Relations Between Roots and Coefficients

JEE Main Mathematics · Quadratic Equations · 25 questions, latest first

• Consider the quadratic equation (n^2-2 n+2) x^2-3 x+(n^2-2 n+2)^2=0, n ∈ R. Let α be the minimum value of the product of its…20266 April, Shift 2 · Q2
• Let one root of the quadratic equation in x: (k^2-15 k+27) x^2+9(k-1) x+18=0 be twice the other. Then the length of the latus…20266 April, Shift 1 · Q2
• Let α, β be the roots of the equation x^2-x+p=0 and γ, δ be the roots the equation x^2-4 x+q=0; p, q ∈ Z. If α, β, γ, δ are in…20265 April, Shift 2 · Q1
• Let a, b ∈ C. Let α, β be the roots of the equation x^2+a x+b=0. If β-α=√11 and β^2-α^2=3 i √11, then (β^3-α^3)^2 is equal to:20265 April, Shift 1 · Q1
• Let α, β be the roots of the equation x^2-3 x+r=0, and (α)/2, 2 β be the roots of the equation x^2+3 x+r=0. If the roots of the…20262 April, Shift 2 · Q1
• 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+1)=4 n for some n ∈ N. Then…20262 April, Shift 1 · Q1
• If α, β, where α<β, are the roots of the equation λ x^2-(λ+3) x+3=0 such that 1/(α)-1/(β)=1/3, then the sum of all possible…202628 January, Shift 1 · Q3
• Let α, β be the roots of the quadratic equation 12 x^2-20 x+3 λ=0, λ ∈ Z. If 1/2 ≤|β-α| ≤ 3/2, then the sum of all possible…202622 January, Shift 2 · Q3
• Let α and β be the roots of x^2+√3 x-16=0, and γ and δ be the roots of x^2+3 x-1=0. If P_n= α^n+β^n and Q_n=γ^n+δ^n, then…20253 April, Shift 1 · Q3
• Let P_n=α^n+β^n, n ∈ N. If P_10=123, P_9=76, P_8=47 and P_1=1, then the quadratic equation having roots 1/(α) and 1/(β) is:20252 April, Shift 1 · Q17
• Let α, β be the roots of the equation x^2-a x-b=0 with Im(α)<Im(β). Let P_n=α^n-β^n. If P_3=-5 √7 i, P_4=-3 √7 i, P_5=11 √7 i and…202523 January, Shift 2 · Q22
• Let α_θ and β_θ be the distinct roots of 2 x^2+( cos θ) x-1=0, θ ∈(0,2 π). If m and M are the minimum and the maximum values of…202522 January, Shift 2 · Q2
• Let α, β; α>β, be the roots of the equation x^2-√2 x-√3=0. Let P_n=α^n-β^n, n ∈ N. Then (11 √3-10 √2) P_10+(11 √2+10) P_11-11…20249 April, Shift 2 · Q3
• Let α, β be the roots of the equation x^2+2 √2 x-1=0. The quadratic equation, whose roots are α^4+β^4 and 1/10(α^6+β^6), is:20249 April, Shift 1 · Q2
• Let α, β be roots of x^2+√2 x-8=0. If U_n=α^n+β^n, then (U_10+√2 U_9)/(2 U_8) is equal to ____.20246 April, Shift 2 · Q21
• Let α, β be the distinct roots of the equation x^2-(t^2-5 t+6) x+1=0, t ∈ R and a_n=α^n+β^n. Then the minimum value of…20246 April, Shift 1 · Q4
• If 2 and 6 are the roots of the equation a x^2+b x+1=0, then the quadratic equation, whose roots are 1/(2 a+b) and 1/(6 a+b), is:20244 April, Shift 1 · Q3
• Let α and β be the roots of the equation p x^2+q x-r=0, where p ≠ 0. If p, q and r be the consecutive terms of a non constant…20241 February, Shift 2 · Q4
• For 0<c<b<a, let (a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0 and α ≠ 1 be one of its root. Then, among the two statements (I) If α…202431 January, Shift 1 · Q8
• Let α, β ∈ N be roots of the equation x^2-70 x+λ=0, where (λ)/2, (λ)/3 ∉ N. If λ assumes the minimum possible value, then…202430 January, Shift 1 · Q22
• Let α, β be the roots of the equation x^2-x+2=0 with Im(α)>Im(β). Then α^6+α^4+β^4-5 α^2 is equal to ____.202429 January, Shift 1 · Q21
• If α, β are the roots of the equation, x^2-x-1=0 and S_n=2023 α^n+2024 β^n, then:202427 January, Shift 2 · Q3
• If a and b are the roots of the equation x^2-7 x-1=0, then the value of (a^21+b^21+a^17+b^17)/(a^19+b^19) is equal to ____.202311 April, Shift 1 · Q24
• Let α, β, γ be the three roots of the equation x^3+b x+c=0. If β γ=1=-α, then b^3+2 c^3-3 α^3-6 β^3-8 γ^3 is equal to20238 April, Shift 1 · Q3
• Let a ∈ R and let α, β be the roots of the equation x^2+60^(1/4) x+a=0 If α^4+β^4=-30, then the product of all possible values of…202325 January, Shift 2 · Q81