GATE CS 2016 Set 2 Q.5 (GA Q.5): Quantitative Aptitude

In a quadratic function, the value of the product of the roots $(\alpha, \beta)$ is 4. Find the value of

\frac{\alpha^n + \beta^n}{\alpha^{-n} + \beta^{-n}}

$n^4$

$4^n$

$2^{2n-1}$

$4^{n-1}$

Official answer and solution

Solution

Rewrite the denominator using positive exponents and simplify the algebraic fraction.

Let $\alpha$ and $\beta$ be the roots of the quadratic function. We are given:

\alpha \beta = 4

We need to evaluate the expression:

E = \frac{\alpha^n + \beta^n}{\alpha^{-n} + \beta^{-n}}

Express the denominator using common positive powers:

\alpha^{-n} + \beta^{-n} = \frac{1}{\alpha^n} + \frac{1}{\beta^n} = \frac{\beta^n + \alpha^n}{\alpha^n \beta^n} = \frac{\alpha^n + \beta^n}{(\alpha \beta)^n}

Substitute this back into $E$ :

E = \frac{\alpha^n + \beta^n}{\frac{\alpha^n + \beta^n}{(\alpha \beta)^n}} = (\alpha \beta)^n

Substitute the given product of roots $\alpha \beta = 4$ :

E = 4^n

Thus, the expression evaluates to $4^n$ .

Final answer

Common trap

Trying to solve for individual roots $\alpha$ and $\beta$ , which cannot be uniquely determined from the product alone.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

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