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Question

Mathematics Question on complex numbers

If the conjugate of a complex number zz is 1i−1\frac{1}{i - 1} , then zz is

A

1i−1\frac{1}{i - 1}

B

1i+1\frac{1}{i + 1}

C

−1i−1\frac{-1}{i - 1}

D

−1i+1\frac{ - 1}{i + 1}

Answer

−1i+1\frac{ - 1}{i + 1}

Explanation

Solution

z=1i−1×i+1i+1z=\frac{1}{i-1} \times \frac{i+1}{i+1}
⇒ z=i+1i2−12\Rightarrow \, z=\frac{i+1}{i^{2}-1^{2}}
z=−12×(i+1)z=-\frac{1}{2} \times(i+1)
⇒ zˉ=−12(1−i)×(1+i)(1+i)\Rightarrow \, \bar{z}=-\frac{1}{2}(1-i) \times \frac{(1+i)}{(1+i)}
=−12(1+1)(1+i)=−1(1+i)=-\frac{1}{2} \frac{(1+1)}{(1+i)}=-\frac{1}{(1+i)}