If a, b, and c are in A.P.,then the value of | vedclass

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(C) Let $\Delta = \left|\begin{array}{ccc}x+2 & x+3 & x+a \ x+4 & x+5 & x+b \ x+6 & x+7 & x+c\end{array}ight|$.Applying row operations $R_{1} ig

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$0$

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(C) Let $\Delta = \left|\begin{array}{ccc}x+2 & x+3 & x+a \ x+4 & x+5 & x+b \ x+6 & x+7 & x+c\end{array}ight|$.Applying row operations $R_{1} ightarrow R_{1}-R_{2}$ and $R_{2} ightarrow R_{2}-R_{3}$:$\Delta = \left|\begin{array}{ccc}(x+2)-(x+4) & (x+3)-(x+5) & (x+a)-(x+b) \ (x+4)-(x+6) & (x+5)-(x+7) & (x+b)-(x+c) \ x+6 & x+7 & x+c\end{array}ight|$$\Delta = \left|\begin{array}{ccc}-2 & -2 & a-b \ -2 & -2 & b-c \ x+6 & x+7 & x+c\end{array}ight|$.Since $a, b, c$ are in $A$.$P$.,we have $b-a = c-b$,which implies $a-b = b-c$.Thus,the first two rows $R_{1}$ and $R_{2}$ are identical.Since two rows are identical,the value of the determinant is $0$.

If $a, b,$ and $c$ are in $A$.$P$.,then the value of $\left|\begin{array}{lll}x+2 & x+3 & x+a \\ x+4 & x+5 & x+b \\ x+6 & x+7 & x+c\end{array}\right|$ is

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