Write down the bearing of P from A. Find the exact length of AC.

The bearing is "from A", so we start at point A. Bearings are measured clockwise from the North line.

The diagram shows an angle of $50^\circ$ measured clockwise from the North line at A to the line segment AP.

Bearings are always written with three digits.
The bearing of P from A is $\boxed{\text{}050^\circ\text{}}$.

3. Find the exact length of AC.

We have a right-angled triangle ABC, with $\angle C = 90^\circ$.
The angle $\angle B = 30^\circ$.
The length of the side adjacent to $\angle B$ is $BC = 12$.
We need to find the length of the side opposite to $\angle B$, which is AC.

Since we know the adjacent side and want to find the opposite side relative to $\angle B$, we use the tangent function:

The exact value of $\tan(30^\circ)$ is $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$.

The exact length of AC is $\boxed{\text{}4\sqrt{3} \text{ units}\text{}}$.

4. Q is due East of R and due South of P. The bearing of R from P is 230°. Show this information on the diagram.

Since there is no diagram provided for this question, here's how you would construct and label it:

• Draw point P at the top.
• Since Q is due South of P, draw a vertical line segment downwards from P and label the end point Q. This line segment PQ represents the North-South direction.
• Since Q is due East of R, draw a horizontal line segment to the left from Q and label the end point R. This line segment RQ represents the East-West direction.
• Connect points P and R to form a triangle PQR. The angle at Q ($\angle PQR$) will be $90^\circ$.

• At point P, draw a North line pointing vertically upwards.
• Measure $230^\circ$ clockwise from this North line. Draw a line from P in this direction, which will be the line segment PR.
• The angle between the North line at P and the line PR, measured clockwise, is $230^\circ$.

• The line PQ points South from P. The angle from the North line to the South line (PQ) is $180^\circ$.
• The angle from the South line (PQ) to the line PR, measured clockwise, is $230^\circ - 180^\circ = 50^\circ$. So, $\angle QPR = 50^\circ$.