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1)

Let  S:S1S2S3 , where   

S1={zC:|z|<4},S2 ={zC:In[z1+3i13i]>0}

and   S3={zC:Rez>0}

Area of S is equal to 


A) 10π3

B) 20π3

C) 16π3

D) 32π3

Answer:

Option B

Explanation:

 Here , S=S1S2S3

 S1={zC:|z|<4}

    S1 :x2+y2   < 4   ...........(i)

 and   S2 ={zC:In[z1+3i13i]>0}

 where   z1+i31i3=(x1)+i(y+3)1i3

 =  (x1)(1+i3)+i(y+3)(1+i3)1+3

 =(x1)+i3(x1)+i(y+3)3(y+3)4

=(x13y3)4+i(3x+y)4

       s2:3x+y>0  .......(ii)

  S3={zC:Rez>0}

  s3:x>0     ......(iii)

  s3:x>0     ......(iii)

Since , 

                S=S1S2S3

 1352021274_m9.JPG

 

 Clearly , the shaded region represents the area of sector

      S=12r2θ=12×42×5π6=20π3