If the average of x and frac 1 x (x ne…

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If the average of x and $$\frac{1}{x}$$ (x $$ \ne $$ 0) is M, then the average of x2 and $$\frac{1}{{{x^2}}}$$ is :

  1. A. 1 – M2
  2. B. 1 – 2M2
  3. C. 2M2 – 1
  4. D. 2M2 + 1

Answer: Option C

Here is complete explanation of If the average of x and frac 1 x (x ne….

Solution(By ExamCraze Team)

According to the question,
Average of $$frac{{x + frac{1}{x}}}{2} = M$$
Put x = 1
$$eqalign{ & therefore ,,frac{{1 + frac{1}{1}}}{2} = M cr & Rightarrow M = 1 cr & therefore ,,frac{{{x^2} + frac{1}{{{x^2}}}}}{2} cr & = frac{{{1^2} + frac{1}{{{1^2}}}}}{2} cr & = 1 cr} $$
Now check from the option
Option: (C) 2M2 - 1 (put M = 1)
= 2 × 1 - 1
= 1 (satisfied)

Alternate :
According to the question,
$$eqalign{ & Rightarrow frac{{x + frac{1}{x}}}{2} = M cr & text{Squaring the both sides} cr & Rightarrow {x^2} + frac{1}{{{x^2}}} = {(2M)^2} - 2 cr & Rightarrow {x^2} + frac{1}{{{x^2}}} = 4{M^2} - 2 cr} $$
Required average
$$eqalign{ & = frac{{{x^2} + frac{1}{{{x^2}}}}}{2} cr & = frac{{4{M^2} - 2}}{2} cr & = 2{M^2} - 1 cr} $$

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