If the mean of x and 1- x is M- the mean of x 3 and 1- x 3 isA- 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.1 ...

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Standard X

Mathematics

Mean

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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If the mean of x and 1x is M, the mean of x3 and 1x3 is

The correct option is B M(4M2−3) Given that x+1x2=M ⇒x+1x=2M (x+1x)2=4M2,x2+1x2=4M2−2 Mean of x3 and 1x3 is =x3+1x32=(x+1x)(x2−x.1x+1x2)2 =(x+1x)(x2−1+1x2)2 =2M.(4M2−2−1)2 =M(4M2−3).

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