Inequalities (AQA A Level Maths: Pure): Exam Questions

The cross section of a tunnel is in the shape of the region defined by the inequalities

$y\le 5-\frac{x^2}{5}$

$y\ge 0$

where $x$ and$y$ are measured in metres.

Sketch the region on the axes below and label it $R$.

Write down the maximum height and maximum width of the tunnel.

The equation

$x^2+kx+4=0$

where $k$ is a constant, has no real solutions.

Find the possible value(s) of $k$.

A stone is fired vertically upwards from ground level.

The vertical distance above the ground, $d$ metres at time $t$ seconds after launch, is given by

$d\left(t\right)=12t-4.9t^2$

Find the length of time that the stone spends at a height greater than $2$ metres above the ground.

Give your answer to 3 significant figures.

On the axes below sketch the region defined by the inequalities

$y+x>x^2$
$5y<20-4x$
$y-1\ge 0$

Label your region $R$.

Using algebra, solve the inequality

$x^2-5x>6$

Find the three inequalities that define the region $R$ shown in the diagram below.

On the axes below sketch the region defined by the inequalities

$x+2y>3$
$y\le x+4$
$y+3x<8$

Label your region $R$.

Given that

$x^2+3x>4$

and that

$4x+1>4$

find the possible values of $x$.

The equation

$k{x}^2+2kx+4=0$

where $k$ is a constant has two distinct real roots.

Find the possible values of $k$.

Using algebra, solve the inequality

${(x+2)}^2>5$

Use algebra to solve the inequality

$\frac{5}{3x^2+2}\le 2$

The equation

$(kx{)}^2+(k-2)x+1=0$

where $k$ is a constant, has two distinct real roots. 

Find the possible values of $k$.

Find the values of $x$ that satisfy both of the inequalities below.

$x^2+4x-3\le 2-x^2-5x$

$8-2x^2\le 2x(2x+1)$

Give your answer in set notation.

An electronics company produces cables.

The company sells cables individually for $(40-c)$ pence each.

Write down an expression, in terms of $c$, for the total income (in pence) made from selling $c$ cables.

The cost to the company of making $c$ cables is $(200+10c)$ pence.

By forming and solving a suitable inequality, find the minimum number of cables the company must sell in order to make a profit.

Find the values of $t$ that satisfy both of the inequalities below.

$t^2-2t-15<0$

$t^2+14\le 9t$

Use algebra to solve the inequality

$\frac{4x^2-11}{(x+1{)}^2}\ge 4$

On the axes below sketch the region defined by the inequalities

$x^2-9\le y$

$y\le (2+x)(2-x)$

Label this region $R$.

Use algebra to solve the inequality

$-6\le x^2+3x-4\le 6$

writing your answer in set notation.

Find the three inequalities that define the region $R$ shown in the diagram below.

The equation 

$(k+1)t^2+2(k+2)t=3(k+3)$

has real roots.

Find the possible values of $k$.

A stone is projected vertically upwards from an initial height of $2$ metres above the ground.

• The vertical height of the stone above its initial position, $d_1$ metres, at time $t$ seconds after launch, is given by

 $d_1\left(t\right)=13.2t-4.9t^2$

At the same time, a second stone is projected vertically upwards from an initial height of $2.3$ metres above the ground.

• The vertical height of the second stone above its initial position, $d_2$ metres, at time $t$ seconds after launch, is given by

 $d_2\left(t\right)=13t-4.9t^2$

Find the length of time during which both stones are greater than $4$ metres above the ground.

Give your answer to 3 significant figures.

A company produces $x$ chairs and$y$ tables in a day. 

They sell every chair and every table they produce. 

Due to the manufacturing processes involved, the number of chairs and tables they can make in a day are limited by the following five inequalities:

$y\le x+20$

$y\ge 3x-45$

$y\le -2x+80$

$x\ge 0$

$y\ge 0$

Explain the significance of the inequalities $x\ge 0$ and $y\ge 0$ in the context.

On the axes below, sketch the region within which the company can produce $x$ chairs and$y$ tables per day.

The company’s profit, $\pounds P$ per day, is given by the formula

$P=3x+2y$

A financial advisor tells the company that their maximum profit is found when $x$ and $y$ lie on a vertex of the region found in part (b), but did not say which vertex.

Use this information to find the number of chairs and tables the company should make in order to maximise its daily profit.

Show your working clearly.