Question

If $\Delta_{1} = \left| \begin{matrix} a_{1}^{2} + b_{1} + c_{1} & a_{1}a_{2} + b_{2} + c_{2} & a_{1}a_{3} + b_{3} + c_{3} \\ b_{1}b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2}b_{3} + c_{3} \\ c_{3}c_{1} & c_{3}c_{2} & c_{3}^{2} \end{matrix} \right|$&

$\Delta_{2} = \left| \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} \right|$, then $\frac{\Delta_{1}}{\Delta_{2}}$ equals-

• A. a₁b₂c₃
• B. a₁a₂a₃
• C. a₃b₂c₁
• D. a₁b₁c₁ + a₂b₂c₂ + a₃b₃c₃

Answer 1

Answer: (A)

a₁b₂c₃

Answer 2

Taking C₃ common from R₃ & applying

R₂ → R₂ – R₃, R₁ → R₁ –R₃

a_{1}^{2} + b_{1} & a_{1}a_{2} + b_{2} & a_{1}a_{3} + b_{3} \\ b_{1}b_{2} & b_{2}^{2} & b_{2}b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} \right|$$ Taking b₂ common from R₂, R₁ → R₁ – R₂ $$\Delta_{1} = a_{1}b_{2}c_{3}\left| \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} \right|$$ $$\frac{\Delta_{1}}{\Delta_{2}} = a_{1}b_{2}c_{3}$$