Leaving Cert Mathematics Complex Numbers: The complex number $z_1$ is show

Question

The complex number $z_1$ is shown on the Argand diagram below.

(a) Using the Argand diagram:

(i) write down the values of $z_2$ and $ar{z_1}$, where $ar{z_1}$ is the complex conjugate of $z_1$

$z_1 = -2 - 3i$

$ar{z_1} = -2 + 3i$

(ii) plot and label $ar{z_1}$ on the Argand diagram above.

$z_2 = -5 + 3i$ and $z_3 = 4 - 2i$, where $i^2 = -1$.

(b) Plot and label $z_2$ and $z_3$ on the Argand diagram on the previous page.

(c) Write $z_2 - z_3$ in the form $a + bi$, where $a, b \in \mathbb{R}$, $i^2 = -1$, and hence find $|z_2 - z_3|$.

$z_2 - z_3 = -5 + 3i - (4 - 2i) = -9 + 5i$

$|z_2 - z_3| = | -9 + 5i | = \sqrt{(-9)^2 + (5)^2} = \sqrt{81 + 25} = \sqrt{106}$

(d) Investigate if $z_3 = 4 - 2i$ is a solution of the equation $z^2 + 2iz - 7i = 0$.

Conclusion: ____________________

The complex number $z_1$ is shown on the Argand diagram below - Leaving Cert Mathematics - Question 1 - 2022