Mathematics Extension 2

Using part (i), or otherwise, show that for all non-negative real numbers $a,b,c$,
$\sqrt{abc}\le\frac{a^2+b^2+c^2+a+b+c}{6}.$

Given vectors $\vec a=\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}$ and $\vec b=\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix}$, show that
$|a_1b_1+a_2b_2+a_3b_3|\le\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}.$

The point $P(x,y,z)$ lies on the sphere of radius $1$ centred at the origin $O$. Using $\overrightarrow{OP}=x\vec i+y\vec j+z\vec k$ and the triangle inequality, or otherwise, show that
$|x|+|y|+|z|\ge1.$

As in part (i), $P(x,y,z)$ lies on the unit sphere centred at the origin. Using part (ii), or otherwise, show that
$|x|+|y|+|z|\le\sqrt3.$

Consider the proposition: “If $2^n-1$ is not prime, then $n$ is not prime”. Given that each of the following statements is true, which statement disproves the proposition?

A. $2^5-1$ is prime
B. $2^6-1$ is divisible by $9$
C. $2^7-1$ is prime
D. $2^{11}-1$ is divisible by $23$

Consider the statement: “If $n$ is even, then if $n$ is a multiple of $3$, then $n$ is a multiple of $6$”. Which of the following is the negation of this statement?

A. $n$ is odd and $n$ is not a multiple of $3$ or $6$.
B. $n$ is even and $n$ is a multiple of $3$ but not a multiple of $6$.
C. If $n$ is even, then $n$ is not a multiple of $3$ and $n$ is not a multiple of $6$.
D. If $n$ is odd, then if $n$ is not a multiple of $3$ then $n$ is not a multiple of $6$.

Given $u^2>gR$, prove that there are two distinct values of $\theta$ for which the particle will land at $x=R$.

By considering the right-angled triangle below, or otherwise, prove that
$\frac{a+b}{2}\geq\sqrt{ab},$
where $a>b\geq0$.

Prove that
$p^2+4q^2\geq4pq.$

Prove that for any integer $n>1$, $\log_n(n+1)$ is irrational.

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 contrapositive of $P$.

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

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