Question

The value of the determinant $\left|\begin{array}{cccc}\sqrt{13}& +\sqrt{3}& 2\sqrt{5}& \sqrt{5}\\ \sqrt{15}& +\sqrt{26}& 5& \sqrt{10}\\ 3& +\sqrt{65}& \sqrt{15}& 5\end{array}\right|$ is equal to

• A. $5\sqrt{3}(\sqrt{6}-5)$
• B. $5\sqrt{3}(\sqrt{6}-\sqrt{5})$
• C. $5(\sqrt{6}-5)$
• D. $\sqrt{3}(\sqrt{6}-\sqrt{5})$

Answer 1

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

First term :

$(\sqrt{13}+\sqrt{3})(25-\sqrt{150})$

$(\sqrt{13}+\sqrt{3})(25-5\sqrt{6})$

$+25\sqrt{3}-5\sqrt{78}+25\sqrt{13}-5\sqrt{18}$

Second term :

$-(\sqrt{15}+\sqrt{26})(10\sqrt{5}-5\sqrt{3})$

$-50\sqrt{3}+15\sqrt{5}-10\sqrt{130}+5\sqrt{78}$

Third term :

$3+\sqrt{65}(10\sqrt{2}-5\sqrt{5})$

$30\sqrt{2}-15\sqrt{5}+10\sqrt{130}-25\sqrt{13}$

$⟹5\sqrt{3}(\sqrt{6}-5)$