In the following diagram, NS is the Prime Meridian. The positions of towns A and B are 30°S, 30°W) and (30°N, 30°W) respectively. Find the distance between A and B in nautical miles. If the local time at B is 09 35 hours, what is the local time at the Prime Meridian?

Here are the solutions to the questions:

1. (a) Given that $R = \{x : 0 \le x < 10, x \text{ is an even number}\}$, list the elements of set R.

Step 1: Identify the conditions for the elements of set R.
The elements must be even numbers, and they must be greater than or equal to 0 and strictly less than 10.

Step 2: List the even numbers that satisfy these conditions.
The even numbers in the range $0 \le x < 10$ are $0, 2, 4, 6, 8$.

$$R = \{0, 2, 4, 6, 8\}$$

The elements of set R are $\boxed{{0, 2, 4, 6, 8}}$.

1. (b) A point $F(-4, 1)$ is mapped onto a point $G(5, -4)$ by a translation $T$. Express $T$ as a column vector.

Step 1: Understand the relationship between the original point, translation vector, and image point.
If a point $F$ is translated by a vector $T$ to an image point $G$, then $F + T = G$.
Let $F = \begin{bmatrix} -4 \\ 1 \end{bmatrix}$ and $G = \begin{bmatrix} 5 \\ -4 \end{bmatrix}$. Let the translation vector be $T = \begin{bmatrix} a \\ b \end{bmatrix}$.

Step 2: Set up the vector equation.

$$\begin{bmatrix} -4 \\ 1 \end{bmatrix} + \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 5 \\ -4 \end{bmatrix}$$

Step 3: Solve for the components of the translation vector.
For the x-component:

$$-4 + a = 5$$ $$a = 5 + 4 = 9$$

For the y-component:

$$1 + b = -4$$ $$b = -4 - 1 = -5$$

The translation vector $T$ is $\boxed*{\begin{bmatrix} 9 \\ -5 \end{bmatrix}*}$.

2. In the following diagram, NS is the Prime Meridian. The positions of towns A and B are $(30^\circ S, 30^\circ W)$ and $(30^\circ N, 30^\circ W)$ respectively.