To the equation `2^(2pi//cos^(-1)x) - (a + (1)/(2)) 2^(pi/cos^(-1)x) -a^(2) = 0` has only one real root, then

Options

A. `1 le a le 3`
B. `a ge 1`
C. `a le -3`
D. `a ge 3`

Explanation

`1 le (pi)/(cos^(-1) x) lt oo`
`rArr 2 le 2^((pi)/(cos^(-1)x)) lt oo`
Hence, 2 should lie between or on the roots of
`t^(2) - (a + (1)/(2)) t - a^(2) = 0, " where " t = 2^(pi//cos^(-1)x)`
`rArr f(2) le 0 rArr a^(2) + 2a -3 ge 0`
`rArr a in (-oo, -3] uu [1, oo)`