The complex number $2+2i$ is denoted by $u.$

Sketch an Argand diagram showing the points representing the complex numbers $1, i$ and $u.$ Shade the region whose points represent the complex numbers $z$ which satisfy both the inequalities $\left|z-1\right|⩽\left|z-i\right|$ and $\left|z-u\right|⩽1.$

The complex number $2+2i$ is denoted by $u.$

Using your diagram, calculate the value of $\left|z\right|$ for the point in this region for which arg $z$ is least.

A certain substance is formed in a chemical reaction. The mass of substance formed $t$ seconds after the start of the reaction is $x$ grams. At any time the rate of formation of the substance is proportional to $\left(20-x\right)$. When $t=0, x=0$ and $\frac{\mathrm{d}x}{ \mathrm{d}t}=1$.

Show that $x$ and $t$ satisfy the differential equation $\frac{dx}{dt}=0.05\left(20-x\right)$.

Find, in any form, the solution of this differential equation $\frac{dx}{dt}=0.05\left(20-x\right)$. When $t=0$, $x=0$ and $\frac{dx}{dt}=1$.

Find, in any form, the solution of this differential equation $\frac{dx}{dt}=0.05\left(20-x\right)$. When $t=0$, $x=0$ and $\frac{dx}{dt}=1$.

Find $x$ when $t=10$, giving your answer correct to $1$ decimal place.

(Use $\ln 20=2.996, e^{2.496}=12.134$)

Find, in any form, the solution of this differential equation $\frac{dx}{dt}=0.05\left(20-x\right)$. When $t=0$, $x=0$ and $\frac{dx}{dt}=1$.

State what happens to the value of $x$ as $t$ becomes very large.

The line $L_1$ has 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 $L_1.$

Write down a vector equation of the line $L_2$.

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 $L_1$.

In a certain country the government charges tax on each litre of petrol sold to motorists. The revenue per year is $R$ million dollars when the rate of tax is $x$ dollars per litre. The variation of $R$ with $x$ is modelled by the differential equation $\frac{\mathrm{d}R}{\mathrm{d}x}=R\left(\frac{1}{x}-0.57\right),$

where $R$ and $x$ are taken to be continuous variables. When $x=0.5, R=16.8$.

Solve the differential equation and obtain an expression for $R$ in terms of $x.$

(Use $\ln \left(16.8\right)=2.82, \ln \left(0.5\right)=-0.693$)