Algebra Toolkit (AQA GCSE Further Maths): Exam Questions

Factorise fully $6x^2+26xy–20y^2$

Write $2x^2-16x+13$ in the form $a{(x+b)}^2+c$ where $a,b$ and $c$ are integers.

Factorise fully  $12p{q}^{3}r–18p{q}^2{r}^2+24p{q}^{2}r$

The radius of a sphere, in cm, is $\frac{3k}{2}$

The volume of the sphere, in cm³, is $972\pi$

Work out the value of $k$.

Factorise fully ${\left(p+6\right)}^{11}-{\left(p+6\right)}^{10}$

The diagram shows a solid hemisphere.

The diameter is $12a$ cm

The volume is $486\pi$ cm³

Work out the value of $a$.

Show that $\left(x+1\right)\left(x+3\right)\left(x+4\right)-x\left(x^2+7x+11\right)$

can be written in the form $(x+a)(x+b)$ where $a$ and $b$ are positive integers.

$P=4x$ and $Q=7x$

$P$ increases by 25%
$Q$ decreases by 40%

Now, $P$ is 28 greater than $Q$.

Work out the value of $x$.

Factorise fully  $6{(y+3)}^5+4{(y+3)}^4$

Give your answer in its simplest form.

Do not attempt to expand ${(y+3)}^5\mathrm{or}{(y+3)}^4$

$A=2-5xB=3x-1C=x^2$

Show that ${(2A+3B)}^2\equiv A+B+C$

You are given that $x^2+6x+2\equiv {\left(x+h\right)}^2+k$

Work out the values of $h$ and $k$.

$h$ = .....................

$k$ = .....................

Write down the coordinates of the minimum point on the curve $y=x^2+6x+2$

Solve the equation $x^2+6x+2=0$

Give your answers in the form $a\pm \sqrt{b}$

Factorise fully $x^4-81$

The $x^2$ term in the expansion of $\left(3x+4\right)\left(x^2+px+5\right)$ is $-23x^2$

Work out the value of $p$.

A cylinder has base radius$x$cm and height $y$ cm

A hemisphere has radius $6y$ cm

The cylinder and hemisphere have equal volumes.

Work out the value of $\frac{x}{y}$

You must show your working.

$3x^2+2bx+8a$ can be written in the form $3{(x+a)}^2+b+2$

Work out the two possible pairs of values of $a$ and $b$.

$$\begin{gathered} a=..........b=.......... \\ a=..........b=.......... \end{gathered}$$

By multiplying both sides of the equation by $x^{\frac{1}{2}}$

Solve   $2x^{\frac{3}{2}}-3x^{\frac{1}{2}}=7x^{-\frac{1}{2}}$    for  $x>0$

Give your answer to 3 significant figures.

You must show your working.

Solve $4{\left(x-5\right)}^2=k^2$ where $k$ is a constant.

Give your answers in their simplest form in terms of $k$.

A cone has base radius $r$ cm, perpendicular height $h$ cm and slant height $l$ cm

The curved surface area is $60\pi$ cm²

$l=3r$

Work out the value of $h$.

Give your answer in the form  $a\sqrt{10}$ where $a$ is an integer greater than 1 You must show your working.