Let's break down and solve each part of the question as shown in the image.

Part a)

We are asked to solve the following equations in the form $x + yi$, where $i$ represents the imaginary unit.

i) $\sqrt{\frac{36}{81}} + 2i^2$

1. Simplify the real part: $\sqrt{\frac{36}{81}} = \frac{6}{9} = \frac{2}{3}$

2. Simplify the imaginary part: $2i^2 = 2(-1) = -2$

Thus, the result is: $-2 + \frac{2}{3}$

In the form $x + yi$, this becomes: $\boxed{-2 + \frac{2}{3}i}$

ii) $(6 + 2i) - (5 - 3i)$

1. Subtract the real and imaginary parts separately: $(6 - 5) + (2i + 3i) = 1 + 5i$

Thus, the solution is: $\boxed{1 + 5i}$

iii) $\frac{5 - 3i}{3 + 2i}$

1. Multiply the numerator and denominator by the conjugate of the denominator: $\frac{5 - 3i}{3 + 2i} \times \frac{3 - 2i}{3 - 2i} = \frac{(5 - 3i)(3 - 2i)}{(3 + 2i)(3 - 2i)}$

2. Simplify the denominator: $(3 + 2i)(3 - 2i) = 9 - 4i^2 = 9 + 4 = 13$

3. Expand the numerator: $(5 - 3i)(3 - 2i) = 15 - 10i - 9i + 6i^2 = 15 - 19i + 6(-1) = 15 - 19i - 6 = 9 - 19i$

4. Final result: $\frac{9 - 19i}{13} = \frac{9}{13} - \frac{19}{13}i$

Thus, the solution is: $\boxed{\frac{9}{13} - \frac{19}{13}i}$

Part b) $z = -4 - 3i$

We are asked to calculate the modulus and argument of $z$, and illustrate the Argand diagram.

Modulus

The modulus is given by: $|z| = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Argument

The argument $\theta$ is calculated using: $\theta = \text{arg}(z) = \tan^{-1} \left( \frac{-3}{-4} \right) = \tan^{-1}\left(\frac{3}{4}\right) = 36.87^\circ$ Since the complex number is in the third quadrant, we add $180^\circ$: $\theta = 180^\circ + 36.87^\circ = 216.87^\circ$

Thus: $\boxed{|z| = 5 \quad \text{and} \quad \text{arg}(z) = 216.87^\circ}$

Would you like further details on any part of the solution, or any questions about this?

Here are 5 related questions:

1. How do you represent complex numbers on the Argand diagram?
2. What is the significance of the modulus and argument in polar form?
3. How would you multiply two complex numbers in polar form?
4. Can you explain the geometric interpretation of the argument?
5. What is the relation between a complex number's conjugate and its argument?

Tip: When dealing with complex division, always multiply by the conjugate to simplify the denominator.