Question

Question 1(Multiple Choice Worth 2 points)
(01.07 MC)

Simplify the given expression below:

(7 − 3i) ⋅ (2 − i)

4 + i
14 + 3i
11 − 13i
14 − 10i
Question 2(Multiple Choice Worth 2 points)
(01.07 MC)

Simplify the given expression below:

(4 + 2i) − (1 − 7i)

5 − 5i
5 + 9i
3 + 9i
3 − 5i
Question 3(Multiple Choice Worth 2 points)
(01.07 MC)

Simplify the given expression below:

(−3 + 2i) ⋅ (2 + i)

−8 + i
−6 + 3i
−3 + 4i
−1 + 2i
Question 4(Multiple Choice Worth 2 points)
(01.07 LC)

Simplify the given expression below:

(3 + 4i) + (5 − 2i)

−2 + 6i
2 − 2i
7 + 3i
8 + 2i
Question 5(Multiple Choice Worth 2 points)
(01.07 MC)

Evaluate (10 – 4i) ÷ (5 + i).

Fifty-four twelfths minus fifteen twelfths i
Twenty-three twelfths minus fifteen twelfths i
Fifty-four thirteenths minus fifteen thirteenths i
Twenty-three thirteenths minus fifteen thirteenths i

Answer 1

Let's simplify each expression step by step.

### Question 1
**Expression**: (7 − 3i) ⋅ (2 − i)

To simplify this, use the distributive property (FOIL):
- First: \(7 \cdot 2 = 14\)
- Outer: \(7 \cdot (-i) = -7i\)
- Inner: \(-3i \cdot 2 = -6i\)
- Last: \(-3i \cdot (-i) = 3i^2 = -3\) (since \(i^2 = -1\))

Combine these results:
\[ 14 - 7i - 6i - 3 = 14 - 13i - 3 = 11 - 13i \]

**Answer**: 11 − 13i

### Question 2
**Expression**: (4 + 2i) − (1 − 7i)

Distributing the negative:
\[ (4 + 2i) - 1 + 7i = 4 - 1 + 2i + 7i = 3 + 9i \]

**Answer**: 3 + 9i

### Question 3
**Expression**: (−3 + 2i) ⋅ (2 + i)

Using distributive property (FOIL):
- First: \(-3 \cdot 2 = -6\)
- Outer: \(-3 \cdot i = -3i\)
- Inner: \(2i \cdot 2 = 4i\)
- Last: \(2i \cdot i = 2i^2 = -2\)

Combine these results:
\[ -6 - 2 - 3i + 4i = -8 + i \]

**Answer**: −8 + i

### Question 4
**Expression**: (3 + 4i) + (5 − 2i)

Combining like terms:
\[ (3 + 5) + (4i - 2i) = 8 + 2i \]

**Answer**: 8 + 2i

### Question 5
**Expression**: (10 − 4i) ÷ (5 + i)

Multiply the numerator and the denominator by the conjugate of the denominator:
\[
(10 - 4i)(5 - i) \text{ and } (5 + i)(5 - i)
\]

Calculating the denominator:
\[
(5 + i)(5 - i) = 25 - i^2 = 25 + 1 = 26
\]

Calculating the numerator:
- First: \(10 \cdot 5 = 50\)
- Outer: \(10 \cdot (-i) = -10i\)
- Inner: \(-4i \cdot 5 = -20i\)
- Last: \(-4i \cdot (-i) = 4i^2 = -4\)

Combine:
\[ 50 - 4 - 10i - 20i = 46 - 30i \]

Now, put it all together:
\[
\frac{46 - 30i}{26} = \frac{46}{26} - \frac{30}{26}i = \frac{23}{13} - \frac{15}{13}i
\]

**Answer**: Twenty-three thirteenths minus fifteen thirteenths i

### Summary of Answers
1. 11 − 13i
2. 3 + 9i
3. −8 + i
4. 8 + 2i
5. Twenty-three thirteenths minus fifteen thirteenths i