Question

If the mean of $x$ and $\frac{1}{x}$ is M, the mean of $x^3$ and $\frac{1}{x^3}$ is _____.

• A. $\frac{M^2-3}{2}$
• B. $M(4M^2-3)$
• C. $M^3$
• D. $M^3+3$

Answer 1

The correct option is B $M(4M^2-3)$

Given that,
$\frac{x+\frac{1}{x}}{2}=M$
$\Rightarrow x+\frac{1}{x}=2M$
${(x+\frac{1}{x})}^2=4M^2,x^2+\frac{1}{x^2}=4M^2-2$

Mean of $x^3$ and $\frac{1}{x^3}$ is
$=\frac{x^3+\frac{1}{x^3}}{2}=\frac{(x+\frac{1}{x})(x^2-x.\frac{1}{x}+\frac{1}{x^2})}{2}$
$=\frac{(x+\frac{1}{x})(x^2-1+\frac{1}{x^2})}{2}$
$=\frac{2M.(4M^2-2-1)}{2}$
$=M(4M^2-3)$