Question

For $x=2\sqrt{3}+2\sqrt{2},$ if $x+\frac{1}{x}$ is $\frac{5\sqrt{3}+3\sqrt{2}}{m}$ then $m$ is :

• A. $2$
• B. $4$
• C. $8$
• D. $5$

Answer 1

The correct option is A $2$

$x=2\sqrt{3}+2\sqrt{2}$
$\frac{1}{x}=\frac{1}{2\sqrt{3}+2\sqrt{2}}$

Rationalising the denominator, we get

$\frac{1}{x}=\frac{1}{2\sqrt{3}+2\sqrt{2}}\times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}$

$=\frac{\sqrt{3}-\sqrt{2}}{2(3-2)}$

$=\frac{\sqrt{3}-\sqrt{2}}{2}$

$x+\frac{1}{x}=2\sqrt{3}+2\sqrt{2}+\frac{\sqrt{3}-\sqrt{2}}{2}$

$=\frac{5\sqrt{3}+3\sqrt{2}}{2}$

Hence, $m=2$.