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. ${\displaystyle {\int }_0^a}$ B. C. D.

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

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

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 , the displacement of the particle, in terms of .

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 are and perpendicular?

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

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

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.

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 .

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 where is the acceleration due to gravity. (Do NOT prove this.)

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

By expanding $(e^{i\theta}+e^{-i\theta})^4$, show that

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

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

Consider two lines in three dimensions given by:
$\mathbf{r}=\left(\begin{array}{c}3\\ -1\\ 7\end{array}\right)+{\lambda }_1\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right)\text{and}\mathbf{r}=\left(\begin{array}{c}3\\ -6\\ 2\end{array}\right)+{\lambda }_2(\begin{array}{c}-2\\ \end{array}$ By equating components, find the point of intersection of the two lines.

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 the particle has its maximum displacement of from the central point of motion. Find an equation for the displacement of the particle in terms of .

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

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

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.

A particle starts from rest and falls through a resisting medium so that its acceleration, in $\mathrm{m/s^2}$, is modelled by
$a=10\bigl(1-(kv)^2\bigr),$ where is the velocity of the particle in and . Find the velocity of the particle after seconds.

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