Nature of Roots & Formation of Equations: JEE Questions & Solutions

Let α and β be the roots of the equation x² + 2ax + (3a + 10) = 0 such that α < 1 < β. Then the set of all possible values of a is:

The smallest positive integral value of a, for which all the roots of x⁴-ax²+9=0 are real and distinct, is equal to

If α,β where α<β, are the roots of the equation λ x²-(λ+3)x+3=0 such that 1/α-1/β=1/3, then the sum of all possible values of λ is:

Let α₁ and β₁ be the distinct roots of 2x² + (cosθ)x - 1 = 0, θ ∈ (0, 2π). If m and M are the minimum and the maximum values of α₁⁴ + β₁⁴,…

If the equation a(b - c)x² + b(c - a)x + c(a - b) = 0 has equal roots, where a + c = 15 and b = 36/5, then a² + c² is equal to.

Let α, β be the roots of the equation x² - ax - b = 0 with Im(α) < Im(β). Let Pₙ = αⁿ - βⁿ. If P₃ = -5√7 i, P₄ = -3√7 i, P₅ = 11√7 i, P₆ =…

The product of all the rational roots of the equation (x² - 9x + 11)² - (x - 4)(x - 5) = 3, is equal to:

If for some α, β; α ≤ β, α + β = 8 and sec²(tan⁻¹ α) + csc²(cot⁻¹ β) = 36, then α² + β is:

Let f: ℝ \ {0} → (-∞, 1) be a polynomial of degree 2, satisfying f(x)f( 1/x) = f(x) + f( 1/x). If f(K) = -2K, then the sum of squares of…

If the set of all a ∈ ℝ, for which the equation 2x² + (a - 5)x + 15 = 3a has no real root, is the interval (α, β), and X = { x ∈ ℤ: α < x <…

If the set of all a ∈ ℝ \ {1}, for which the roots of the equation (1 - a)x² + 2(a - 3)x + 9 = 0 are positive is (-∞, -α] ∪ [β, γ), then 2α…

Let α and β be the roots of x² + √3x - 16 = 0, and γ and δ be the roots of x² + 3x - 1 = 0. If Pₙ = αⁿ + βⁿ and Qₙ = γⁿ + δⁿ, then (P₂₅ +…

Consider the equation x² + 4x - n = 0, where n ∈ [20, 100] is a natural number. Then the number of all distinct values of n, for which the…

Let the set of all values of p ∈ ℝ, for which both the roots of the equation x² - (p + 2)x + (2p + 9) = 0 are negative real numbers, be the…

The number of real roots of the equation x|x-2| + 3|x-3| + 1 = 0 is:

Let α, β ∈ ℕ be roots of the equation x² - 70x + λ = 0, where λ/2, λ/3 ∉ ℕ. If λ assumes the minimum possible value, then…

For 0 < c < b < a, let (a + b - 2c)x² + (b + c - 2a)x + (c + a - 2b) = 0 and α ≠ 1 be one of its roots. Then, among the two statements (I)…

Let S be the set of positive integral values of a for which (ax² + 2(a + 1)x + 9a + 4)/(x² - 8x + 32) < 0, ∀ x ∈ ℝ. Then, the number of…

Let α and β be the roots of the equation px² + qx - r = 0, where p ≠ 0. If p, q, r are consecutive terms of a non-constant G.P. and 1/α +…

Let α, β; α > β, be the roots of the equation x² - √2x - √3 = 0. Let Pₙ = αⁿ - βⁿ, n ∈ N. Then (11√3 - 10√2)P₁₀ + (11√2 + 10)P₁₁ - 11P₁₂ is…

Let a ∈ ℝ and let α, β be the roots of the equation x² + 60^(1/4) x + a = 0. If α⁴ + β⁴ = -30, then the product of all possible values of a…

Let λ ≠ 0 be a real number. Let α, β be the roots of the equation 14x² - 31x + 3λ = 0 and α, γ be the roots of the equation 35x² - 53x + 4λ…

Let α₁, α₂, …, α₇ be the roots of the equation x⁷ + 3x⁵ - 13x³ - 15x = 0 and |α₁| ≥ |α₂| ≥ … ≥ |α₇|. Then α₁ α₂ - α₃ α₄ + α₅ α₆ is equal to:

The number of integral values of k, for which one root of the equation 2x² - 8x + k = 0 lies in the interval (1, 2) and its other root lies…

The sum of all the roots of the equation |x² - 8x + 15| - 2x + 7 = 0 is:

Let α, β, γ be the three roots of the equation x³ + bx + c = 0. If βγ = 1 = -α, then b³ + 2c³ - 3α³ - 6β³ - 8γ³ is equal to:

If a and b are the roots of the equation x² - 7x - 1 = 0, then the value of (a²¹ + b²¹ + a¹⁷ + b¹⁷)/(a¹⁹ + b¹⁹) is equal to:

The set of all a ∈ ℝ for which the equation x|x - 1| + |x + 2| + a = 0 has exactly one real root is:

Let α, β be the roots of the equation x² - √2 x + 2 = 0. Then α¹⁴ + β¹⁴ is equal to: