Question

The value of determinant $\Delta =\left|\begin{array}{ccc}{x}^2+4& 2x& 1-x\\ 5x+1& 2x& 1\\ x+2& x& 2\end{array}\right|,(x\ne 0)$ is equal to

• A. $12x^2-9x$
• B. $12x^2+12x$
• C. $-12x^2+9x$
• D. $-12x^2-9x$

Answer 1

The correct option is C $-12x^2+9x$

Given : $\Delta =\left|\begin{array}{ccc}{x}^2+4& 2x& 1-x\\ 5x+1& 2x& 1\\ x+2& x& 2\end{array}\right|$
Put $x=1$ in the above determinant,
$\Delta =\left|\begin{array}{ccc}5& 2& 0\\ 6& 2& 1\\ 3& 1& 2\end{array}\right|$
Expanding along $C_3,$ we get
$\Rightarrow \Delta =0+-1(5-6)+2(10-12)=-3$

Putting $x=1$ in given options,
Only one option satisfies i.e. $-12x^2+9x.$