Question

Let $Z$ be a complex number and $\overline{Z}$ denotes the conjugate of $Z$. If $2Z-3\overline{Z}=\frac{-27+23i}{1+i}$, then which of the following is/are correct?

• A. $Re\left(Z\right)=2$
• B. $Im\left(Z\right)=5$
• C. $Re\left(Z\right)=5$
• D. $Im\left(Z\right)=2$

Answer 1

Answer: (A), (B)

The correct options are
A $Re\left(Z\right)=2$
B $Im\left(Z\right)=5$

Given equation : $2Z-3\overline{Z}=\frac{-27+23i}{1+i}$
Assuming $Z=x+iy\Rightarrow \overline{Z}=x-iy$
Now,
$2Z-3\overline{Z}=3(Z-\overline{Z})-Z=3\left(2iy\right)-(x+iy)=-x+5iy$
And
$\frac{-27+23i}{1+i}=\frac{-27+23i}{1+i}\times \frac{1-i}{1-i}=\frac{-27+27i+23i+23}{1+1}=-2+25i$

Therefore,
$-x+5iy=-2+25i\Rightarrow x=2,y=5$