Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20265 AprilMorning Shift
Exam
JEE Main
Year
2026
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2026 · 5 April · Morning Shift · Mathematics · Quadratic Equation

Let a, b ∈ ℂ . Let α, β be the roots of the equation x^2 + ax + b =...

Let a,b∈Ca, b \in \mathbb{C}. Let α,β\alpha, \beta be the roots of the equation x2+ax+b=0x^2 + ax + b = 0. If β−α=11\beta - \alpha = \sqrt{11} and β2−α2=3i11\beta^2 - \alpha^2 = 3i\sqrt{11}, then (β3−α3)2(\beta^3 - \alpha^3)^2 is equal to:

Options

  1. A160160
  2. B176176
  3. C194194
  4. D187187

Correct answer

B. 176176

Step-by-step solution

Given β−α=11\beta - \alpha = \sqrt{11} and β2−α2=3i11\beta^2 - \alpha^2 = 3i\sqrt{11}

β+α=β2−α2β−α=3i1111=3i\beta + \alpha = \dfrac{\beta^2 - \alpha^2}{\beta - \alpha} = \dfrac{3i\sqrt{11}}{\sqrt{11}} = 3i

Using the identity (β−α)2=(β+α)2−4αβ(\beta - \alpha)^2 = (\beta + \alpha)^2 - 4\alpha\beta

(11)2=(3i)2−4αβ(\sqrt{11})^2 = (3i)^2 - 4\alpha\beta

11=−9−4αβ11 = -9 - 4\alpha\beta

4αβ=−20⇒αβ=−54\alpha\beta = -20 \Rightarrow \alpha\beta = -5

Now, β3−α3=(β−α)(β2+αβ+α2)\beta^3 - \alpha^3 = (\beta - \alpha)(\beta^2 + \alpha\beta + \alpha^2)

β3−α3=(β−α)((β+α)2−αβ)\beta^3 - \alpha^3 = (\beta - \alpha)((\beta + \alpha)^2 - \alpha\beta)

β3−α3=11((3i)2−(−5))\beta^3 - \alpha^3 = \sqrt{11}((3i)^2 - (-5))

β3−α3=11(−9+5)=−411\beta^3 - \alpha^3 = \sqrt{11}(-9 + 5) = -4\sqrt{11}

Squaring both sides, we get:

(β3−α3)2=(−411)2=16×11=176(\beta^3 - \alpha^3)^2 = (-4\sqrt{11})^2 = 16 \times 11 = 176

Answer: 176176
Practice Quadratic Equation on Quantrex Academy →

More from Quadratic Equation

Match each entry in List-I to the correct entry in List-II and choose the correct option.… JEE Advanced 2026Let a, b, c be positive integers in arithmetic progression such that the equation ax^2 +… JEE Advanced 2026Consider the quadratic equation (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0 , n ∈ ℝ .… JEE Main 2026Let one root of the quadratic equation in x : (k^2 - 15k + 27)x^2 + 9(k-1)x + 18 = 0 be… JEE Main 2026Let e_1 and e_2 be two distinct roots of the equation x^2 - ax + 2 = 0 . Let the sets \ a… JEE Main 2026Let α, β be the roots of the equation x^2 - x + p = 0 and γ, δ be the roots the equation… JEE Main 2026Let A, B , where A, B ∈ (-(π)/(2), (π)/(2)) , be the roots of the quadratic equation x^2… JEE Main 2026If α=1 and β=1+i 2 , where i= -1 are two roots of the equation x^3+ax^2+bx+c=0 , a,b,c ∈… JEE Main 2026If the quadratic equation (λ+2)x^2-3λ x+4λ=0 , λ ≠ -2 , has two positive roots, then the… JEE Main 2026Let α, β be the roots of the equation x^2 - 3x + r = 0 , and (α)/(2), 2β be the roots of… JEE Main 2026Let a, b, c ∈ \ 1, 2, 3, 4\ . If the probability, that ax^2 + 2 2 \,bx + c > 0 for all x… JEE Main 2026Let S=\ x^ 3 +a x^ 2 +b x+c: a, b, c ∈ ~N. and .a, b, c ≤ 20\ be a set of polynomials.… JEE Main 2026If α, β , where α<β , are the roots of the equation λ x^ 2 -(λ+3) x+3=0 such that… JEE Main 2026The smallest positive integral value of a , for which all the roots of x^ 4 -a x^ 2 +9=0… JEE Main 2026The sum of all the real solutions of the equation _ (x+3) (6 x^ 2 +28 x+30)=5-2 _ (6… JEE Main 2026A building construction work can be completed by two masons A and B together in 22.5… JEE Main 2026 Full Quadratic Equation list Quadratic Equation JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs