Question

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

• 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

Answer: (A), (C), (D)

The correct options are
A has a maximum value of $2$
C $16$ different $D$ are possible
D has a minimum value of $-2$

$({\sin }^{-1}x+{\sin }^{-1}w)({\sin }^{-1}y+{\sin }^{-1}z)={\pi }^2$
$\therefore {\sin }^{-1}x+{\sin }^{-1}w={\sin }^{-1}y+{\sin }^{-1}z=\pi$
${\sin }^{-1}x+{\sin }^{-1}w={\sin }^{-1}y+{\sin }^{-1}z=-\pi$
$\therefore x=y=z=w=1$ or $x=y=z=w=-1$
Hence, the maximum value of
$\left|\begin{array}{cc}{x}^{N_1}& y^{N_2}\\ z^{N_3}& w^{N_4}\end{array}\right|=\left|\begin{array}{cc}1& -1\\ 1& 1\end{array}\right|=2$ and minimum value $\left|\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right|=-2.$
Also, there are $16$ different determinants as each palce value is either $1$ or $-1.$