On a sketch of an Argand diagram, shade the region whose points represent the complex numbers $z$ which satisfy the inequality $|z-3\mathrm{i}|⩽2$. Find the greatest value of $\arg z$ for points in this region.

Relative to the origin $O,$ the position vectors of the points $A$ and $B$ are given by $\overrightarrow{OA}=\left(\begin{array}{r}-5\\ 0\\ 3\end{array}\right)$ and $\overrightarrow{OB}=\left(\begin{array}{l}1\\ 7\\ 2\end{array}\right)$
The line $AB$ is perpendicular to the line $L$ with vector equation:

$\overrightarrow{r}=\left(\begin{array}{r}4\\ 2\\ -3\end{array}\right)+\mu \left(\begin{array}{c}m\\ 3\\ 9\end{array}\right)$

Show that the line $AB$ and the line $L$ do not intersect.

The line $L_1$ has a vector equation

$r=-2\hat{j}+\hat{k}+\lambda \left(\hat{i}+3\hat{k}\right)$

The line $L_2$ passes through the point $P\left(-2, 1, -1\right)$ and is parallel to $L_1.$

Find the shortest distance from $P$ to the line $L_1$.

The line $L_1$ has vector equation $\mathbf{r}=\left(\begin{array}{r}-2\\ 3\\ 0\end{array}\right)+\lambda \left(\begin{array}{r}1\\ -1\\ 2\end{array}\right).$
The line $L_2$ has vector equation $\mathbf{r}=\left(\begin{array}{r}-4\\ 5\\ m\end{array}\right)+\mu \left(\begin{array}{r}-1\\ 0\\ 1\end{array}\right).$

In the case where $m=2,$ show that $L_1$ and $L_2$ do not intersect.

The line $L_1$ has vector equation $\mathbf{r}=\left(\begin{array}{r}-2\\ 3\\ 0\end{array}\right)+\lambda \left(\begin{array}{r}1\\ -1\\ 2\end{array}\right).$
The line $L_2$ has vector equation $\mathbf{r}=\left(\begin{array}{r}-4\\ 5\\ m\end{array}\right)+\mu \left(\begin{array}{r}-1\\ 0\\ 1\end{array}\right).$

Find the value of $m$ in the case where $L_1$ and $L_2$ intersect.

On a sketch of an Argand diagram, shade the region whose points represent the complex numbers $z$ which satisfy the inequality $|z-3\mathrm{i}|⩽2$. Find the greatest value of $\arg z$ for points in this region.

Liquid is flowing into a small tank which has a leak. Initially the tank is empty and $t$ minutes later, the volume of liquid in the tank is $V{ \mathrm{cm}}^3.$ The liquid is flowing into the tank at a constant rate of $80{ \mathrm{cm}}^3$ per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to $kV{ \mathrm{cm}}^3$ per minute where $k$ is a positive constant.

Write down a differential equation describing this situation and solve it to show that $V=\frac{1}{k}\left(80-80{\mathrm{e}}^{-kt}\right).$

Liquid is flowing into a small tank which has a leak. Initially the tank is empty and $t$ minutes later, the volume of liquid in the tank is $V{ \mathrm{cm}}^3.$ The liquid is flowing into the tank at a constant rate of $80{ \mathrm{cm}}^3$ per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to $kV{ \mathrm{cm}}^3$ per minute where $k$ is a positive constant and $V=\frac{1}{k}\left(80-80{\mathrm{e}}^{-kt}\right).$

It is observed that $V=500$ when $t=15$, so that $k$ satisfies the equation $k=\frac{4-4{\mathrm{e}}^{-15k}}{25}.$
Use an iterative formula, based on this equation, to find the value of $k$ correct to $2$ significant figures. Use an initial value of $k=0.1$ and show the result of each iteration to $4$ significant figures.