Given the complex numbers $z_1 = 3 + 2i$ and $z_2 = 4 + 3i$, let's find:

(i) $z_1z_2$ (2 marks)

To multiply two complex numbers, we use the distributive property:

$z_1z_2 = (3 + 2i)(4 + 3i)$

Now, distribute the terms:

$z_1z_2 = 3 \times 4 + 3 \times 3i + 2i \times 4 + 2i \times 3i$

Simplifying each term:

$z_1z_2 = 12 + 9i + 8i + 6i^2$

Since $i^2 = -1$:

$z_1z_2 = 12 + 17i + 6(-1)$

$z_1z_2 = 12 + 17i - 6$

$z_1z_2 = 6 + 17i$

Answer for (i):

$z_1z_2 = 6 + 17i$

(ii) $\frac{z_1}{z_2}$ (2 marks)

To divide two complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator:

$\frac{z_1}{z_2} = \frac{3 + 2i}{4 + 3i} \times \frac{4 - 3i}{4 - 3i}$

This simplifies to:

$\frac{z_1}{z_2} = \frac{(3 + 2i)(4 - 3i)}{(4 + 3i)(4 - 3i)}$

First, calculate the denominator using the difference of squares:

$(4 + 3i)(4 - 3i) = 4^2 - (3i)^2 = 16 - 9(-1) = 16 + 9 = 25$

Next, calculate the numerator:

$(3 + 2i)(4 - 3i) = 3 \times 4 + 3 \times (-3i) + 2i \times 4 + 2i \times (-3i)$

$= 12 - 9i + 8i - 6i^2$

Substitute $i^2 = -1$:

$= 12 - 9i + 8i + 6 = 18 - i$

Now, divide by the denominator:

$\frac{z_1}{z_2} = \frac{18 - i}{25} = \frac{18}{25} - \frac{1}{25}i$

Answer for (ii):

$\frac{z_1}{z_2} = \frac{18}{25} - \frac{1}{25}i$

Would you like more details or have any questions?

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Here are five related questions:

1. How do you find the modulus and argument of a complex number like $z_1$ or $z_2$?
2. What is the geometric interpretation of the multiplication of complex numbers?
3. How do you express the product $z_1z_2$ in polar form?
4. What is the conjugate of a complex number, and how is it used in division?
5. How do you add or subtract two complex numbers?

Tip: When dividing complex numbers, always multiply by the conjugate of the denominator to simplify the expression.