Question

If $f\left(x\right)=\left|\begin{array}{ccc}1& 2x& 3x^2\\ 3& a& 27\\ 1& 3& 9\end{array}\right|$ and ${\int }_0^{3}f\left(x\right)dx=0$, then a is equal to

• A. 3
• B. 6
• C. 9
• D. any real number

Answer 1

The correct option is D any real number

${\int }_0^{3}f\left(x\right)dx=\left|\begin{array}{ccc}{\int }_0^{3}dx& {\int }_0^{3}2xdx& {\int }_0^{3}3x^{2}dx\\ 3& a& 27\\ 1& 3& 9\end{array}\right|$
$=\left|\begin{array}{ccc}3& 9& 27\\ 3& a& 27\\ 1& 3& 9\end{array}\right|$
${\int }_0^{3}f\left(x\right)dx=0$
$\therefore a\in$ any real number.