A) \[5(\sqrt{6}-5)\]
B) \[5\sqrt{3}(\sqrt{6}-5)\]
C) \[\sqrt{5}(\sqrt{6}-\sqrt{3})\]
D) \[\sqrt{2}(\sqrt{7}-\sqrt{5})\]
E) \[3(\sqrt{5}-\sqrt{2})\]
Correct Answer: B
Solution :
\[\left| \begin{matrix} \sqrt{13}+\sqrt{3} & 2\sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5 \\ \end{matrix} \right|\] \[=\left| \begin{matrix} \sqrt{13} & 2\sqrt{5} & \sqrt{5} \\ \sqrt{26} & 5 & \sqrt{10} \\ \sqrt{65} & \sqrt{15} & 5 \\ \end{matrix} \right|+\left| \begin{matrix} \sqrt{3} & 2\sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \\ \end{matrix} \right|\] \[=\sqrt{13}.\sqrt{5}.\sqrt{5}\left| \begin{matrix} 1 & 2 & 1 \\ \sqrt{2} & \sqrt{5} & \sqrt{2} \\ \sqrt{5} & \sqrt{3} & \sqrt{5} \\ \end{matrix} \right|\] \[\sqrt{3}.\sqrt{5}.\sqrt{5}\left| \begin{matrix} 1 & 2 & 1 \\ \sqrt{5} & \sqrt{5} & \sqrt{2} \\ \sqrt{3} & \sqrt{3} & \sqrt{5} \\ \end{matrix} \right|\] \[=0+5\sqrt{3}\left| \begin{matrix} -1 & 2 & 1 \\ 0 & \sqrt{5} & \sqrt{2} \\ 0 & \sqrt{3} & \sqrt{5} \\ \end{matrix} \right|=5\sqrt{3}(\sqrt{6}-5)\]
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