Proof (OCR AS Maths A: Pure): Exam Questions

Factorise $n^2+3n+2$.

Determine whether $n^3+3n^2+2n$ is odd or even, where $n$ is a natural number.

Explain your answer clearly.

Use algebra to prove that the sum of any three consecutive even numbers is a multiple of 6.

Use algebra to prove that the square of an odd number is always odd.

Let $m$ be a natural number in the range $5<m<10$.

Use proof by exhaustion to show that all possible values of $m^2$ differ from a multiple of 5 by 1.

"All integers of the form $2^n-1$, where $n$ is a positive non-square integer less than 10, are prime."

Disprove this statement by means of a counter example.

Prove that

• if $n$ is odd, then $n^3+6n^2+8n$ is odd,

• if $n$ is even, then $n^3+6n^2+8n$ is even.

Two non-zero rational numbers, $a$ and $b$, are given by $a=\frac{m}{n}$ and $b=\frac{p}{q}$ where $m$, $n$, $p$ and $q$ are non-zero integers with no common factors.

Determine whether

(i) $ab$ is rational

(ii) $\frac{a}{b}$ is rational

A function is given by

$\mathrm{f}\left(x\right)=\frac{9x^2+12x+4}{5}$

Show that $\mathrm{f}\left(x\right)\ge 0$ for all real values of $x$.

Use algebra to prove that the positive difference between a positive integer and its cube is the product of three consecutive integers.

Use algebra to prove that the sum of two rational numbers is rational.