Question

If the maximum and minimum values of the determinant
$\left|\begin{array}{ccc}1+{\cos }^{2}x& {\sin }^{2}x& \cos 2x\\ {\cos }^{2}x& 1+{\sin }^{2}x& \cos 2x\\ {\cos }^{2}x& {\sin }^{2}x& 1+\cos 2x\end{array}\right|$
are $\alpha$ and $\beta$ respectively, then which of the following is correct?

• A. ${\alpha }^2+{\beta }^{101}=10$
• B. ${\alpha }^3-{\beta }^{99}=26$
• C. $2{\alpha }^2-18{\beta }^{11}=0$
• D. All the above

Answer 1

The correct option is D All the above

$\Delta =\left|\begin{array}{ccc}1+{\cos }^{2}x& {\sin }^{2}x& \cos 2x\\ {\cos }^{2}x& 1+{\sin }^{2}x& \cos 2x\\ {\cos }^{2}x& {\sin }^{2}x& 1+\cos 2x\end{array}\right|$

Apply $R_1\to R_1-R_2,R_2\to R_2-R_3$
$\Delta =\left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ {\cos }^{2}x& {\sin }^{2}x& 1+\cos 2x\end{array}\right|$

$=1(1+\cos 2x+{\sin }^{2}x)+1(0+{\cos }^{2}x)$

$\Rightarrow \Delta =2+\cos 2x-1\le \cos 2x\le 1\Rightarrow 1\le 2+\cos 2x\le 3\therefore \alpha =3,\beta =1$