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If $a,b,c$ are in $A.P$., then the value of $\left| {\,\begin{array}{*{20}{c}}{x + 2}&{x + 3}&{x + a}\\{x + 4}&{x + 5}&{x + b}\\{x + 6}&{x + 7}&{x + c}\end{array}\,} \right|$ is
$x - (a + b + c)$
$9{x^2} + a + b + c$
$a + b + c$
$0$
Solution
(d) Let $A = \left| {\,\begin{array}{*{20}{c}}{x + 2}&{x + 3}&{x + a}\\{x + 4}&{x + 5}&{x + b}\\{x + 6}&{x + 7}&{x + c}\end{array}\,} \right|$
Applying ${C_2} \to {C_2} – {C_1},$ we get,
==> $A = \,\left| {\,\begin{array}{*{20}{c}}{x + 2}&1&{x + a}\\{x + 4}&1&{x + b}\\{x + 6}&1&{x + c}\end{array}\,} \right|$
= ${R_Z}→ {R_2} – R_1$ and ${R_3} → {R_3} – {R_1} $
==> $A = \left| {\,\begin{array}{*{20}{c}}{x + 2}&1&{x + a}\\2&0&{b – a}\\4&0&{c – a}\end{array}\,} \right|\, = \, – 1\,(2c – 2a – 4b + 4a)$
= $2(2b – c – a)$
$\because$ $a, b, c $ are in $ A = 0.$