JEE Mathematics 2023 Question with Solution

Step-by-step solution

Common mistakes

• Using $\alpha + \beta$ and $\alpha\beta$ incorrectly from the quadratic. For $x^2 - \sqrt{5}x + 2 = 0$, we must use $\alpha + \beta = \sqrt{5}$ and $\alpha\beta = 2$ from Vieta's formulas.

• Trying to find the roots explicitly and making sign errors with surds. It is safer to use identities for $\alpha^2 + \beta^2$ and then build up to $\alpha^4 + \beta^4$.

• Computing $\alpha^4 + \beta^4$ as $\left(\alpha + \beta\right)^4$. This is incorrect because powers do not distribute over addition in that way; use symmetric identities instead.