Question

The value of determinant $\Delta =\left|\begin{array}{ccc}x& -x^2& -x^3\\ -x^2& x^2& -x\\ -x^3& -x& x^3\end{array}\right|,(\Delta \ne 0,x\ne 0)$ is equal to

• A. $-x^3[1+x+x^2+x^3+x^4+x^5]$
• B. $-x^3[1+x^2+x^3+x^4+x^5]$
• C. $-x^3[1+x^3+x^4+x^5]$
• D. $-x^3[1+x+x^2+x^3+x^4]$

Answer 1

The correct option is C $-x^3[1+x^3+x^4+x^5]$

Given : $\Delta =\left|\begin{array}{ccc}x& -x^2& -x^3\\ -x^2& x^2& -x\\ -x^3& -x& x^3\end{array}\right|$
Put $x=1$ in the above determinant,
$\Rightarrow \Delta =\left|\begin{array}{ccc}1& -1& -1\\ -1& 1& -1\\ -1& -1& 1\end{array}\right|$
$=1(1-1)+1(-1-1)-1(1+1)=-4$
Putting $x=1,$ in the given options
$\therefore$ Only one option satisfies above result i.e. $-x^3[1+x^3+x^4+x^5]$