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

In this question you should show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Using algebra, solve the inequality

writing your answer in set notation.

Solve the inequality .

Solve the inequality .

The equation , where  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

Label this region R.

Find the values of  that satisfy the inequalities

Solve the inequality , 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  and 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  cables is  pence.

The total income from selling all  cables is  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, m at  seconds after launch, is given by

How long does the stone remain m above the ground?

Given that and are integers such that

show that

Figure 3 shows a sketch of a curve and a straight line .

Given that

• has equation where is a quadratic expression in

• cuts the -axis at 0 and 6

• cuts the -axis at 60 and intersects at the point

use inequalities to define the region shown shaded in Figure 3.

Solve the inequality .

Solve the inequality .

The equation , where  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

Label this region R.

Find the values of  that satisfy the inequalities

Solve the inequality .

Find the values of  that satisfy the inequalities

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  and 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  cables at a total cost of  pence.
The cables can be sold for  pence each.

Show that the total income from selling  cables is  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 m.
It’s height, above its starting position, m at time  seconds after launch, is given by

How long does the stone remain m above the ground?

Figure 4 shows a sketch of the graph of , where

Find all the values of for which

Figure 1 shows a sketch of a curve with equation and a straight line .

The curve meets at the points (-2, 13) and (0, 25) as shown.

The shaded region is bounded by and as shown in Figure 1.

Given that

• is a quadratic function in

• (-2, 13) is the minimum turning point of

use inequalities to define .

Solve the simultaneous inequalities

and
.

Solve the inequality .

The equation  has real roots.

Find the possible values of .

On the axes below show the region satisfied by the inequalities

Label this region R.

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

Solve the inequality , giving your answer in set notation.

Solve the inequality , 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  and  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  cables at a total cost of  pence.
The cables can then be sold for  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  m.
It’s height, above it’s starting position, m, at time  seconds after launch, is given by

At the same time a second stone is projected upwards from a height of m.
It’s height, above its starting position, is given by

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

A company produces chairs and 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  and  are appropriate.

On the axes below show the region within which the company can produce  chairs and tables per day.

The company’s profit, , per day, is given by the formula .
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.