1. **State the problem:** We are given three complex numbers $z_1 = -1 + 2i$, $z_2 = 2 + 3i$, and $z_3 = 4 - i$. We need to plot $z_2$ and $z_3$ on the Argand diagram and find the modulus of $z_1$, denoted $|z_1|$. 2. **Plotting on the Argand diagram:** - The Argand diagram is a Cartesian plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. - $z_2 = 2 + 3i$ corresponds to the point $(2, 3)$. - $z_3 = 4 - i$ corresponds to the point $(4, -1)$. - These points should be plotted and labeled accordingly. 3. **Formula for modulus of a complex number:** The modulus of a complex number $z = a + bi$ is given by: $$|z| = \sqrt{a^2 + b^2}$$ 4. **Calculate the modulus of $z_1$:** Given $z_1 = -1 + 2i$, here $a = -1$ and $b = 2$. $$|z_1| = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$$ 5. **Final answer:** The modulus of $z_1$ is $\boxed{\sqrt{5}}$. This is the length of the vector from the origin to the point $(-1, 2)$ on the Argand diagram.