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Question

Let z be a complex number such that z+|z|=3+i. Then which of the following is/are true?

A
sin3θ+cos3θ=91125, where θ=argz
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B
|z|=35
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C
Re(logz)=log(53)
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D
z¯z=2i
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Solution

The correct options are
A sin3θ+cos3θ=91125, where θ=argz
C Re(logz)=log(53)
Let z=x+iy
Given z+|z|=3+i
x+iy+x2+y2=3+i
x+x2+y2=3 and y=1
x+x2+1=3
x2+1=(3x)2x=43
z=43+i
|z|=169+1=53

tanθ=14/3=34 (x>0,y>0,z lies in Q1).
sinθ=35, cosθ=45sin3θ+cos3θ=91125

We know logz=log|z|+iθ , where θ=argz
Re(logz)=log|z|=log(53)
z¯z=(43+i)(43i)=2i

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