Question : The mean of $x$ and $\frac{1}{x}$ is $N$. Then the mean of $x^2$ and $\frac{1}{x^2}$ is:
Option 1: $N^2$
Option 2: $2N^2-1$
Option 3: $N^2-2$
Option 4: $4N^2-2$
Correct Answer: $2N^2-1$
Solution :
Mean of $x$ and $\frac{1}{x}$ is $N$.
So, $\frac{x+\frac{1}{x}}{2}$ = $N$
Squaring both sides,
⇒ $\frac{(x+\frac{1}{x})^2}{2^2}$ = $N^2$
⇒ $\frac{x^2+\frac{1}{x^2}+2×x×\frac{1}{x}}{4}$ = $N^2$
⇒ $x^2+\frac{1}{x^2}$ = $4N^2-2$
So, the mean of $x^2$ and $\frac{1}{x^2}$ = $\frac{x^2+\frac{1}{x^2}}{2}$ = $2N^2-1$
Hence, the correct answer is $2N^2-1$.
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