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यदि $a, b, c$ समांतर श्रेढ़ी में हों तो सारणिक
$\left|\begin{array}{lll}x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c\end{array}\right|$ का मान होगा|:
$1$
$x$
$2x$
$0$
Solution
$\Delta=\left|\begin{array}{lll}x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c\end{array}\right|$
$=\left|\begin{array}{llc}x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+(a+c) \\ x+4 & x+5 & x+2 c\end{array}\right|$ $(2 b=a+c \text { as } a, b, \text { and } c \text { are in } A P)$
Applying $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{3} \rightarrow R_{3}-R_{2},$ we have:
$\Delta=\left|\begin{array}{ccc}-1 & -1 & a-c \\ x+3 & x+4 & x+(a+c) \\ 1 & 1 & c-a\end{array}\right|$
Applying $R_{1} \rightarrow R_{1}+R_{3},$ we have:
$\Delta=\left|\begin{array}{ccc}0 & 0 & 0 \\ x+3 & x+4 & x+a+c \\ 1 & 1 & c-a\end{array}\right|$
Here, all the elements of the first row ( $\mathrm{R}_{1}$ ) are zero.
Hence, we have $\Delta=0$ The correct
answer is $D$.
