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Question

Let zC with Im(z)=10 and it satisfies
2zn2z+n=2i1 for some natural number n. Then

A
n=20 and Re(z)=10
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B
n=20 and Re(z)=10
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C
n=40 and Re(z)=10
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D
n=40 and Re(z)=10
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Solution

The correct option is D n=40 and Re(z)=10
Let z=x+10i
2(x+10i)n2(x+10i)+n=2i1(2xn)+20i=(2i1)[(2x+n)+20i]
On comparing real and imaginary part, we have -
2xn=2(20)(2x+n) and 20=2(2x+n)202xn=402xn and 20=4x+2n204x=40 and 4x+2n=40x=10 and 40+2n=40n=40n=40 and Re(z)=10

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