1.2 Inequalities & Simultaneous Equations

By eliminating  from the equations

show that

Hence solve the simultaneous equations
                   

By eliminating  from the equations

show that 

Hence solve the simultaneous equations
                 

giving  and in the form   where  and  are rational numbers.

are simultaneous equations, where  is a constant.

Given that the simultaneous equations have exactly one solution, find the value of the constant .

Find the solution to the simultaneous equations for the value of  that you found in part (a).

A firework is launched inside a large shed with a sloping roof. In relation to the horizontal distance from the point it was launched, the height of the firework, can be modelled by the quadratic equation

The sloping roof of the shed can be modelled with the equation

Determine whether, according to the model, the firework will hit the roof of the shed before escaping out the open end of the shed on the right of the diagram.

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.

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?

By eliminating from the equations

show that .

Hence solve the simultaneous equations
                

By eliminating from the equations

                 
show that .

Hence solve the simultaneous equations
                

giving  and  in the form , where and are integers.

are simultaneous equations, where  is a constant.

By eliminating  from the equations show that .

By considering the discriminant of find the value of  for which the simultaneous equations have only one solution.

Find the solution to the simultaneous equations for the value of  that you found in part (b).

Solve the inequality .

The equation , where  is a constant, has two distinct real roots.

Find the possible value(s) of k.

Find the values of  that satisfy the inequalities

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.

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?

Solve the simultaneous equations

By eliminating  from the equations
                 

show that  

Hence solve the simultaneous equations

giving and  in the form where  and  are rational numbers and  is a prime number.

are simultaneous equations, where  is a constant.

Find the respective sets of values for for which the simultaneous equations have one, two, and no solutions.

Given that the simultaneous equations have exactly one solution, find all possible pairs  that might correspond to that solution. Give all your values for  and in the form where  is a rational number.

The goal in a video game is to have a unicorn leap as far as possible in a horizontal direction without being destroyed by the death ray that is being fired overhead.  You hack into the game code and find that the height of the unicorn, , is being modelled in relation to the horizontal distance from the point it jumps by the quadratic equation where   is a parameter that can be controlled by the player’s actions, and  is the horizontal distance in metres.  You also find that the path of the death ray is being modelled by the equation

The value of  can never be less than zero, and if the path of the unicorn crosses or touches the path of the death ray, the unicorn is considered to have been destroyed.

Ignoring the problem of the death ray, explain why the parameter  represents the horizontal distance leapt by the unicorn.

Your friend’s personal best in the game is a leap of 21.5 m without the unicorn being destroyed. He is determined to keep playing until his unicorn has leapt 22 m safely.  Determine whether or not your friend has a chance of reaching this goal.

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.

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?