SSC Sample Paper Mock Test-8 SSC CGL Tear-II Paper-1

If \[x=\frac{\sqrt{3}+1}{\sqrt{3}-1}\] and \[y=\frac{\sqrt{3}-1}{\sqrt{3}+1}\] the value of \[{{x}^{2}}+{{y}^{2}}\]is

A) 14

B) 13

C) 15

D) 10

Correct Answer: A

Solution :

\[x=\frac{\sqrt{3}+1}{\sqrt{3}-1}=\frac{\sqrt{3}+1}{\sqrt{3}-1}\times \frac{(\sqrt{3}+1)}{(\sqrt{3}+1)}\]

\[=\frac{{{(\sqrt{3}+1)}^{2}}}{3-1}=\frac{3+1+2\sqrt{3}}{2}\]

\[=\frac{4+2\sqrt{3}}{2}=2+\sqrt{3}\]

Similarly, \[y=\frac{\sqrt{3}-1}{\sqrt{3}+1}=2-\sqrt{3}\]

\[\therefore \]\[{{x}^{2}}+{{y}^{2}}={{(2+\sqrt{3})}^{2}}+{{(2-\sqrt{3})}^{2}}\]

\[=4+3+4\sqrt{3}+4+3-4\sqrt{3}=14\]