Let a, b ∈ C. Let α, β be the roots of the equation x^2 + ax + b = 0. If β - α = √11 and β^2 - α^2 = 3i√11, then (β^3 - α^3)^2 is equal to:

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q1 – JEE-MAIN 2026

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Home Maths Complex & QuadraticQuestion #76

Let

a, b \in C

. Let

α, β

be the roots of the equation

x^{2} + a x + b = 0

. If

β - α = \sqrt{11}

and

β^{2} - α^{2} = 3 i \sqrt{11}

, then

(β^{3} - α^{3})^{2}

is equal to:

Clue 1 of 3: Algebraic Identities

Recognize that the given expressions involve differences of powers, which can often be factored using standard algebraic identities.

🧠 Full Solution Path

Step 1: Determine Sum of Roots (

β + α

)✦ Active

We are given $β - α = \sqrt{11}$ and $β^{2} - α^{2} = 3 i \sqrt{11}$ . Factor the second equation using the difference of squares identity:

(β - α) (β + α) = 3 i \sqrt{11}

Substitute the value of $β - α$ :

(\sqrt{11}) (β + α) = 3 i \sqrt{11}

Dividing by $\sqrt{11}$ gives:

β + α = 3 i

Step 2: Calculate Product of Roots (

α β

)○ Expand

We use the identity $4 α β = (β + α)^{2} - (β - α)^{2}$ . Substitute the values found in Step 1 and the given value:

4 α β = (3 i)^{2} - (\sqrt{11})^{2}

Simplify the expression:

4 α β = 9 i^{2} - 11 = - 9 - 11 = - 20

Therefore, the product of the roots is:

α β = - 5

Step 3: Evaluate

(β^{3} - α^{3})^{2}

○ Expand

Use the algebraic identity for the difference of cubes: $β^{3} - α^{3} = (β - α) (β^{2} + α β + α^{2})$ . We can rewrite $β^{2} + α^{2}$ as $(β + α)^{2} - 2 α β$ . So, the identity becomes:

β^{3} - α^{3} = (β - α) ((β + α)^{2} - α β)

Substitute the values $β - α = \sqrt{11}$ , $β + α = 3 i$ , and $α β = - 5$ :

β^{3} - α^{3} = (\sqrt{11}) ((3 i)^{2} - (- 5))

Simplify the expression:

β^{3} - α^{3} = (\sqrt{11}) (- 9 + 5) = (\sqrt{11}) (- 4) = - 4 \sqrt{11}

Finally, square the result:

(β^{3} - α^{3})^{2} = (- 4 \sqrt{11})^{2} = (- 4)^{2} (\sqrt{11})^{2} = 16 \times 11 = 176

💡 Teacher's Secret Hint

Ensure careful handling of complex numbers and negative signs during squaring.

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Maths Question 1 – JEE-MAIN 2026

Clue 1 of 3: Algebraic Identities

🧠 Full Solution Path