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

Question

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, $ x-(a+b+c) $, $ 9 x^{2}+a+b+c $, $ 00 $, $ a+b+c $

MARKS App · Accepted Answer

Given $ \left|\begin{array}{ccc}x+2 & x+3 & x+a \ x+4 & x+5 & x+b \ x+6 & x+7 & x+6\end{array}\right| $ $ R_{1} \rightarrow R_{1}-R_{2} ; R_{2} \rightarrow R_{2}-R_{3} $ $ \Delta=\left|\begin{array}{ccc}-2 & -2 & a-b \ -2 & -2 & b-c \ x+6 & x+7 & x+c\end{array}\right| $ Since, a, b, c are in A.P. then $ a-b=b-c $ So, $ R_{1} \equiv R_{2} $ Therefore, $ \Delta=0 $

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