Question

If the mean of $x$ and $\frac{1}{x}$ is $M$, then the mean of $x^2$ and $\frac{1}{x^2}$ is

• A. $2M^2-1$
• B. $2M^2+1$
• C. $2M^2\times 1$
• D. $2M^2-2$

Answer 1

The correct option is A $2M^2-1$

Given

the mean of $x$ and $\frac{1}{x}$ is $M$.
$\frac{x+\frac{1}{x}}{2}=M$
$\Rightarrow x+\frac{1}{x}=2M$
Squaring on both the sides,
${(x+\frac{1}{x})}^2=4M^2$
$\Rightarrow x^2+\frac{1}{x^2}=4M^2-2$

$\Rightarrow \frac{x^2+\frac{1}{x^2}}{2}=2M^2-1$

Therefore, the mean of$x^2$ and $\frac{1}{x^2}$ is $2M^2-1$.