The homework contains three questions: 1. The construction below is required in constructing angle POQ. What is the size of angle POQ? 2. A car travels 160 miles on 4 gallons of diesel. Find the avera

Here are the solutions to the questions.

Question 1: The construction below is required in constructing angle POQ. What is the size of angle POQ?

Step 1: Observe the construction. The construction shows a method to bisect an angle.

Step 2: The initial angle being bisected is a right angle ($90^\circ$), indicated by the perpendicular construction marks.

Step 3: Bisecting a $90^\circ$ angle means dividing it into two equal parts.

$$\frac{90^\circ}{2} = 45^\circ$$

The final answer is $\boxed{\text{A. } 45^\circ}$.

Question 2: A car travels 160 miles on 4 gallons of diesel. Find the average mile per gallon.

Step 1: Divide the total distance by the total fuel consumed.

$$\text{Average mile per gallon} = \frac{\text{Total miles}}{\text{Total gallons}}$$ $$\text{Average mile per gallon} = \frac{160 \text{ miles}}{4 \text{ gallons}}$$

Step 2: Calculate the result.

$$\text{Average mile per gallon} = 40 \text{ miles/gallon}$$

The final answer is $\boxed{\text{A. } 40 \text{ miles}}$.

Question 3: Find the gradient of the line in the equation $4y - 3x - 2 = 0$.

Step 1: Rearrange the equation into the slope-intercept form $y = mx + c$.

$$4y - 3x - 2 = 0$$ $$4y = 3x + 2$$

Step 2: Divide all terms by 4.

$$y = \frac{3}{4}x + \frac{2}{4}$$ $$y = \frac{3}{4}x + \frac{1}{2}$$

Step 3: Identify the gradient ($m$), which is the coefficient of $x$.
The gradient is $m = \frac{3}{4}$.
The final answer is $\boxed{\text{C. } \frac{3}{4}}$.

Question 4: Simplify $(2a + 3)(b + 2)$.

Step 1: Use the distributive property (FOIL method) to multiply the binomials.

$$(2a + 3)(b + 2) = 2a(b) + 2a(2) + 3(b) + 3(2)$$

Step 2: Perform the multiplications.

$$= 2ab + 4a + 3b + 6$$

The final answer is $\boxed{\text{C. } 2ab + 4a + 3b + 6}$.

Question 5: At what distance did the bus driver drove from Town A to Town B?

Step 1: Locate Town A and Town B on the distance axis of the graph.
Town A is at 0 km.
Town B is at 4 km.

Step 2: The distance driven from Town A to Town B is the difference between their distances from the origin.

$$\text{Distance} = 4 \text{ km} - 0 \text{ km} = 4 \text{ km}$$

The final answer is $\boxed{\text{A. } 4 \text{ km}}$.

Question 6: The bus driver drove from Town A to Town B within how many hours?

Step 1: Identify the time the bus left Town A and arrived at Town B from the graph.
Left Town A (0 km) at 8:00 a.m.
Arrived at Town B (4 km) at 9:00 a.m.

Step 2: Calculate the duration of the journey.

$$\text{Duration} = 9:00 \text{ a.m.} - 8:00 \text{ a.m.} = 1 \text{ hour}$$

The final answer is $\boxed{\text{B. } 1 \text{ hour}}$.

Question 7: The bus driver drove from Town C back to Town A in what time?

Step 1: Identify the time the bus left Town C and arrived back at Town A from the graph.
Left Town C (6 km) at 9:30 a.m.
Arrived back at Town A (0 km) at 11:00 a.m.

Step 2: Calculate the duration of the journey.

$$\text{Duration} = 11:00 \text{ a.m.} - 9:30 \text{ a.m.} = 1 \text{ hour } 30 \text{ minutes}$$

The final answer is $\boxed{\text{A. } 1 \text{ hour } 30 \text{ minutes}}$.

Question 8: Find the value of $\frac{3e}{5} + \frac{e}{2}$ when $e = 10$.

Step 1: Substitute $e = 10$ into the expression.

$$\frac{3(10)}{5} + \frac{10}{2}$$

Step 2: Simplify each term.

$$\frac{30}{5} + 5$$ $$6 + 5$$

Step 3: Add the values.

$$11$$

The final answer is $\boxed{\text{B. } 11}$.

Question 9: The equation of a straight line is $y + 2x - 2 = 0$. What is the gradient?

Step 1: Rearrange the equation into the slope-intercept form $y = mx + c$.

$$y + 2x - 2 = 0$$ $$y = -2x + 2$$

Step 2: Identify the gradient ($m$), which is the coefficient of $x$.
The gradient is $m = -2$.
The final answer is $\boxed{\text{A. } -2}$.

Question 10: Factorize $\frac{1}{2}ty^2 + \frac{1}{4}ty$.

Step 1: Identify the greatest common factor (GCF) of the terms.
The numerical GCF of $\frac{1}{2}$ and $\frac{1}{4}$ is $\frac{1}{4}$.
The variable GCF of $ty^2$ and $ty$ is $ty$.
So, the GCF is $\frac{1}{4}ty$.

Step 2: Factor out the GCF from each term.

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