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

Solve the inequality $3x+4\le 5(x-1)$.

Solve the inequality $x^2-5x>6$.

The equation $k{x}^2+2kx+4=0$, where $k$ is a constant, has two distinct real roots.

Find the possible value(s) of k.

On the axes below show the region satisfied by the inequalities

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

Label this region R.

Find the values of $x$ that satisfy the inequalities

$x^2+3x>4$
$4x+1>4$

Solve the inequality $-2\le 3x-4\le 5$, giving your answer in set notation.

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

On the axes below show the region satisfying the inequalities

Given that $x$ and$y$ are in metres write down the height and the maximum width of the tunnel.

Write down the inequalities that define the region R shown in the diagram below.

The total cost to a company manufacturing $c$ cables is $(100+5c)$ pence.

The total income from selling all $c$ cables is $(30c-c^2)$ pence.

What is the minimum number of cables the company needs to sell in order to recover their costs?

A stone is projected vertically upwards from ground level.

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

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

How long does the stone remain $2$ m above the ground?

Solve the inequality ${(x+2)}^2>5$.

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.

On the axes below show the region satisfied by the inequalities

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

Label this region R.

Find the values of $x$ that satisfy the inequalities

$x^2+x<2$
$x^2<4$

Solve the inequality $-2\le x^2-4\le 5$.

Find the values of $x$ that satisfy the inequalities

$x^2+4x-3\le 2-x^2-5x$
$8-2x^2\le 2x(2x+1)$

Give your answer in set notation.

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

On the axes below show the region satisfying the inequalities

Given that $x$ and$y$ are in metres, write down the height and the maximum width of the tunnel.

Find the area of the cross-section of the tunnel.

Write down the inequalities that define the region R shown in the diagram below.

An electronics company can produce $c$ cables at a total cost of $(200+10c)$ pence.
The cables can be sold for $(40-c)$ pence each.

Show that the total income from selling $c$ cables is $(40c-c^2)$ pence

What is the minimum number of cables the company needs to sell in order to make a profit?

A stone is projected vertically upwards from a height of $1.5$ m. It’s height, above its starting position, $d$ m at time $t$ seconds after launch, is given by

$d\left(t\right)=16t-4.9t^2$

How long does the stone remain $3$ m above the ground?

Solve the simultaneous inequalities

$t^2-2t-15<0$ and
$t^2+14\le 9t$.

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

The equation $(k+1)t^2+2(k+2)t=3(k+3)$ has real roots.

Find the possible values of $k$.

On the axes below show the region satisfied by the inequalities

$x^2-9\le y$
$y\le (2+x)(2-x)$

Label this region R.

Write down the equation(s) of any line(s) of symmetry of the region R.

Solve the inequality $-6\le x^2+3x-4\le 6$, giving your answer in set notation.

Solve the inequality $2x^2+1\le x^2+10x-8<2x^2-7x+52$, giving your answer in interval notation.

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

On the axes below show the region satisfying the inequalities

Given that $x$ and $y$ are in metres, write down the height and the maximum width of the tunnel.

Using a semi-circle of radius 6, estimate the area of the cross-section of the tunnel.

Given that the tunnel is to be 20 m in length estimate the volume of earth that will need to be removed in order to build the tunnel.

Write down the inequalities that define the region R shown in the diagram below.

An electronics company can produce $c$ cables at a total cost of $(160+12c)$ pence.
The cables can then be sold for $(38-c)$ pence each.

Find the minimum and maximum number of cables the company needs to sell in order to make a profit?

How many cables does the company need to sell to make the maximum profit?

A stone is projected vertically upwards from a height of $2$ m.
It’s height, above it’s starting position, $d_1$ m, 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 upwards from a height of $2.3$ m.
It’s height, above its starting position, is given by

 $d_2\left(t\right)=13t-4.9t^2$

For how long are both stones simultaneously at least $4$ m above the ground?

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 inequalities:

Briefly explain why the inequalities $x\ge 0$ and $y\ge 0$ are appropriate.

On the axes below show 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 $P=3x+2y$. Given that the maximum profit lies on a vertex of the region found in part (b), find the number of chairs and tables the company should make in order to maximise its daily profit.