Write down the solutions to .
Sketch the graph of , clearly showing the coordinates of the points where the graph intercepts the -axis.
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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.
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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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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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The equation , where is a constant, has no real roots.
Find the possible value(s) of k.
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Substitute into the equation in order to solve the equations simultaneously.
Clearly state which values of correspond to which values of from your solutions.
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Solve the simultaneous equations
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Solve the simultaneous equations
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Solve the simultaneous equations
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By eliminating from the equations
show that .
Hence solve the simultaneous equations
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By eliminating from the equations
show that .
Hence solve the simultaneous equations
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By eliminating from the equations
show that .
Hence solve the simultaneous equations
giving and in the form , where and are integers.
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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).
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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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Find the values of that satisfy the inequalities
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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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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 simultaneous equations
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By eliminating from the equations
show that
Hence solve the simultaneous equations
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By eliminating from the equations
show that
Hence solve the simultaneous equations
giving and in the form where and are rational numbers.
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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).
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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.
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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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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 equations
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Solve the simultaneous equations
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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.
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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.
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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.
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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.
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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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