Question

$\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{15}+\sqrt{26}& 5& \sqrt{10}\\ \sqrt{65}+3& \sqrt{15}& 5\end{array}\right|$ equals -

• A. $0$
• B. $5\sqrt{3}(\sqrt{6}-5)$
• C. $5\sqrt{3}(5-\sqrt{6})$
• D. none of these

Answer 1

The correct option is B $5\sqrt{3}(\sqrt{6}-5)$

we can take $\sqrt{5}$ common from both column $2$ and $3$,

$=5\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3}& 2& 1\\ \sqrt{15}+\sqrt{26}& \sqrt{5}& \sqrt{2}\\ \sqrt{65}+3& \sqrt{3}& \sqrt{5}\end{array}\right|$

$C\to C_1-\sqrt{3}{C}_2$

$=5\left|\begin{array}{ccc}\sqrt{13}-\sqrt{3}& 2& 1\\ \sqrt{26}& \sqrt{5}& \sqrt{2}\\ \sqrt{65}& \sqrt{3}& 5\end{array}\right|$

$C_1\to C_1-\sqrt{13}{C}_3$

$=5\left|\begin{array}{ccc}-\sqrt{3}& 2& 1\\ 0& \sqrt{5}& \sqrt{2}\\ 0& \sqrt{3}& \sqrt{5}\end{array}\right|$

Expanding along $C_1=5(-\sqrt{13})(5-\sqrt{6})$

$=5\sqrt{3}(\sqrt{6}-5)$