Question : If $p = 5 + 2\sqrt6$, then $\frac{\sqrt p - 1}{\sqrt p}$ is:
Option 1: $1 + \sqrt2 - \sqrt3$
Option 2: $1 - \sqrt2 + \sqrt3$
Option 3: $ - 1 + \sqrt2 - \sqrt3$
Option 4: $1 - \sqrt2 - \sqrt3$
Correct Answer: $1 + \sqrt2 - \sqrt3$
Solution :
$p = 5 + 2\sqrt{6}$
= $(\sqrt{3})^2 + (\sqrt{2})^2 + 2\sqrt{6}$
= $(\sqrt{3} + \sqrt{2})^2 $
Now, $\sqrt{p}=\sqrt{3} + \sqrt{2}$
So, $\frac{1}{\sqrt{p}} = \sqrt{3} - \sqrt{2}$ (by rationalisation)
Thus, $\frac{\sqrt{p} - 1}{\sqrt{p}} = \frac{\sqrt{p}}{\sqrt{p}} - \frac{{1}}{\sqrt{p}} = 1 - \frac{{1}}{\sqrt{p}}$
Putting the value we get,
$1 - (\sqrt{3} - \sqrt{2})=1 - \sqrt{3} + \sqrt{2}$
Hence, the correct answer is $1 + \sqrt{2} - \sqrt{3}$.
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