Let's go through the problems you uploaded.

Problem 5:

We are given three complex numbers:

• $z_1 = 2 - i$
• $z_2 = 2i$
• $z_3 = 1 - i$

We need to identify which operations among the options are correct:

• Option A: $z_1 + z_2 = 2 - i + 2i = 2 + i$. This is true.
• Option B: $z_1 - z_3 = (2 - i) - (1 - i) = 2 - i - 1 + i = 1$. This simplifies to just 1, so false.
• Option C: $z_1 \times z_2 = (2 - i) \times 2i = 2 \times 2i - i \times 2i = 4i - 2i^2 = 4i + 2 = 2 + 4i$. This is false.
• Option D: $z_2 + z_3 = 2i + (1 - i) = 1 + i$. This is true.
• Option E: $z_3 \times z_1 = (1 - i)(2 - i) = 2 - i - 2i + i^2 = 2 - 3i - 1 = 1 - 3i$. This is true.

So, the correct options are A, D, and E.

Problem 6:

We are given the complex number $z = -3 - 3i$, and we need to determine which statements are correct:

• Option A: The polar form of $z$. First, we find the modulus and argument:

  • $|z| = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$.
  • The argument $\theta = \tan^{-1}\left(\frac{-3}{-3}\right) = \tan^{-1}(1) = 45^\circ$ (but it's in the third quadrant, so $180^\circ + 45^\circ = 225^\circ$). Thus, the polar form is $z = 3\sqrt{2}(\cos 225^\circ + i\sin 225^\circ)$, so false (the modulus in the statement is wrong).

• Option B: Exponential form $z = 3\sqrt{2}e^{i225^\circ}$. This is true.

• Option C: The complex number as an ordered pair $z = (-3, -3)$. This is true.

• Option D: The conjugate $\bar{z} = -3 + 3i$. This is true.

• Option E: The principal argument $\arg(z) = 135^\circ$. This is false; the correct argument is $225^\circ$.

So, the correct options are B, C, and D.

Would you like more details or further explanation on any of these answers?

Here are 5 related questions to help expand your understanding:

1. How do you convert a complex number into polar form?
2. What is the general process to find the argument of a complex number?
3. How do you perform multiplication of two complex numbers?
4. Why is the conjugate of a complex number important in mathematics?
5. What are some applications of complex numbers in real-world problems?

Tip: Always remember to adjust the angle of a complex number based on its quadrant when finding the argument.