Question

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

• A. $\alpha +{\beta }^{99}=4$
• B. ${\alpha }^3-{\beta }^{17}=26$
• C. $({\alpha }^{2n}-{\beta }^{2n})$ is always an even integer for $n\in N$
• D. a triangle can be constructed having it's sides as $\alpha ,\beta$ and $\alpha -\beta$

Answer 1

The correct option is A $\alpha +{\beta }^{99}=4$

Given $\alpha ,\beta$ are maximum and minimum values of $\Delta$ x

$\left|\begin{array}{ccc}1+{sin}^{2}x& {Cos}^{2}x& sin2x\\ {sin}^{2}x& 1+{cos}^{2}x& sin2x\\ {sin}^{2}x& {cos}^{2}x& 1+sin2x\end{array}\right|$

$R_1\to R_1-R_3$

$R_2\to R_2-R_3$

$\Delta x=\left|\begin{array}{ccc}1& 0& -1\\ 0& 1& -1\\ {sin}^{2}x& {cos}^{2}x& 1+sin2x\end{array}\right|$

$\Delta x=1+\sin 2x+{\cos }^{2}x+{\sin }^{2}x$

$\Delta x=2+\sin 2x$

$\alpha =(\Delta x{)}_{max}=3$

$\beta =(\Delta x{)}_{min}=1$

$\alpha +{\beta }^{99}=3+(1{)}^{99}=49(A)$