Question

If $\Delta _{1}=\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{vmatrix},\Delta _{2}=\begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$ then

• A. $\Delta _{1}+\Delta _{2}=0$
• B. $\Delta _{1}=\Delta _{2}$
• C. $\Delta _{1}+2\Delta _{2}=0$
• D. $\Delta _{1}-2\Delta _{2}=0$

Answer 1

Answer: (B)

$\Delta _{2}=\begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}\frac{a b c}{a b c}$
$=\frac{1}{abc}\begin{vmatrix} a & a^{2} & abc \\ b & b^{2} & abc \\ c & c^{2} & abc \end{vmatrix}=\begin{vmatrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{vmatrix}=\Delta _{1}$