Mathematics Extension 2

Solve the quadratic equation $z^2-3z+4=0$, where $z$ is a complex number. Give your answers in Cartesian form.

In the diagram below, $OABCD$ is a solid figure where $|\overrightarrow{OA}|=|\overrightarrow{OB}|=4$ units and $|\overrightarrow{OD}|=3$ units. The edge $\overrightarrow{OD}$ is vertical, $\overrightarrow{DC}$ is parallel to $\overrightarrow{OB}$ and $|\overrightarrow{DC}|=1$ unit. The base, $OAB$, is horizontal and $\angle AOB=90^\circ$. Unit vectors $\vec i$, $\vec j$ and $\vec k$ are parallel to $\overrightarrow{OA}$, $\overrightarrow{OB}$ and $\overrightarrow{OD}$ respectively. The midpoint of $\overline{AB}$ is $M$ and the point $N$ on $\overline{BC}$ is such that $\overrightarrow{NC}=2\overrightarrow{BN}$.

Express vectors $\overrightarrow{MD}$ and $\overrightarrow{ON}$ in terms of , and .

A particle moves in simple harmonic motion described by $\ddot x=-9(x-4)$. Find the period and the central point of motion.

Given $a\in\mathbb{Z}$, prove that if $3a^2-4a+5$ is even, then $a$ is odd.

Find the angle between the vectors $\mathbf a=\mathbf i+2\mathbf j-3\mathbf k$ and $\mathbf b=-\mathbf i+4\mathbf j+2\mathbf k$, giving your answer to the nearest degree.

A polygonal number is an integer which can be represented as a series of dots arranged
in the shape of a regular polygon. Triangular numbers, square numbers and pentagonal
numbers are examples of polygonal numbers.

For an $r$-sided regular polygon, where $r\in\mathbb{Z}^+$, $r\geq 3$, the $n$th polygonal number $P_r(n)$
is given by

where $n\in\mathbb{Z}^+$. Hence, the $n$th triangular number can be expressed as

The $n$th polygonal number, $P_r(n)$, can be represented by the series

where $r\in\mathbb{Z}^+$, $r\ge 3$.

Use mathematical induction to prove that

where $n\in\mathbb{Z}^+$.

Hence or otherwise, evaluate

Let $\alpha$ be a real number and suppose that $z$ is a complex number such that

You may assume that

for all positive integer $n$. Let

Hence or otherwise, find all solutions of

The complex number $2+i$ is a zero of $P(z)=z^4-3z^3+cz^2+dz-30$, where $c,d\in\mathbb R$. Explain why $2-i$ is also a zero of $P(z)$.

Let

and

for $n=0,1,2,3,\ldots$.

Show that $I_n=1-nI_{n-1}$ for $n=1,2,3,\dots$.

Let

and

for $n=0,1,2,3,\ldots$.

Show that

for $n=0,1,2,3,\ldots$.

Using the same inclined-plane model, given that the object is initially at rest, find its velocity $\mathbf v(t)$ in terms of $g$, $\theta$, $t$ and $\mathbf i$.

Let $\alpha$ be a real number and suppose that $z$ is a complex number such that

You may assume that

for all positive integer $n$. Let

Show that

Prove that $\sqrt{23}$ is irrational.

Prove that for all real numbers $x$ and $y$, where $x^2+y^2\ne0$, $\frac{(x+y)^2}{x^2+y^2}\le2.$

Let $\alpha$ be a real number and suppose that $z$ is a complex number such that

You may assume that

for all positive integer $n$. Let

Find the ninth roots of unity.

Find the cube roots of $2-2i$. Give your answer in exponential form.

An object of mass $m$ slides down a smooth inclined plane. The plane makes an angle $\theta$ with the horizontal. The normal reaction is $\mathbf R$, gravity is $\mathbf g$ with magnitude $g$, and $\mathbf i,\mathbf j$ are unit vectors parallel and perpendicular to the plane. Show that the resultant force is $\mathbf F=-(mg\sin\theta)\mathbf i.$

Show that

By making the substitution $u=a-x$, show that

Let

and

for $n=0,1,2,3,\ldots$.

Hence find the value of $J_3$ in simplest exact form.

Another line, $\ell_2$, is defined by the vector equation

Find the possible values of $a$ when the angle between $\ell_1$ and $\ell_2$ is $\frac{\pi}{4}$.

Find real numbers $a$, $b$ and $c$ such that

Use De Moivre’s Theorem to solve the equation