Question

If the determinant $\left|\begin{array}{ccc}cos2x& si{n}^{2}x& cos4x\\ si{n}^{2}x& cos2x& co{s}^{2}x\\ cos4x& co{s}^{2}x& cos2x\end{array}\right|$ is expanded in powers of $\sin x$ then the constant term in the expansion is

• A. $1$
• B. $2$
• C. $-1$
• D. none of these

Answer 1

The correct option is C $-1$

Let $\left|\begin{array}{ccc}cos2x& si{n}^{2}x& cos4x\\ si{n}^{2}x& cos2x& co{s}^{2}x\\ cos4x& co{s}^{2}x& cos2x\end{array}\right|=A+B\sin x+C{\sin }^{2}x+.....$
Substitute $x=0$, we get
$\left|\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 1& 1& 1\end{array}\right|=A$
$\Rightarrow A=-1$