On an Argand diagram, sketch the loci of points representing complex numbers $w$ and $z$ such that $|w-1-2\mathrm{i}|=1$ and $\arg (z-1)=\frac{3}{4}\pi .$

On an Argand diagram, sketch the loci of points representing complex numbers $w$ and $z$ such that $|w-1-2\mathrm{i}|=1$ and $\arg (z-1)=\frac{3}{4}\pi .$

Calculate the least value of $\left|w-z\right|$ for points on these loci.

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.

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 $\mathbf{r}=\left(\begin{array}{r}-4\\ 5\\ 2\end{array}\right)+\mu \left(\begin{array}{r}-1\\ 0\\ 1\end{array}\right).$

Find the acute angle between $L_1$ and $L_2.$.

The number of birds of a certain species in a forested region is recorded over several years. At time $t$ years, the number of birds is $N,$ where $N$ is treated as a continuous variable.
The variation in the number of birds is modelled by $\frac{\mathrm{d}N}{ \mathrm{d}t}=\frac{N(1800-N)}{3600}.$ It is given that $N=300$ when $t=0.$

Find an expression for $N$ in terms of $t.$

The number of birds of a certain species in a forested region is recorded over several years. At time $t$ years, the number of birds is $N,$ where $N$ is treated as a continuous variable.
The variation in the number of birds is modelled by $\frac{\mathrm{d}N}{ \mathrm{d}t}=\frac{N(1800-N)}{3600}.$ It is given that $N=300$ when $t=0.$

According to the model, how many birds will there be after a long time?

Showing your working, find the square roots of the complex number $1-\left(2\sqrt{6}\right)i$. Give your answers in the form $x+\mathrm{i}y,$ where $x$ and $y$ are exact.

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.