Mathematics Extension 2

By considering the roots of $z^3=4\sqrt{2}(1+i)$, show that

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

Sketch the intersection of the following:

Use De Moivre’s Theorem to solve the equation

The only force acting on a particle moving in a straight line is a resistance $m\lambda(c+v)$ acting in the same line. The mass of the particle is $m$, its velocity is $v$, and $\lambda$ and $c$ are positive constants. The particle starts to move with velocity $u>0$ and comes to rest after $T$ seconds. After half the time has elapsed, the particle's velocity is a quarter of its initial velocity.

Show that

A particle is moving in simple harmonic motion with centre around the origin, starting at $x=m$, where $m>0$. The displacement equation is given by $x=a\cos(nt+\alpha)$. After 1 second, the particle is at $x=r$, where $r>m$ and after another second, it returns to $x=r$.

Show that

Given $a$, $b$, $c$ are positive real numbers, hence or otherwise, prove that

Given $a$, $b$, $c$ are positive real numbers, hence or otherwise, prove that

Given $a$, $b$, $c$ are positive real numbers, prove that

Prove by contradiction that there are no rational solutions to the equation

$z_1$ and $z_2$ are two complex numbers representing the two points $A$ and $B$ in the Argand diagram. $z_3$ is a complex number representing the point $C$ such that $|AB|:|AC|=1:4$. $z_4$ is a complex number representing the point $D$, such that $|OB|:|OD|=1:k$, for some constant $k$ and $O$ is the origin. The points $A$, $B$ and $C$ are collinear.

Find $z_3$ in terms of $z_1$ and $z_2$.

A particle of unit mass is moving in horizontal motion, subject to a resistance force of $v^2+v^3$, where $v$ is the object’s velocity. The particle has initial velocity $v_0$, where $v_0>0$.

Show that the time $T$ taken to travel the distance $s$ is

$z_1$ and $z_2$ are two complex numbers representing the two points $A$ and $B$ in the Argand diagram. $z_3$ is a complex number representing the point $C$ such that $|AB|:|AC|=1:4$. $z_4$ is a complex number representing the point $D$, such that $|OB|:|OD|=1:k$, for some constant $k$ and $O$ is the origin. The points $A$, $B$ and $C$ are collinear.

Given that $z_2-z_1$ and $z_4-z_3$ are perpendicular, prove that

where $\theta$ is the angle between $z_1$ and $z_2$.

A particle of unit mass is moving in horizontal motion, subject to a resistance force of $v^2+v^3$, where $v$ is the object’s velocity. The particle has initial velocity $v_0$, where $v_0>0$.

Find the distance $s$ travelled by the particle when its velocity is

A particle of unit mass is moving in horizontal motion, subject to a resistance force of $v^2+v^3$, where $v$ is the object’s velocity. The particle has initial velocity $v_0$, where $v_0>0$.

Show that if the particle starts at the origin, then

satisfies the equation of motion.

A particle of unit mass is thrown vertically downwards with an initial velocity of $v_0$. It experiences a resistive force of magnitude $kv^2$, where $v$ is its velocity. Taking downwards as the positive direction, the equation of motion of the particle is given by

Let $V$ be the terminal velocity of the particle.

Explain why

A particle of unit mass is thrown vertically downwards with an initial velocity of $v_0$. It experiences a resistive force of magnitude $kv^2$, where $v$ is its velocity. Taking downwards as the positive direction, the equation of motion of the particle is given by

Let $V$ be the terminal velocity of the particle.

Show that

What is the unit vector in the same direction as $\vec a=2\mathbf i+\mathbf j-3\mathbf k$?

A. $-\frac1{14}\begin{bmatrix}2\\1\\-3\end{bmatrix}$ B. C. D.

The converse of “if it miaows it’s a cat” is

A. If it’s a cat it miaows.

B. If it’s not a cat it doesn’t miaow.

C. If it doesn’t miaow it isn’t a cat.

D. If it’s a cat it doesn’t miaow.

Let $R$ be the region in the complex plane defined by $1<\operatorname{Re}(z)\le3$ and $\frac{\pi}{6}\le\operatorname{Arg}(z)<\frac{\pi}{3}$. Which diagram best represents $R$?

The contrapositive of “if it miaows it’s a cat” is

A. If it’s a cat it miaows.

B. If it’s not a cat it doesn’t miaow.

C. If it doesn’t miaow it isn’t a cat.

D. If it’s a cat it doesn’t miaow.

Given that $z^n=a+bi$ and $|z|=1$, where $a$ and $b$ are real, which of the following is equal to $z^{-n}$?

A. $a+bi$

B. $a-bi$

C. $-a+bi$

D. $-a-bi$

The following proof aims to establish that $-4=0$.

Let $a=-4$.

$\Rightarrow a^2=16$ and $4a+4=-12$ (Line 1)

$\Rightarrow a^2+4a+4=4$ (Line 2)

$\Rightarrow (a+2)^2=2^2$ (Line 3)

$\Rightarrow a+2=2$ (Line 4)

$\Rightarrow a=0$

At which line is the implication incorrect?

A. Line 1
B. Line 2
C. Line 3
D. Line 4

Consider the statement:

“If $z$ is even, then $x$ and $y$ are either both even or both odd.”

Which of the following is equivalent?

A. If $x$ and $y$ are either both even or both odd, then $z$ is even.

B. If $z$ is odd, then exactly one of $x$ and $y$ is even.

C. If $x$ and $y$ are either both even or both odd, then $z$ is odd.

D. If exactly one of $x$ and $y$ is even, then $z$ is odd.