Mathematics Extension 2

Let $z_1$ be a complex number and let
$z_2=e^{i\pi/3}z_1.$
The diagram shows points $A$ and $B$, which represent $z_1$ and $z_2$, respectively, in the Argand plane.

(i) Explain why triangle $OAB$ is an equilateral triangle.

Let $OPQR$ be a parallelogram with $\overrightarrow{OP}=\mathbf p$ and $\overrightarrow{OR}=\mathbf r$. The point $S$ is the midpoint of $QR$ and $T$ is the intersection of $PR$ and $OS$, as shown in the diagram.

(iii) Show that
$\overrightarrow{OT}=\frac23\mathbf r+\frac13\mathbf p.$

Using parts (ii) and (iii), or otherwise, prove that $T$ is the point that divides the interval $PR$ in the ratio $2:1$.

Using the proposition $P$ from part (i), write down the contrapositive of $P$.

In the set of integers, let $P$ be the proposition: “If $k+1$ is divisible by $3$, then $k^3+1$ is divisible by $3”.

(i) Prove that the proposition $P$ is true.

Using the proposition $P$ from part (i), write down the converse of $P$ and state, with reasons, whether this converse is true or false.

The point $C$ divides the interval $AB$ so that
$\frac{CB}{AC}=\frac mn.$
The position vectors of $A$ and $B$ are $\mathbf a$ and $\mathbf b$, respectively, as shown in the diagram.

(i) Show that
$\overrightarrow{AC}=\frac n{m+n}(\mathbf b-\mathbf a).$

Using the same definitions as in part (i), prove that
$\overrightarrow{OC}=\frac m{m+n}\mathbf a+\frac n{m+n}\mathbf b.$

Given that $v<\dfrac{gm}{k}$, show that when
$t=\frac{3m}{k}\ln2,$
the velocity of the larger mass is
$\frac{gm}{2k}.$

Let
$I_n=\int_0^{\pi/2}\sin^{2n+1}(2\theta)\,d\theta,\qquad n=0,1,\ldots.$

(i) Prove that
$I_n=\frac{2n}{2n+1}I_{n-1},\qquad n\geq1.$

Using the result of part (i), deduce that
$I_n=\frac{2^{2n}(n!)^2}{(2n+1)!}.$

Let
$J_n=\int_0^1x^n(1-x)^n\,dx,\qquad n=0,1,2,\ldots.$

Using the result of part (ii), or otherwise, show that
$J_n=\frac{(n!)^2}{(2n+1)!}.$

Prove that
$\bigl(2^n n!\bigr)^2\leq(2n+1)!.$

Two masses, $2m$ kg and $4m$ kg, are attached by a light string. The string is placed over a smooth pulley as shown. The two masses are at rest before being released and $v$ is the velocity of the larger mass at time $t$ seconds after they are released. The force due to air resistance on each mass has magnitude $kv$, where $k$ is a positive constant.

(i) Show that
$\frac{dv}{dt}=\frac{gm-kv}{3m}.$