Let z₁ = 3 - 4i and z₂ = 1 + 2i, where i² = -1 - Leaving Cert Mathematics - Question 1 - 2013

Answer: Yes.

Reason: The distance from the origin to z₁ is greater than the distance from the origin to z₂.

To find |z₁| and |z₂|:

1. |z₁| = |3 - 4i| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.
2. |z₂| = |1 + 2i| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}.

By comparison, we have 5 > \sqrt{5}, thus |z₁| > |z₂|.

To find \frac{1}{z₂} where z₂ = 1 + 2i:

\frac{1}{z₂} = \frac{1}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{1 - 2i}{1^2 + (2)^2} = \frac{1 - 2i}{1 + 4} = \frac{1 - 2i}{5} = \frac{1}{5} - \frac{2}{5}i.

Thus, \frac{1}{z₂} = \frac{1}{5} - \frac{2}{5}i.