Question

If $\left|\begin{array}{ccc}2x^2+x& 3x-5& 4x+3\\ x+5& -x^2& 6x-6\\ 2x-3& 5x+6& -3x\end{array}\right|=a{x}^5+b{x}^4+c{x}^3+d{x}^2+ex+f\forall x\in R,$ then the value of $f$ is

• A. $0$
• B. $-8$
• C. $14$
• D. $5$

Answer 1

The correct option is A $0$

As, the given expression is true for all $x.$
Put $x=0$ on both sides, we get
$f=\left|\begin{array}{ccc}0& -5& 3\\ 5& 0& -6\\ -3& 6& 0\end{array}\right|$
$\Rightarrow f$ is a skew-symmetric determinant of odd order.
$\Rightarrow f=0$
$(\because$ determinant of odd order skew-symmetric matrix is $0)$