Which complex number lies in the region $2 < |z - 1| < 3$? A - HSC - SSCE Mathematics Extension 2 - Question 3 - 2017 - Paper 1

Calculate the modulus:
$|z - 1| = |(1 + 3i) - 1| = |3i| = 3$
Since $3 < 3$ does not hold, this option is also not valid.

Calculate the modulus:
$|z - 1| = |(2 + i) - 1| = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}$
Since $\sqrt{2} \approx 1.414$, it does not satisfy the region because $2 < \sqrt{2}$.

Calculate the modulus:
$|z - 1| = |(3 - i) - 1| = |2 - i| = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$
Since $2 < \sqrt{5} < 3$ holds true, this complex number lies in the required region.