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

We will check which of the given complex numbers satisfies the distance conditions:

1. Option A: $1 + \sqrt{3}i$

   • Calculate $|1 + \sqrt{3}i - 1| = |\sqrt{3}i| = \sqrt{3}$ (not in the range)

2. Option B: $1 + 3i$

   • Calculate $|1 + 3i - 1| = |3i| = 3$ (on the outer boundary)

3. Option C: $2 + i$

   • Calculate $|2 + i - 1| = |1 + i| = \sqrt{2}$ (not in the range)

4. Option D: $3 - i$

   • Calculate $|3 - i - 1| = |2 - i| = \sqrt{4 + 1} = \sqrt{5}$ (in the range)

Thus, only Option D has a distance that lies within $2 < |z - 1| < 3$.

From the examination of the complex numbers, the only option that satisfies the condition $2 < |z - 1| < 3$ is D: $3 - i$. Therefore, the correct answer is Option D.