Question

Find the value of: $\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{15}+\sqrt{26}& 5& \sqrt{10}\\ 3+\sqrt{65}& \sqrt{15}& 5\end{array}\right|$

• A. $15\sqrt{2}-25\sqrt{3}$
• B. $15\sqrt{5}-25\sqrt{6}$
• C. $25\sqrt{2}+15\sqrt{3}$
• D. $0$

Answer 1

The correct option is A $15\sqrt{2}-25\sqrt{3}$

$\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{15}+\sqrt{26}& 5& \sqrt{10}\\ 3+\sqrt{65}& \sqrt{15}& 5\end{array}\right|$
$=\left|\begin{array}{ccc}\sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{15}& 5& \sqrt{10}\\ 3& \sqrt{15}& 5\end{array}\right|+\left|\begin{array}{ccc}\sqrt{13}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{26}& 5& \sqrt{10}\\ \sqrt{65}& \sqrt{15}& 5\end{array}\right|$
$=5\sqrt{3}\left|\begin{array}{ccc}1& 2& 1\\ \sqrt{5}& \sqrt{5}& \sqrt{2}\\ \sqrt{3}& \sqrt{3}& \sqrt{5}\end{array}\right|+5\sqrt{13}\left|\begin{array}{ccc}1& 2& 1\\ \sqrt{2}& \sqrt{5}& \sqrt{2}\\ \sqrt{5}& \sqrt{3}& \sqrt{5}\end{array}\right|$
$=5\sqrt{3}\left|\begin{array}{ccc}1& 2& 1\\ \sqrt{5}& \sqrt{5}& \sqrt{2}\\ \sqrt{3}& \sqrt{3}& \sqrt{5}\end{array}\right|$
$=5\sqrt{3}\left[1\right(5-\sqrt{6})-2(5-\sqrt{6})+1(\sqrt{15}-\sqrt{15}\left)\right]$
$=5\sqrt{3}(-5+\sqrt{6})=-25\sqrt{3}+15\sqrt{2}$