#### If $\sqrt{2}= 1\cdot 4142$ = 1.4142 then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$  is equal to :(A)2.4142 (B)5.8282 (C)0.4142 (D)0.1718

Solution

We have, $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$
We have to rationalize it
$\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}\times \frac{\sqrt{2}-1}{\sqrt{2}-1}}$           [Multiplying numerator and denominator by $\sqrt{2}-1$]

= $\frac{\sqrt{\left ( \sqrt{2}-1 \right )\times\left ( \sqrt{2} -1\right ) }}{\sqrt{\left ( \sqrt{2} \right )^{2}-\left ( 1 \right )^{2}}}$      [$\because$ (a – b) (a + b) = a2 – b2]

= $\frac{\sqrt{\left ( \sqrt{2}-1 \right )^{2}}}{1}$

$\sqrt{\left ( \sqrt{2}-1 \right )^{2}}$
=$\sqrt{2}-1$
=$1\cdot 4142-1$
=0.4142

Hence option C is correct.