If the roots of the equation x^{2} – x – A = 0 are real and the sum of the fourth powers of the roots is 337. What is the value of A?

Option 3 : 12

RBI Grade B 2020: Full Mock Test

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**Calculation:**

Let the roots of the equation be a and b.

∴ a + b = 1 and ab = -A and a^{4} + b^{4} = 337

Squaring a + b = 1 on both sides.

⇒ (a + b)^{2} = 1^{2}

⇒ a^{2} + b^{2} - 2ab = 1

⇒ a^{2} + b^{2} = 1 + 2A

[Squaring the equation on both sides]

⇒ (a^{2} + b^{2})^{2} = (1 + 2A)^{2}

⇒ a^{4} + b^{4} + 2a^{2}b^{2} = (1 + 2A)^{2} [a^{2}b^{2} = P^{2}]

⇒ a^{4} + b^{4} = 1 + 4A^{2} + 4A – 2A^{2}

⇒ a^{4} + b^{4} = 1 + 2A^{2} + 4A

[a^{4} + b^{4} = 337]

⇒ 337 = 1 + 2A^{2} + 4A

⇒ 2A^{2} + 4A = 336

⇒ A^{2} + 2A – 168 = 0

⇒ A^{2} + 14A – 12A – 168 = 0

⇒ A(A + 14) – 12(A + 14) = 0

⇒ (A + 14) (A - 12) = 0

∴ A = -14 and A = 12.

A = -14 is not real root

**∴ the value of A is 12. **