Question

If $a,b,c$ are the consecutive integers then the determinant $\left|\begin{array}{c}-a^{2}abac\\ ab-b^{2}bc\\ acbc-c^2\end{array}\right|$ is divisible by

• A. 36
• B. 144
• C. 48
• D. none of these

Answer 1

Answer: (A), (B), (C)

36
144
48

$\Delta =\left|\begin{array}{c}-a^{2}abac\\ ab-b^{2}bc\\ acbc-c^2\end{array}\right|$
$=abc\left|\begin{array}{c}-aaa\\ b-bb\\ cc-c\end{array}\right|$
$=a^2{b}^2{c}^2\left|\begin{array}{ccc}-1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right|$
$R_1\to R_1+R_2,R_2\to R_2+R_3$
$=a^2{b}^2{c}^2\left|\begin{array}{ccc}0& 0& 2\\ 2& 0& 0\\ 1& 1& -1\end{array}\right|$
$\Delta =4a^2{b}^2{c}^2$
Now, let $a=1$ then $b=2,c=3$
$\Delta =4\times 4\times 9$
Hence, divisible by $36,144,48$