If `(sin^(-1)x+sin^(-1)w)(sin^(-1)y+sin^(-1)z)=pi^2,` then `D=|x^(N_1)y^(N_3)z^(N_3)w^(N_4)|(N_1,N_2,N_3,N_4 in N)` has a maximum value of 2 different D are possible has a minimum value of `-2`

Options

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

A, C, D

Explanation

`(sin^(-1) x + sin^(-1) w) (sin^(-1) y + sin^(-1) z) = pi^(2)`
`:. Sin^(-1) x + sin^(-1) w = sin^(-1) y + sin^(-1) z = pi`
or `sin^(-1) x + sin^(-1) w = sin^(-1) y + sin^(-1) z = -pi`
`:. x =y = z = w = 1 " or " x = y = z = w = -1`
Hence, the maximum value of `|(x^(N_(1)),y^(N_(2))),(z^(N_(3)),w^(N_(4)))|=|(1,-1),(1,1)|=` minimum value `|(-1,1),(1,1)| = -2
Also, there are 16 diefferent determinants as each place value is either 1 or -1