If the mean of x and 1- x is M- the mean of x 3 and 1- wA- M 2-3-2B- M 4 M 2-3C- M 3D- M 3-3

The correct option is B M(4M^2 3)Given that, x+1/x/2=M ⇒ x+1/x=2M ( x+1/x )^2=4M^2, x^2+1/x^2=4M^2 2 Mean of x^3 and 1/x^3 is =x^3+1/x^3/2= ( x+1/x ) ( x^2 x. ...

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Byju's Answer

Standard X

Mathematics

Mean for Ungrouped Data

If the mean o...

Question

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

\frac{M^{2} - 3}{2}

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M (4 M^{2} - 3)

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M^{3}

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M^{3} + 3

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Solution

The correct option is B $M (4 M^{2} - 3)$
Given that,
$\frac{x + \frac{1}{x}}{2} = M$
$\Rightarrow x + \frac{1}{x} = 2 M$
${(x + \frac{1}{x})}^{2} = 4 M^{2}, x^{2} + \frac{1}{x^{2}} = 4 M^{2} - 2$

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

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