Solve the inequalities:
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Solve the inequalities:
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Solve the inequalities:
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Write down the solutions to .
Sketch the graph of , clearly showing the coordinates of the points where the graph intercepts the -axis.
Hence, or otherwise, solve the inequality .
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Find the discriminant for the quadratic function .
Write down the number of real solutions to the equation .
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On the axes below, show the region bounded by the inequalities
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Sketch the graph of and hence solve the inequality .
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Write down, in terms of , the discriminant of .
Hence find the values of for which the equation has two real and distinct solutions.
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Write down the three inequalities that define the region shown in the diagram below.
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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?
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The equation , where is a constant, has no real roots.
Find the possible value(s) of k.
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Solve the inequality , giving your answer in set notation.
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Solve the inequality .
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Solve the inequality .
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Solve the inequality .
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The equation , where is a constant, has two distinct real roots.
Find the possible value(s) of k.
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On the axes below show the region satisfied by the inequalities
Label this region R.
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Find the values of that satisfy the inequalities
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Solve the inequality , giving your answer in set notation.
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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.
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Write down the inequalities that define the region R shown in the diagram below.
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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?
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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?
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Solve the inequality .
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Solve the inequality .
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The equation , where is a constant, has two distinct real roots. Find the possible values of k.
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On the axes below show the region satisfied by the inequalities
Label this region R.
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Find the values of that satisfy the inequalities
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Solve the inequality .
Find the values of that satisfy the inequalities
Give your answer in set notation.
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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.
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Write down the inequalities that define the region R shown in the diagram below.
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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?
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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?
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Solve the simultaneous inequalities
and
.
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Solve the inequality .
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The equation has real roots.
Find the possible values of .
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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.
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Solve the inequality , giving your answer in set notation.
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Solve the inequality , giving your answer in interval notation.
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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.
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Write down the inequalities that define the region R shown in the diagram below.
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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?
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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?
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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.
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