A person walks `2sqrt(2)` units away from origin in south west direction `(S45^(@)W)` to reach `A`, then walks `sqrt(2)` units in south east direction `(S45^(@)E)` to reach `B`. From `B` he travel is `4` units horizontally towards east to reach `C`. Then he travels along a circular path with centre at origin through an angle of `2pi//3` in anti-clockwise direction to reach his destination `D`.
Let the complex number `Z` represents `C` in argand plane. then `arg(Z)=`

Options

A. `-(pi)/(6)`
B. `(pi)/(4)`
C. `-(pi)/(4)`
D. `(pi)/(3)`

Answer

C

Explanation

`(c )`
`Z_(1)=2sqrt(2)e^(-i3pi//4)=-2-2i`
`(Z_(2)-(-2-2i))/(O-(-2-2i))=(1)/(2)xxe^(-ipi//2)` (rotation at `A`)
`impliesZ_(2)=-(1+i)(2-i)=-1-3i`
`:. Z_(3)=3-3i`
`(Z_(4)-0)/(3-3i-0)=e^(i2pi//3)impliesZ_(4)=3(1-i)omega`(rotation at `O`)