Question

If $({\sin }^{-1}x+{\sin }^{-1}w)({\sin }^{-1}y+{\sin }^{-1}z)={\pi }^2$, then $D=\left|\begin{array}{cc}{x}^{N_1}& y^{N_2}\\ z^{N_3}& w^{N_4}\end{array}\right|(N_1,N_2,N_3,N_4ϵN)$

• A. has a maximum value of 2
• B. has a minimum value of 0
• C. 16 different $D$ are possible
• D. has a minimum value of -2

Answer 1

The correct option is B has a minimum value of 0

$({\sin }^{-1}x+{\sin }^{-1}w)({\sin }^{-1}y+{\sin }^{-1}z)={\pi }^2$
Is only true when
${\sin }^{-1}x={\sin }^{-1}w=sy={\sin }^{-1}z=\frac{\pi }{2}$
or ${\sin }^{-1}x={\sin }^{-1}w={\sin }^{-1}y={\sin }^{-1}z=-\frac{\pi }{2}$
Gives $x=y=w=z=1$ or $x=y=w=z=-1$

Hence, $D=\left|\begin{array}{cc}{1}^{N_1}& 1^{N_2}\\ 1^{N_3}& 1^{N_4}\end{array}\right|=\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|=0$