If `z` is a complex number lying in the fourth quadrant of Argand plane and `|[k z//(k+1)]+2i|>sqrt(2)` for all real value of`k(k!=-1),` then range of `"a r g"(z)` is `(pi/8,0)` b. `(pi/6,0)` c.`(pi/4,0)` d. none of these

Options

A. `(-(pi)/(8),0)`
B. `(-(pi)/(6),0)`
C. `(-(pi)/(4),0)`
D. None of these

Answer

C

Explanation

`kz//(K =1)` represents any point lying on the line joining origin and z.

Given, `|(kz)/(k+1)+2i|gtsqrt(2)`
Hence, `kz//(k+1)` sholud lie outside the circle `|z + 2i| gt sqrt(2)`. So, z shloud lie in the shaded region.
`therefore - (pi)/(4) lt arg(z) lt0`