• # question_answer If $\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ {{(a+\lambda )}^{2}} & {{(b+\lambda )}^{2}} & {{(a+\lambda )}^{2}} \\ {{(a-\lambda )}^{2}} & {{(b-\lambda )}^{2}} & {{(c+\lambda )}^{2}} \\ \end{matrix} \right|=$ $k\lambda \left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix} \right|,\lambda \ne 0$then k is equal to:   [JEE Main Online Paper ( Held On 12 Apirl  2014 ) A) $4\lambda abc$                             B) $-4\lambda abc$ C) $4{{\lambda }^{2}}$                     D) $-4{{\lambda }^{2}}$

Let $\Delta =\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ {{(a+\lambda )}^{2}} & {{(b+\lambda )}^{2}} & {{(c+\lambda )}^{2}} \\ {{(a-\lambda )}^{2}} & {{(b-\lambda )}^{2}} & {{(c-\lambda )}^{2}} \\ \end{matrix} \right|$ Apply ${{R}_{2}}\to {{R}_{2}}-{{R}_{3}}$ $\Delta =\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ {{(a+\lambda )}^{2}}-{{(a-\lambda )}^{2}} & {{(b+\lambda )}^{2}}-{{(b-\lambda )}^{2}} & {{(c+\lambda )}^{2}}-{{(c-\lambda )}^{2}} \\ {{(a-\lambda )}^{2}} & {{(b-\lambda )}^{2}} & {{(a-\lambda )}^{2}} \\ \end{matrix} \right|$$=\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ 4a\lambda & 4b\lambda & 4c\lambda \\ {{(a-\lambda )}^{2}} & {{(b-\lambda )}^{2}} & {{(a-\lambda )}^{2}} \\ \end{matrix} \right|$ $(\because {{(x+y)}^{2}}-{{(x-y)}^{2}}=4xy)$ Taking out 4 common from ${{R}_{2}}$ $=4\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a\lambda & b\lambda & c\lambda \\ {{a}^{2}}+{{\lambda }^{2}}-2a\lambda & {{b}^{2}}+{{\lambda }^{2}}-2b\lambda & {{c}^{2}}+{{\lambda }^{2}}-2c\lambda \\ \end{matrix} \right|$Apply ${{R}_{3}}\to [{{R}_{3}}-({{R}_{1}}-2{{R}_{2}})]$$=4\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a\lambda & b\lambda & c\lambda \\ {{\lambda }^{2}} & {{\lambda }^{2}} & {{\lambda }^{2}} \\ \end{matrix} \right|$ Taking out $\lambda$ common from ${{R}_{2}}$and ${{\lambda }^{2}}$from ${{R}_{3}}.$ $=4\lambda ({{\lambda }^{2}})\left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix} \right|$$=k\lambda \left| \begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix} \right|$ $\Rightarrow$$k=4{{\lambda }^{2}}$