Question

If $\left|\begin{array}{ccc}x-1& 2x-1& x^2+1\\ -x& 2x-2& -3x\\ -2x^2& 4x-1& 3x-3\end{array}\right|=\sum _{i=0}^n{a}_i{x}^i\forall x\in R,n\le 10,$ then which of the following is correct?

• A. $\sum _{i=0}^n{a}_i=1$
• B. $\sum _{i=0}^n{a}_i=-1$
• C. $\sum _{i=0}^n{a}_i=0$
• D. $\sum _{i=0}^n{a}_i=10$

Answer 1

The correct option is C $\sum _{i=0}^n{a}_i=0$

Given : $\left|\begin{array}{ccc}x-1& 2x-1& x^2+1\\ -x& 2x-2& -3x\\ -2x^2& 4x-1& 3x-3\end{array}\right|=\sum _{i=0}^n{a}_i{x}^i$
As, above expression is true for all $x.$
Put $x=1$ on both sides, we get
$\sum _{i=0}^n{a}_i=\left|\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right|$
$R.H.S.$ is a skew-symmetric determinant of odd order.
$\Rightarrow \sum _{i=0}^n{a}_i=0$
$(\because$ determinant of odd order skew-symmetric matrix is $0)$