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3 problems match “imaginary unit”. Clear

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1. Dividing complex numbers ●○○○○ 10 ptsCompute $\dfrac{20+10i}{1-3i}$. Write the result in the form $a+bi$ and find $a$ and $b$.
2. Maximum modulus on a locus ●●○○○ 20 ptsGiven a complex number $z$ such that $\frac{z-\mathrm{i}}{z-1}$ is a purely imaginary number, where $\mathrm{i}$ is the imaginary unit, find the maximum value of $|z|$. Round to two decimal places.
3. Complex sequence with conjugate recurrence ●●●○○ 50 ptsThe complex number sequence $\{z_n\}$ satisfies $z_1=1$ and $z_{n+1}=\overline{z_n}+1+ni$ for $n=1,2,3,\dots$, where $i$ is the imaginary unit and $\overline{z_n}$ is the conjugate of $z_n$. Find the real part and the imaginary part of $z_{2015}$.