ (3) $1$ (4) $5$ 54. $Find number of real roots of equation e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0 is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 55. $If the roots of x^{3} - k x^{2} + 14 x - 8 = 0 are in geometric progression, then k is equal to$ (1) $-3$ (2) $7$ (3) $4$ (4) $0$ 56. $If z_{1} and z_{2} are the roots of the equation x^{2} + 2 x + 2 = 0, then \frac{- 2 ^{11} ( z _{1} + 1 + 3 i ) ^{11}}{2 ^{5} ( z _{2} + 1 - 3 i ) ^{11}} is equal to$ (1) $64$ (2) $32$ (3) $162$ (4) $82$ 57. $Let φ (x) = \frac{x}{( x ^{2} + 1 ) ( x + 1 )} . If a, b and c are the roots of the equation x^{3} - 3 x + λ = 0, (λ \neq = 0) . Then, φ (a) φ (b) φ (c) =$ (1) $λ$ (2) $\frac{- λ}{( λ + 2 ) ( λ ^{2} + 16 )}$ (3) $\frac{λ}{( λ + 2 )}$ (4) $\frac{λ}{( λ + 2 ) ( λ ^{2} + 16 )}$ 58. $Let the set C = ((x, y) ∣ x^{2} - 2^{y} = 2023, x, y \in N) . Then \sum_{(x, y) = C} (x + y) is equal to______$ 59. $For a natural number n, let a_{n} = 1 9^{n} - 1 2^{n} . Then, the value of \frac{31 α _{9} - α _{10}}{57 α _{8}} is$ 60. $Let α, β \in N be roots of equation x^{2} - 70 x + λ = 0, where \frac{λ}{2}, \frac{λ}{3} \in / N . If λ assumes the minimum possible value, then \frac{( α - 1 + β - 1 ) ( λ + 35 )}{\| α - β \|} is equal to:$