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.

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

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

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

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

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

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

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”.

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

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

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

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

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 is a positive constant.

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