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

We calculate: $|z - 1| = |(1 + \sqrt{3}i) - 1| = |\sqrt{3}i| = \sqrt{3}$ Since \sqrt{3} \approx 1.732, this does not satisfy the inequality.

We calculate: $|z - 1| = |(1 + 3i) - 1| = |3i| = 3$ Since 3 is on the outer boundary, this does not satisfy the inequality < 3.

We calculate: $|z - 1| = |(2 + i) - 1| = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}$ Since \sqrt{2} \approx 1.414, this does not satisfy the inequality.

We calculate: $|z - 1| = |(3 - i) - 1| = |2 - i| = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$ Since 2 < \sqrt{5} < 3, this satisfies the inequality.