Mathematics Extension 2

What is the maximum value of $|e^{i\theta}-2|+|e^{i\theta}+2|$ for $0\leq\theta\leq2\pi$?

A. $\sqrt5$
B. $4$
C. $2\sqrt5$ D.

Which of the following is equal to $\displaystyle\int_0^{2a}f(x)\,dx$?

A. $\displaystyle\int_0^a[f(x)-f(2a-x)]\,dx$
B. $\displaystyle\int_0^a[f(x)+f(2a-x)]\,dx$
C. $\displaystyle2\int_0^a f(x-a)\,dx$
D. $\displaystyle\int_0^a\frac12f(2x)\,dx$

Consider the complex numbers $w=-1+4i$ and $z=2-i$. Evaluate $|w|$.

Use integration by parts to evaluate
$\int_1^e x\ln x\,dx.$

A particle starts at the origin with velocity $1$ and acceleration given by
$a=v^2+v,$
where $v$ is the velocity of the particle. Find an expression for $x$, the displacement of the particle, in terms of $v$.

Consider the two vectors $\mathbf u=-2\mathbf i-\mathbf j+3\mathbf k$ and $\mathbf v=p\mathbf i+\mathbf j+2\mathbf k$. For what values of $p$ are $\mathbf u-\mathbf v$ and $\mathbf u+\mathbf v$ perpendicular?

Solve $z^2+3z+(3-i)=0$, giving your answer(s) in the form $a+bi$, where $a$ and $b$ are real.

Consider the complex numbers $w=-1+4i$ and $z=2-i$. Evaluate $w\bar{z}$.

Given $u^2>gR$, prove that there are two distinct values of $\theta$ for which the particle will land at $x=R$.

Find the velocity of the box after the first three seconds.

Show that the net force horizontally is approximately $53.2$ newtons.

A 50-kilogram box is initially at rest. The box is pulled along the ground with a force of $200$ newtons at an angle of $30^\circ$ to the horizontal. The box experiences a resistive force of $0.3R$ newtons, where $R$ is the normal force, as shown in the diagram. Take $g=10\,\mathrm{m/s^2}$.

(i) By resolving the forces vertically, show that $R=400$.

Using the projectile model from part (i), show that the Cartesian equation of the path of flight is
$y=\frac{-gx^2}{2u^2}\left(\tan^2\theta-\frac{2u^2}{gx}\tan\theta+1\right).$

A particle is projected from the origin with initial velocity $u\,\mathrm{m/s}$ at an angle $\theta$ to the horizontal. The particle lands at $x=R$ on the $x$-axis. The acceleration vector is
$\mathbf a=\begin{pmatrix}0\\-g\end{pmatrix},$
where $g$ is the acceleration due to gravity. (Do NOT prove this.)

(i) Show that the position vector $\mathbf r(t)$ of the particle is given by
$\mathbf r(t)=\begin{pmatrix}ut\cos\theta\\ut\sin\theta-\frac12gt^2\end{pmatrix}.$

A particle is undergoing simple harmonic motion with period $\dfrac{\pi}{3}$. The central point of motion of the particle is at $x=\sqrt3$. When $t=0$ the particle has its maximum displacement of $2\sqrt3$ from the central point of motion. Find an equation for the displacement $x$ of the particle in terms of $t$.

Hence, or otherwise, find
$\int_0^{\pi/2}\cos^4\theta\,d\theta.$

By expanding $(e^{i\theta}+e^{-i\theta})^4$, show that
$\cos^4\theta=\frac18\bigl(\cos(4\theta)+4\cos(2\theta)+3\bigr).$

Show that for any integer $n$,
$e^{in\theta}+e^{-in\theta}=2\cos(n\theta).$

Prove that
$p^2+4q^2\geq4pq.$

By considering the right-angled triangle below, or otherwise, prove that
$\frac{a+b}{2}\geq\sqrt{ab},$
where $a>b\geq0$.

Consider two lines in three dimensions given by:
$\mathbf r=\begin{pmatrix}3\\-1\\7\end{pmatrix}+\lambda_1\begin{pmatrix}1\\2\\1\end{pmatrix} \quad \text{and} \quad \mathbf r=\begin{pmatrix}3\\-6\\2\end{pmatrix}+\lambda_2\begin{pmatrix}-2\\1\\3\end{pmatrix}.$
By equating components, find the point of intersection of the two lines.

Prove by mathematical induction that, for $n\geq2$,
$\frac1{2^2}+\frac1{3^2}+\cdots+\frac1{n^2}<\frac{n-1}{n}.$

Prove that for any integer $n>1$, $\log_n(n+1)$ is irrational.

Prove that $z_1^2+z_2^2=z_1z_2.$