10 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 4
- 1
2 \end{array} \right) + \mu \left( \begin{array} { l } 5
4
8 \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
Prove that lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.

1. A spherical ball of ice of radius 12 cm is placed in a bucket of water.

In a model of the situation,

• the ball remains spherical as it melts
• \(t\) minutes after the ball of ice is placed in the bucket, its radius is \(r \mathrm {~cm}\)
• the rate of decrease of the radius of the ball of ice is inversely proportional to the square of the radius
• the radius of the ball of ice is 6 cm after 15 minutes

Using the model and the information given,

1. find an equation linking \(r\) and \(t\),
2. find the time taken for the ball of ice to melt completely.
3. On Diagram 1 on page 27, sketch a graph of \(r\) against \(t\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{032d2541-9905-4570-9584-9a144b02fde5-27_662_728_1959_671} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}