Without expanding evaluate the determinant `|[(a^x+a^(-x))^2,(a^x-a^(-x))^2, 1],[(a^y+a^(-y))^2,(a^y-a^(-y))^2, 1],[(a^z+a^(-z))^2,(a^z-a^(-z))^2, 1]|,` where `a>0` and `x , y , z in R`

Options

A. 1
B. `-1`
C. 0
D. none of these

Answer

C

Explanation

Applying `C_(1) rarr C_(1) - C_(2)`, we get
`Delta = |(4,(a^(x) - a^(-x))^(2),1),(4,(a^(y) -a^(-y))^(2),1),(4,(a^(z) - a^(-z))^(2) ,1)| [ "Using ": (a +b)^(2) - (a-b)^(2) = 4ab]`
`rArr Delta = 4 |(1,(a^(x) -a^(-x))^(2),1),(1,(a^(y) -a^(-y))^(2),1),(1,(a^(z) -a^(-z))^(2) ,1)| " " ["Taking out 4 common from " C_(1)]`
`rArr Delta = 4 xx 0 = 0 " " [ :' C_(1) and C_(3) " are identical"]`