If Δ1= a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2+c1b22+c2b2b3+c3c3c1c3c2c32 and Δ2=a1b1c1a2b2c2a3b3c3 then the value of Δ1Δ2 is

Δ1=a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2+c1b22+c2b2b3+c3c3c1c3c2c32 =c3a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2+c1b22+c2b2b3+c3c1c2c3 R2→R2−R1 =c3a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2b22b2b3c1c2c3 =b2c3a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2b3c1c2c3 R1→R1−R2−R3 =b2c3a12a1a2a1a3b1b2b3c1c2c3=a1b2c3a1a2a3b1b2b3c1c2c3=a1b2c3a1b1c1a2b2c2a3b3c3=a1b2c3Δ2⇒Δ1Δ2=a1b2c3

If Δ1= a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2+c1b22+c2b2b3+c3c3c1c3c2c32 and Δ2=a1b1c1a2b2c2a3b3c3 then the value of Δ1Δ2 is

Δ1=a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2+c1b22+c2b2b3+c3c3c1c3c2c32 =c3a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2+c1b22+c2b2b3+c3c1c2c3 R2→R2−R1 =c3a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2b22b2b3c1c2c3 =b2c3a12+b1+c1a1a2+b2+c2a1a3+b3+c3b1b2b3c1c2c3 R1→R1−R2−R3 =b2c3a12a1a2a1a3b1b2b3c1c2c3=a1b2c3a1a2a3b1b2b3c1c2c3=a1b2c3a1b1c1a2b2c2a3b3c3=a1b2c3Δ2⇒Δ1Δ2=a1b2c3

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If $Δ_{1} =$ $|\begin{matrix} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{3} c_{1} & c_{3} c_{2} & c_{3}^{2} \end{matrix}|$ and $Δ_{2} = |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|$ then the value of $\frac{Δ_{1}}{Δ_{2}} i s$

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$a_{3} b_{2} c_{1}$

$a_{1} b_{2} c_{3}$

$a_{2} b_{1} c_{3}$

$a_{1} b_{1} c_{1}$

answer is C.

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Detailed Solution

$Δ_{1} = |\begin{matrix} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{3} c_{1} & c_{3} c_{2} & c_{3}^{2} \end{matrix}| = c_{3} |\begin{matrix} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}| R_{2} \to R_{2} - R_{1} = c_{3} |\begin{matrix} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} b_{2} & b_{2}^{2} & b_{2} b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}| = b_{2} c_{3} |\begin{matrix} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}| R_{1} \to R_{1} - R_{2} - R_{3} = b_{2} c_{3} |\begin{matrix} a_{1}^{2} & a_{1} a_{2} & a_{1} a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}|$

$= a_{1} b_{2} c_{3} |\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}| = a_{1} b_{2} c_{3} |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = a_{1} b_{2} c_{3} Δ_{2} \Rightarrow \frac{Δ_{1}}{Δ_{2}} = a_{1} b_{2} c_{3}$