Let z₁ = 3 + 4 i and z₂ = 1 - 2 i. Give angles in degrees, and use the principal argument, which…

(a) $\dfrac{z_1}{z_2} = \dfrac{(3 + 4 i)\overline{1 - 2 i}}{|z_2|^2}$: the top is $-5 + 10 i$ and the bottom is $5$, so $\dfrac{z_1}{z_2} = -1 + 2 i$: $a = -1$ and $b = 2$.

(b) $|z_1| = 5$ and $|z_2| = \sqrt{5}$, so $|z_1 z_2| = 5 \sqrt{5} = 11.18$.

(c) $-1 + 2 i$ lies in the second quadrant. Its angle with the negative real axis is $\tan^{-1}\dfrac{2}{|-1|} = 63.43^\circ$, so the argument is $180^\circ - 63.43^\circ = 116.57^\circ$.

(d) $\overline{z_1} = 3 - 4 i$ and $2z_2 = 2 - 4 i$, so $\overline{z_1} - 2z_2 = 1$ and its modulus is $1$.