Question:
The value of $\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|$ is
Correct Option: 1
Solution:
$\mathrm{C}_{1} \rightarrow \mathrm{c}_{1}-\mathrm{c}_{2}, \mathrm{c}_{2} \rightarrow \mathrm{c}_{2}-\mathrm{c}_{3}$
$=\left|\begin{array}{ccc}(a+2) a & a+1 & 1 \\ (a+3)(a+1) & a+2 & 1 \\ (a+4)(a+2) & a+3 & 1\end{array}\right|$
$R_{2} \rightarrow R_{2}-R_{1} \quad \& R_{3} \rightarrow R_{3}-R_{1}$
$=\left|\begin{array}{ccc}a^{2}+2 a & a+1 & 1 \\ 2 a+3 & 1 & 0 \\ 4 a+8 & 2 & 0\end{array}\right|$
$=6-8=-2$
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