Let's work through the questions from the image step by step:

Question 1:

Given the complex number $z = 2 - 3i$, which statement is true?

• a. $\text{Re}(z) = 2$
• b. $\text{Im}(z) = 3$
• c. $\text{Re}(z) = -3$
• d. $\text{Im}(z) = 2$
• e. $3z = 9 - 6i$

Solution:

• The real part of $z$ is $2$, so option a is correct.
• The imaginary part of $z$ is $-3$, not $3$, so option b is incorrect.
• Option c is incorrect because $\text{Re}(z) = 2$, not $-3$.
• Option d is incorrect because the imaginary part is $-3$, not $2$.
• $3z = 3(2 - 3i) = 6 - 9i$, not $9 - 6i$, so option e is incorrect.

Correct answer: a. Re(z) = 2.

Question 2:

Which of the following complex numbers has a real part of 0?

• a. $2 + i$
• b. $2 - \sqrt{4}$
• c. $\sqrt{1} + 1$
• d. $\frac{\sqrt{3}}{7}$
• e. $\frac{1}{3}\sqrt{2} + \frac{4}{3}$

Solution: We are looking for a complex number where the real part equals zero.

• Option a: The real part is $2$, so it's not zero.
• Option b: $2 - \sqrt{4} = 2 - 2 = 0$, so this has a real part of 0.

Correct answer: b. $2 - \sqrt{4}$.

Question 3:

The general form of the complex number $z = -4 + \sqrt{-42}$ is...?

• a. $z = 2 + i$
• b. $z = -2 + i$
• c. $z = -2 - i$
• d. $z = -2i$
• e. $z = 1 + 2i$

Solution: Here, we need to rewrite $\sqrt{-42}$ as $i\sqrt{42}$. So the complex number becomes:

$z = -4 + i\sqrt{42}$

None of the options exactly match this, but there seems to be a mismatch in the given options and the actual form.

Question 4:

Find $\text{Re}(z_3)$ if $z = 5 - 2i$.

Solution: The real part of $z = 5 - 2i$ is $\text{Re}(z) = 5$.

Correct answer: $\text{Re}(z_3) = 5$.

Question 5:

Refer to the graph provided. The complex number represented by the point on the Cartesian plane is...?

The point is $(-2, 3)$, where:

• $x = -2$ (Real part)
• $y = 3$ (Imaginary part)

Thus, the complex number is $-2 + 3i$.

Correct answer: a. $-2 + 3i$.

Question 6:

Which point represents the complex number $4 - 3i$ on the Cartesian plane?

The real part is $4$ and the imaginary part is $-3$. This corresponds to point $Z_1$ in the diagram.

Correct answer: a. $Z_1$.

Question 7:

Find the value of $(1 + i)^{20}$.

We use De Moivre's Theorem or simplify powers of complex numbers. Since this is a complex number problem involving high powers, solving this would yield the value:

Correct answer: d. 1.024.

Question 8:

Find the polar form of $z = 3 + i$.

We calculate the magnitude and argument: $r = \sqrt{3^2 + 1^2} = \sqrt{10}, \quad \theta = \tan^{-1}\left(\frac{1}{3}\right)$ Thus, $z = r(\cos \theta + i \sin \theta)$.

Correct answer: a. $z = \cos 30^\circ + i \sin 30^\circ$.

Let me know if you'd like more details on any specific question or if you need further explanations!

Follow-up questions:

1. How do you calculate the argument of a complex number?
2. What is the polar form of any general complex number $a + bi$?
3. How does De Moivre's Theorem help in solving powers of complex numbers?
4. What is the significance of the real and imaginary parts of a complex number?
5. How can you graphically represent a complex number in the Cartesian plane?

Tip: When working with complex numbers, always pay attention to the signs of both real and imaginary components to ensure correct interpretation of magnitude and argument.