Mathematics Extension 2

The equation of line $l_2$ is
$\frac{x+3}{k}=\frac{4-y}{6}=z-1.$
Find the value of $k$ for which $l_1$ and $l_2$ intersect.

The points $X$, $Y$ and $Z$ bisect $AB$, $BC$ and $CA$ respectively. Show that
$\overrightarrow{AY}+\overrightarrow{BZ}+\overrightarrow{CX}=\vec0.$

Let $J_n=\int_0^1x^ne^{-x}\,dx$, where $n$ is a non-negative integer. Using parts (i) and (iii), show by mathematical induction, or otherwise, that for all $n\ge0$,
$J_n=n!-\frac{n!}{e}\sum_{r=0}^{n}\frac1{r!}.$

Let $J_n=\int_0^1x^ne^{-x}\,dx$, where $n$ is a non-negative integer. Show that
$J_n=nJ_{n-1}-\frac1e,\qquad n\ge1.$

Let $J_n=\int_0^1x^ne^{-x}\,dx$, where $n$ is a non-negative integer. Show that
$J_n\le\frac1{n+1}.$

Use a suitable substitution to show that
$\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx.$

On the Argand diagram, $z$ and $zw$ are represented by $A$ and $B$. Given $z=re^{i\theta}$ and $w=e^{i\pi/3}$, where $r>0$, explain why $OAB$ is an equilateral triangle.

Simplify fully
$\frac1{1+\omega}+\frac1{1+\omega^2},$
given that $\omega$ is a non-real cube root of unity.

A particle is moving so that $v^2=n^2(a^2-x^2)$. Show that the particle is moving in simple harmonic motion.

If the particle is moving at $\sqrt{15}\,\mathrm{m/s}$ when it is $2\,\mathrm m$ to the right of the centre of motion and at $2\sqrt3\,\mathrm{m/s}$ when it is $4\,\mathrm m$ to the right of the centre of motion, find the period and amplitude of the motion.

A particle is moving so that $x=2\sin3t-2\sqrt3\cos3t$. By expressing the motion in the form $A\sin(3t-\alpha)$, find the period and amplitude.

A particle is moving so that $x=2\sin3t-2\sqrt3\cos3t$. Find the initial position and velocity of the particle.

A $20\,\mathrm{kg}$ projectile is launched upwards from the ground at $500\,\mathrm{m/s}$. It experiences a gravitational force of $200\,\mathrm N$ and air resistance, opposite to its motion, of $\frac{v^2}{18}\,\mathrm N$. Letting the ground be $x=0$, show that the acceleration is
$\ddot{x}=-10-\frac{v^2}{360}.$

Show that
$x=180\ln\left(\frac{253600}{3600+v^2}\right)$
and find the maximum height the projectile reaches.

Show that the time taken for the projectile to reach velocity $v\,\mathrm{m/s}$ is
$t=6\tan^{-1}\left(\frac{25}{3}\right)-6\tan^{-1}\left(\frac{v}{60}\right),$
and find the time taken for the projectile to reach its maximum height.

If $x\le1$ and $y\ge1$, show that $x+y\ge1+xy$.

Let
$J_n=\int_0^1x^ne^{-x}\,dx,$
where $n$ is a non-negative integer. Show that
$J_0=1-\frac1e.$

The point $P$ is defined by
$\overrightarrow{AP}=-6\overrightarrow{AB}-8\overrightarrow{AC}.$

Does $P$ lie on the line $AL$? Justify your answer.

Using part (ii), or otherwise, determine the position of $L$ by showing that
$\overrightarrow{BL}=\frac47\overrightarrow{BC}.$

Let $J_n=\int_0^1x^ne^{-x}\,dx$. Using parts (ii) and (iv), prove that
$e=\lim_{n\to\infty}\sum_{r=0}^{n}\frac1{r!}.$

The two non-parallel vectors $\vec u$ and $\vec v$ satisfy $\lambda\vec u+\mu\vec v=\vec0$ for some real numbers $\lambda$ and $\mu$. Show that $\lambda=\mu=0$.

A recurrence relation is defined by
$u_{n+1}=\frac{3u_n-1}{4u_n-1},\qquad u_1=1.$
Use mathematical induction to prove that
$u_n=\frac n{2n-1}\quad\text{for }n\ge1.$

Hence, or otherwise, evaluate
$\int_0^{\pi/4}\frac{\sin x}{\cos x+\sin x}\,dx.$

Show by finding it that the point of intersection between the lines
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\10\\5\end{pmatrix}+\lambda\begin{pmatrix}2\\-3\\1\end{pmatrix}$
and
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}15\\-1\\6\end{pmatrix}+\delta\begin{pmatrix}4\\-1\\-1\end{pmatrix}$
is $\begin{pmatrix}7\\1\\8\end{pmatrix}$.