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Let S=S1 nn S2 nn S3, where s1={z in ...

Let `S=S_1 nn S_2 nn S_3`, where `s_1={z in C :|z|<4}, S_2={z in C :ln[(z-1+sqrt(3)i)/(1-sqrt(31))]>0} and S_3={z in C : Re z > 0}`

A

`(10pi)/(3)`

B

`(20pi)/(3)`

C

`(16pi)/(3)`

D

`(32pi)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
`S_(1): |z| lt 4, ` z lies inside the circle or radius 4 .
`S_(2): sqrt(3)x + y gt 0` z lies above the line `sqrt(3)x + y = 0`
`S_(3)Re(z) gt 0`, z lies to the right of imaginary axis.

Area of region `S_(1)nnS_(2)nnS_(3)`= shaded area
`=(pi xx4^(2))/(4) + (4^(2)xx pi)/(6) =4^(2) pi{(1)/(4)+ (1)/(6)} = (20pi)/(3)`
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