Question

If $\Delta \left(x\right)=\left|\begin{array}{ccc}\alpha +x& \theta +x& \lambda +x\\ \beta +x& \varphi +x& \mu +x\\ \gamma +x& \Psi +x& v+x\end{array}\right|$, then
($S$ denotes the sum of all the cofactors of all elements in $\Delta \left(0\right)$ and dash denotes the derivative.)

• A. $\Delta "\left(x\right)=0$ and $\Delta \left(x\right)=\Delta \left(x\right)-Sx$
• B. $\Delta "\left(x\right)=0$ and $\Delta \left(x\right)=-\Delta \left(x\right)+Sx$
• C. $\Delta "\left(x\right)=1$ and $\Delta \left(x\right)=\Delta \left(x\right)+Sx$
• D. None of these.

Answer 1

The correct option is D None of these.

We have
${\Delta }^{\prime }\left(x\right)=\left|\begin{array}{ccc}1& \theta +x& \lambda +x\\ 1& \varphi +x& \mu +x\\ 1& \Psi +x& \nu +x\end{array}\right|+\left|\begin{array}{ccc}\alpha +x& 1& \lambda +x\\ \beta +x& 1& \mu +x\\ \gamma +x& 1& \nu +x\end{array}\right|+\left|\begin{array}{ccc}\alpha +x& \theta +x& 1\\ \beta +x& \varphi +x& 1\\ \nu +x& \Psi +x& 1\end{array}\right|$
Applying $C_2\to C_2-x{C}_1$ and $C_3\to C_3-x{C}_1$ in first, $C_1\to C_1-x{C}_2$ and
$C_3\to C_3-x{C}_2$ in second and $C_1\to C_1-x{C}_3$ and $C_2\to C_2-x{C}_3$ in third, then
${\Delta }^{\prime }\left(x\right)=\left|\begin{array}{ccc}1& \theta & \lambda \\ 1& \varphi & \mu \\ 1& \Psi & \nu \end{array}\right|+\left|\begin{array}{ccc}\alpha & 1& \lambda \\ \beta & 1& \mu \\ \gamma & 1& \nu \end{array}\right|+\left|\begin{array}{ccc}\alpha & \theta & 1\\ \beta & \varphi & 1\\ \gamma & \Psi & 1\end{array}\right|$
$=S(\text{Sum of all cofactors in}\because \Delta (0)=\left|\begin{array}{ccc}\alpha & \theta & \lambda \\ \beta & \varphi & \mu \\ \gamma & \psi & \nu \end{array}\right|)$
$\therefore \Delta "\left(x\right)=0$ ($\because$ S is constant)
since ${\Delta }^{\prime }\left(x\right)=S$
on integrating
$\Delta \left(x\right)=Sx+c$
$\therefore \Delta \left(0\right)=0+c$
Hence $\Delta \left(x\right)=Sx+\Delta \left(0\right)$