Mathematics Extension 2

$ABCD$ is a parallelogram. $N$ is on $AD$ so that $\overrightarrow{AN}=\frac13\overrightarrow{AD}$. $M$ is on the diagonal $AC$. By letting $\overrightarrow{AM}=\lambda\overrightarrow{AC}$ and $\overrightarrow{MB}=\delta\overrightarrow{NB}$, prove that
$\overrightarrow{AM}=\frac14\overrightarrow{AC}.$

Show that $a^2+9b^2\ge6ab$, where $a$ and $b$ are real numbers.

Hence, if $a>b>c>0$, show that
$a^2+5b^2+9c^2>9bc.$

Hence, or otherwise, show that the point $P$ cannot be lifted to a position $\frac{2h}{3}$ metres above the floor.

The diagrams show two positions of a single piston in the cylinder chamber of a motorcycle. The piston moves vertically, in simple harmonic motion, between a maximum height of $0.17$ metres and a minimum height of $0.05$ metres.

The mass of the piston is $0.8\,\mathrm{kg}$. The piston completes $40$ cycles per second.

What is the resultant force on the piston, in newtons, that produces the maximum acceleration of the piston? Give your answer correct to the nearest newton.

Using the substitution $x=\tan^2\theta$, evaluate
$\int_0^1\sin^{-1}\sqrt{\frac{x}{1+x}}\,dx.$

The complex number $z$ satisfies
$\left|z-\frac4z\right|=2.$

Using the triangle inequality, or otherwise, show that
$|z|\le\sqrt5+1.$

Consider
$I_n=\int_0^1\frac{x^n}{\sqrt{1-x}}\,dx.$
Use integration by parts to show that
$I_n=\frac{2n}{2n+1}I_{n-1}.$

The line has vector equation
$\vec r=\lambda\begin{bmatrix}1\\3\\1\end{bmatrix}.$
For $P(x_1,y_1,z_1)$, the point $Q$ on the line is such that $PQ$ is a minimum. Using vector projection methods, show that
$\overrightarrow{OQ}=\frac{x_1+3y_1+z_1}{11}\begin{bmatrix}1\\3\\1\end{bmatrix}.$

It is given that
$\vec c=\begin{bmatrix}1\\4\\0\end{bmatrix}+t\begin{bmatrix}-1\\1\\5\end{bmatrix}$
does not intersect $\vec r$. Point $T$ is on $\vec c$ and $S$ is a point on $\vec r$ such that $TS$ is a minimum. By first expressing $\vec c$ in parametric form, find an expression for $\overrightarrow{TS}$ in terms of $t$.

Hence, or otherwise, show that
$a^2+5b^2+9c^2\ge3(ab+bc+ac).$

Show by finding it that the only point of intersection between
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\10\\5\end{pmatrix}+\lambda\begin{pmatrix}2\\-3\\1\end{pmatrix}$
and the sphere
$(x-5)^2+(y+2)^2+(z-3)^2=38$
is $\begin{pmatrix}7\\1\\8\end{pmatrix}$.

Show that the line
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}15\\-1\\6\end{pmatrix}+\delta\begin{pmatrix}4\\-1\\-1\end{pmatrix}$
is also tangent to the sphere
$(x-5)^2+(y+2)^2+(z-3)^2=38$
by showing that it is perpendicular to the radius of the sphere at $\begin{pmatrix}7\\1\\8\end{pmatrix}$.

Hence show that
$\cos^7\theta=\frac1{64}\left(\cos7\theta+7\cos5\theta+21\cos3\theta+35\cos\theta\right).$

Prove that
$\frac{\cos\theta+i\sin\theta-1}{\cos\theta+i\sin\theta+1}=i\tan\frac\theta2.$

Find the roots of $w^5=1$. Express your answers in the form $\cos\theta+i\sin\theta$, where $-\pi<\theta\le\pi$.

Using parts (i) and (ii), find the roots of
$\left(\frac{2+z}{2-z}\right)^5=1.$

A square in the Argand plane has vertices $5+5i$, $5-5i$, $-5-5i$ and $-5+5i$. The complex numbers $z_A=5+i$, $z_B$ and $z_C$ lie on the square and form the vertices of an equilateral triangle, as shown in the diagram.

Find the exact value of the complex number $z_B$.

A projectile of mass $M\,\mathrm{kg}$ is launched vertically upwards from a horizontal plane with initial speed $v_0\,\mathrm{m\,s^{-1}}$, where $v_0<100\,\mathrm{m\,s^{-1}}$. The projectile experiences a resistive force of magnitude $0.1Mv$ newtons, where $v\,\mathrm{m\,s^{-1}}$ is the speed of the projectile. The acceleration due to gravity is $10\,\mathrm{m\,s^{-2}}$.

The projectile lands on the horizontal plane $7$ seconds after launch.

Find the value of $v_0$, correct to $1$ decimal place.

It is given that for positive numbers $x_1,x_2,x_3,\ldots,x_n$ with arithmetic mean $A$,
$\frac{x_1x_2x_3\cdots x_n}{A^n}\le1.$
(Do NOT prove this.)

Suppose a rectangular prism has dimensions $a,b,c$ and surface area $S$. Show that
$abc\le\left(\frac S6\right)^{3/2}.$

Using part (i), show that when the rectangular prism with surface area $S$ is a cube, it has maximum volume.

Using Euler’s theorem, show that if $z=e^{i\theta}=\cos\theta+i\sin\theta$, then
$z^n+z^{-n}=2\cos n\theta.$

Hence find
$\int_0^{\pi/2}\cos^7\theta\,d\theta.$

If
$I_n=\int_0^{\pi/2}\cos^n\theta\,d\theta,$
show that
$I_n=\frac{n-1}{n}I_{n-2}.$