The image you have uploaded contains exercises on complex numbers. Let's address each problem one by one.

Exercise 1

Let $z_1$ and $z_2$ be two complex numbers. Prove that: $|z_1 + z_2| \leq |z_1| + |z_2|$

This is known as the triangle inequality for complex numbers.

Exercise 2

Let $z_1$ and $z_2$ be two non-zero complex numbers. Prove that: $\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}$

Exercise 3

Prove that for a complex number $z$: $|z|^2 = z \overline{z}$ where $\overline{z}$ is the complex conjugate of $z$.

Exercise 4

Let $z = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$. Prove that: $|e^{i\theta}| = 1$

Exercise 5

Let $z_1$ and $z_2$ be complex numbers such that $|z_1| = |z_2|$. Prove that the points representing $z_1$ and $z_2$ on the Argand diagram lie on a circle of radius $|z_1|$ centered at the origin.

Exercise 6

If $z$ is a complex number such that $\text{Re}(z) = 0$, show that $z$ is purely imaginary.

Exercise 7

If $z$ is a complex number and $|z| = 1$, show that: $\overline{z} = \frac{1}{z}$

Exercise 8

Let $z$ be a variable complex number such that $|z - 4i| \leq 3$. Show that the locus of $z$ in the Argand diagram lies inside or on a circle centered at $4i$ with a radius of 3.

Exercise 9

Show that $\left| |z_1| - |z_2| \right| \leq |z_1 - z_2|$.

Exercise 10

If $z$ is a non-zero complex number such that $z + \frac{1}{z}$ is real, prove that $z$ lies on the unit circle.

Exercise 11

Find the greatest and least values of ( |z| + \left| \frac{