Question

When the determinant $\left|\begin{array}{ccc}\cos 2x& {\sin }^{2}x& \cos 4x\\ {\sin }^{2}x& \cos 2x& {\cos }^{2}x\\ \cos 4x& {\cos }^{2}x& \cos 2x\end{array}\right|$ is expanded in powers of $\sin x$, then the constant term in the expression is

• A. $1$
• B. $0$
• C. $-1$
• D. $2$

Answer 1

The correct option is C $-1$

$f\left(x\right)=\left|\begin{array}{ccc}1-2{\sin }^{2}x& {\sin }^{2}x& 1-8{\sin }^{2}x(1-{\sin }^{2}x)\\ {\sin }^{2}x& 1-2{\sin }^{2}x& 1-{\sin }^{2}x\\ 1-8{\sin }^{2}x(1-{\sin }^{2}x)& 1-{\sin }^{2}x& 1-2{\sin }^{2}x\end{array}\right|$
The required constant term is
$f\left(0\right)=\left|\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 1& 1& 1\end{array}\right|=-1$