Mathematics Extension 2

Prove that if $a$ is divisible by $5$, then $a^2-5a$ is divisible by $50$.

(You may assume that the product of two consecutive numbers is even.)

Prove by contradiction that $\sqrt2$ is irrational.

Prove for all real $a$ and $b$ that
$a^2+b^2\ge 2ab.$

Hence prove for all $a>0$ that
$a+\frac1a\ge2.$

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

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 $\lambda=\mu=0$.

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

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

Hence, if $a>b>c>0$, show that
$a^2+5b^2+9c^2>9bc.$

The complex number $z$ satisfies
$\left|z-\frac4z\right|=2.$

Using the triangle inequality, or otherwise, show that
$|z|\le\sqrt5+1.$

Show that $a^2+9b^2\ge6ab$, where $a$ and $b$ are real numbers.

Hence, or otherwise, show that
$a^2+5b^2+9c^2\ge3(ab+bc+ac).$

Using part (i), show that when the rectangular prism with surface area $S$ is a cube, it has maximum volume.

It is given that for positive numbers $x_1,x_2,x_3,\ldots,x_n$ with arithmetic mean $A$,
$\frac{x_1x_2x_3\cdots x_n}{A^n}\le1.$
(Do NOT prove this.)

Suppose a rectangular prism has dimensions $a,b,c$ and surface area $S$. Show that
$abc\le\left(\frac S6\right)^{3/2}.$

Prove that
$\frac{\cos\theta+i\sin\theta-1}{\cos\theta+i\sin\theta+1}=i\tan\frac\theta2.$

Given
$S_n=\frac1{3^3}+\frac1{4^3}+\frac1{5^3}+\cdots+\frac1{n^3},\qquad n\ge3,$
use part (i), or otherwise, to prove that
$S_n<\frac1{12}.$

Consider the statement: “For all integers $n$, if $n$ is a multiple of $6$, then $n$ is a multiple of $2$.” Which is its contrapositive?

A. There exists an integer $n$ such that $n$ is a multiple of $6$ and not a multiple of $2$.
B. There exists an integer $n$ such that $n$ is a multiple of $2$ and not a multiple of $6$.
C. For all integers $n$, if $n$ is not a multiple of $2$, then $n$ is not a multiple of $6$.
D. For all integers $n$, if $n$ is not a multiple of $6$, then $n$ is not a multiple of $2$.

Which of the following statements is FALSE?

A. $\forall a,b\in\mathbb R$, $a<b\Rightarrow a^3<b^3$.
B. $\forall a,b\in\mathbb R$, $a<b\Rightarrow e^{-a}>e^{-b}$.
C. $\forall a,b\in(0,+\infty)$, $a<b\Rightarrow\ln a<\ln b$.
D. $\forall a,b\in\mathbb R$, with $a,b\ne0$, $a<b\Rightarrow\frac1a>\frac1b$.

Four cards have either RED or BLACK on one side and either WIN or TRY AGAIN on the other side. The visible cards are: Card 1 WIN; Card 2 BLACK; Card 3 TRY AGAIN; Card 4 RED.
For the statement “If a card is RED, then it has WIN written on the other side.”

What is the minimum number of cards Sam should turn over to check if the statement is true and which cards should be turned over?

A. 1 and 4
B. 3 and 4
C. 1, 2 and 4
D. 1, 3 and 4

Show that the converse of Statement A is true.

Consider Statement A: “If $n^2$ is even, then $n$ is even.” What is the converse of Statement A?

For all non-negative real numbers $x$ and $y$, $\sqrt{xy}\le\frac{x+y}{2}$. (Do NOT prove this.) Using this fact, show that for all non-negative real numbers $a,b,c$,
$\sqrt{abc}\le\frac{a^2+b^2+2c}{4}.$

Show that the triangular numbers $t_1,t_3,t_5$, and so on, are also hexagonal numbers.

Show that the triangular numbers $t_2,t_4,t_6$, and so on, are not hexagonal numbers.