Question

$\text{If}f\left(x\right)=\left|\begin{array}{ccc}1+x^n& (1-x{)}^n& 2+x^n\\ (2+x{)}^n& (2+x{)}^n& 1\\ (3-x{)}^n& 1& 3+x\end{array}\right|$,
$\text{then the constant term in the expansion is}$

• A. $(3^n-1)(1-2^{n+1})$
• B. $(3^n-1)(2^{n+1}-1)$
• C. $3^n(1-2^{n+1}-2^{n+1}-1)$
• D. $2^{n+1}(1-3^n)+(3^n+1)$

Answer 1

The correct option is A $(3^n-1)(1-2^{n+1})$

As $f\left(x\right)$ is a polynomial in $x$ so,
Constant term is obtained by putting the value of $x=0$
$f\left(0\right)=\left|\begin{array}{ccc}1& 1& 2\\ 2^n& 2^n& 1\\ 3^n& 1& 3\end{array}\right|$

$C_1\to C_1-C_2$
$=\left|\begin{array}{ccc}0& 1& 2\\ 0& 2^n& 1\\ 3^n-1& 1& 3\end{array}\right|$
$=(3^n-1)(1-2^{n+1})$