Question

Let a, b, c be such that b$(a+c)\ne 0$. If $\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ (-1{)}^{n+2}a& (-1{)}^{n+1}b& (-1{)}^{n}c\end{array}\right|=0$
Then the value of n is?

• A. $0$
• B. Any even integer
• C. Any odd integer
• D. Any integer

Answer 1

Answer: (C)

Any odd integer

$\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}(-1{)}^{n+2}a& a+1& a-1\\ (-1{)}^{n+1}b& b+1& b-1\\ (-1{)}^{n}c& c-1& c+1\end{array}\right|$
$=\left|\begin{array}{ccc}a+(-1{)}^{n+2}a& a+1& a-1\\ -b+(-1{)}^{n+1}b& b+1& b-1\\ c+(-1{)}^{n}c& c-1& c+1\end{array}\right|$
$=0$ if n is an odd integer.