Discussion

The complex number \(z\) satisfying the equation \(\dfrac{Re(z)}{2 + i} + \dfrac{Im(z)}{1 + 2i} = \dfrac{3}{1 - 2i}\), is

Answer: E

\(z = 4 - 5i\).

Let \(z = a + ib\), so \(Re(z) = a\) and \(Im(z) = b\). Rationalise each fraction; all three denominators have modulus squared 5:

\[ \begin{aligned} \frac{a(2 - i)}{5} + \frac{b(1 - 2i)}{5} &= \frac{3(1 + 2i)}{5} \\ (2a + b) - i(a + 2b) &= 3 + 6i \end{aligned} \]

\[ \begin{aligned} 2a + b &= 3 \\ a + 2b &= -6 \\ \Rightarrow a = 4,\ b &= -5 \end{aligned} \]

Check: \(2(4) - 5 = 3\) and \(4 - 10 = -6\). The options \(5 - 4i\) and \(6 - 5i\) fail the first equation, and \(-4 + 5i\) has both signs reversed.

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