Quadratic Equations and Inequalities questions
963 published questions with worked solutions.
- The sum of the cubes of all the roots of the equation x^4-3x^3-2x^2+3x+1=0 is
- Let α, β be the roots of the equation x² − 3x + r = 0 and 2, 2β be the roots of the equation x² + 3x + r = 0. If the roots of the...
- Let the parabola y = x² + px + q passing through the point (1, −1) be such that the distance between its vertex and the x-axis is...
- Let α, α + 2, α ∈ ℤ, be the roots of the quadratic equation x(x + 2) + (x + 1)(x + 3) + (x + 2)(x + 4) + … + (x + n − 1)(x + n +...
- Let a, b, c ∈ 1, 2, 3, 4 . If the probability, that ax² + 2√2 bx + c > 0 for all x ∈ ℝ, is mn, gcd(m, n) = 1, then m + n is equal...
- If the quadratic equation (λ+2)x² − 3λx + 4λ = 0, λ ≠ −2, has two positive roots, then the number of possible integral values of...
- Let α, β be the roots of the equation x²−x+p=0 and γ, δ be the roots of the equation x²−4x+q=0; p,q∈ℤ. If α, β, γ, δ are in G.P.,...
- Let z₁,z₂∈ℂ be the distinct solutions of the equation z²+4z−(1+12i)=0. Then |z₁|²+|z₂|² is equal to:
- Let a, b ∈ ℂ. Let α, β be the roots of the equation x² + ax + b = 0. If β − α = √11 and β² − α² = 3i√11, then (β³ − α³)² is equal...
- Consider the quadratic equation (n²−2n+2)x²−3x+(n²−2n+2)²=0, n∈ℝ. Let α be the minimum value of the product of its roots and β be...
- Let one root of the quadratic equation in x: (k²−15k+27)x²+9(k−1)x+18=0 be twice the other. Then the length of the latus rectum...
- 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...
- Let α, β be the roots of the equation x² + 2√2x − 1 = 0. The quadratic equation, whose roots are α⁴ + β⁴ and (1/10)(α⁶ + β⁶), is
- Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4....
- Let α, β be the roots of the equation x² – x + 2 = 0 with Im(α) > Im(β). Then α⁶ + α⁴ + β⁴ – 5α² is equal to ______.
- The number of real solution of the equation x(x² + 3|x| + 5|x – 1| + 6|x – 2|) = 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 ( - 1 +...
- Let a, b, c be the lengths of three sides of a triangle satisfying the condition (a² + b²)x² − 2b(a + c)x + (b² + c²) = 0. If the...
- 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 root. Then, among the two...
- Let S be the set of positive integral values of a for which ax^2 + 2(a+1)x + 9a + 4 x^2 - 8x + 32 < 0, ∀ x ∈ ℝ. Then, the number...
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