If one root of the equation $\left|\begin{array}{ccc}7& 6& x^2-13\\ 2& x^2-13& 2\\ x^2-13& 3& 7\end{array}\right|=0 \mathrm{is} x=2$, the sum of all other five roots is
(a) $2\sqrt{15}$               (b) –2                (c) $\sqrt{20}+\sqrt{15}-2$             (d) None of these 

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \left|\begin{array}{ccc}7& 6& {\mathrm{x}}^2-13\\ 2& {\mathrm{x}}^2-13& 2\\ {\mathrm{x}}^2-13& 3& 7\end{array}\right|=0 \\ \mathrm{Expand} \mathrm{along} {\mathrm{R}}_1 \\ 7\left[7\left({\mathrm{x}}^2-13\right)-6\right]-6\left[14-2\left({\mathrm{x}}^2-13\right)\right]+\left({\mathrm{x}}^2-13\right)\left[6-{\left({\mathrm{x}}^2-13\right)}^2\right]=0 \\ \mathrm{If} \mathrm{we} \mathrm{observe} \mathrm{above} \mathrm{equation}, \mathrm{then} \mathrm{it} \mathrm{is} \mathrm{clear} \mathrm{that} \mathrm{it} \mathrm{will} \mathrm{not} \\ \mathrm{generate} \mathrm{term} \mathrm{in} \mathrm{which} \mathrm{we} \mathrm{have} {\mathrm{x}}^5 \mathrm{and} \mathrm{by} \mathrm{observing} \mathrm{last} \mathrm{term} \\ \mathrm{i}.\mathrm{e}. \left({\mathrm{x}}^2-13\right)\left[6-{\left({\mathrm{x}}^2-13\right)}^2\right], \mathrm{it} \mathrm{is} \mathrm{clear} \mathrm{that} \mathrm{highest} \mathrm{exponent} \mathrm{of} \mathrm{x} \mathrm{is} 6. \\ \mathrm{Hence} \mathrm{it} \mathrm{is} 6 \mathrm{degree} \mathrm{equation} \mathrm{with} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^5=0 \\ \mathrm{One} \mathrm{of} \mathrm{the} \mathrm{root} \mathrm{is} 2. \\ \mathrm{Now} \\ \mathrm{Sum} \mathrm{of} \mathrm{roots}=-\frac{\mathrm{coefficient} \mathrm{of} {\mathrm{x}}^5}{\mathrm{coefficient} \mathrm{of} {\mathrm{x}}^6}=-\frac{0}{\mathrm{coefficient} \mathrm{of} {\mathrm{x}}^6}=0 \\ \Rightarrow 2+\mathrm{Sum} \mathrm{of} \mathrm{all} \mathrm{other} \mathrm{five} \mathrm{root}=0 \\ \Rightarrow \mathrm{Sum} \mathrm{of} \mathrm{all} \mathrm{other} \mathrm{five} \mathrm{root}=-2 \left(\mathrm{Answer}\right) \end{gathered}$$

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