Algebraic Proof (AQA GCSE Further Maths): Exam Questions

Show that ${\left(2n+3\right)}^3+n^3$ is divisible by 9 for all integer values of $n$.

$–2<a<0$ and $–1<b<1$

Tick the correct box for each statement.

If ${\left(x-a\right)}^2-q^2+p$ is positive for all values of $x$, which condition below is correct?

Circle your answer.

$x<ax>ap<q^{2}p>q^2$

Prove that the sum of the squares of three consecutive integers is always two more than a multiple of 3.

$5n-1$, $5n$ and $5n+1$ are three consecutive integers, for all integer values of $n$.

The product of the three consecutive integers is added to the middle integer.

Prove that the result is always a cube number.

Prove that  ${\left(3x+5\right)}^2-5x\left(x+10\right)⩾0$  for all values of $x$.

$a,b$ and $c$ are numbers such that

$\begin{array}{rcl}a& <& 0\\ b& >& 1\\ -1& <& c<1\end{array}$

Tick the correct box for each statement.

A function is given by $\mathrm{f}\left(t\right)=38-t\left(12-t\right)$

Prove that, for any input $t$, the function will never give a negative output.