Question

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

• A. ${\alpha }^2+{\beta }^{10}=10$
• B. ${\alpha }^3-{\beta }^{99}=26$
• C. $2{\alpha }^2-18{\beta }^4=0$
• D. All the above

Answer 1

Answer: (D)

All the above

Let $A=\left|\begin{array}{ccc}1+co{s}^{2}x& si{n}^{2}x& \cos 2x\\ co{s}^{2}x& 1+si{n}^{2}x& \cos 2x\\ co{s}^{2}x& si{n}^{2}x& 1+cos2x\end{array}\right|$
So, By row transformation in A,
$r_2\to r_2-r_1$ and $r_3\to r_3-r_1$
$A=\left|\begin{array}{ccc}1+co{s}^{2}x& si{n}^{2}x& cos2x\\ -1& 1& 0\\ -1& 0& 1\end{array}\right|$
So, $A=cos2x\left(1\right)+1(1+co{s}^{2}x+si{n}^{2}x)$
$A=2+cos2x$
So, max value of $A=\alpha =3$
and min value of $A=\beta =1$
So, ${\alpha }^2+{\beta }^{10}=9+1=10$
${\alpha }^3-{\beta }^{99}=27-1=26$
$2{\alpha }^2-18{\beta }^4=18-18=0$
So, all are true