The curve has parametric equations and .
Complete the table below.
t | -2 | -1 | 0 | 1 | 2 |
x | |||||
y |
Plot the graph of C on the axes below.
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The curve has parametric equations and .
Complete the table below.
t | -2 | -1 | 0 | 1 | 2 |
x | |||||
y |
Plot the graph of C on the axes below.
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A company’s logo is created using an arc of a circle as shown in the diagram below.
When the end points of the arc are joined to the origin, they form the minor sector of a circle with angle θ radians at the centre.
The arc is formed using the parametric equations and , with domain .
Write down the size of the angle θ.
Use the formula “l = r θ” to find the length of the arc.
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A curve C has parametric equations and .
Find a Cartesian equation for the curve C in the form .
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A curve is defined parametrically by the equations and .
Describe the shape of the curve.
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A sketch of the graph with parametric equations and is shown below.
Find the value of t when and hence write down the coordinates of the y-axis intercept.
Find the values of t when and hence write down the coordinates of the x-axis intercepts.
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A curve C is defined by the parametric equations and .
Show that the equation of the curve can be given in the Cartesian form , with .
Sketch the graph of , labelling any points where the graph intercepts the coordinate axes.
Given , write down the domain and range of .
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The curve defined parametrically by and , where is a non-zero constant, passes through the point (16 , 5).
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A curve C is defined by the parametric equations and , where .
Show that, where .
Sketch the graph of C, labelling any points where the graph intercepts the coordinate axes.
Determine the domain and range for the function .
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The curve C has parametric equations
and
Complete the table below.
t | -2 | -1 | 0 | 1 | 2 |
x | |||||
y |
Plot the graph of C on the axes below.
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A company’s logo is created using an arc of a circle as shown in the diagram below.
When the end points of the arc are joined to the origin, they form the major sector of a circle with angle radians at the centre.
The arc is formed using the parametric equations
with domain
Find the size of the angle , as well as the length of the arc used in the logo.
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A curve C has parametric equations
Find a Cartesian equation for the curve C in the form .
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A sketch of the graph with parametric equations
is shown below.
Find the value of t whenand hence write down the coordinates of the x-axis intercept.
Find the values of t whenand hence write down the coordinates of the y-axis intercepts.
Find the coordinates of the points where the graph intersects the line with equation .
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A curve C is defined by the parametric equations
Show that the equation of the curve can be given in the Cartesian form , with .
Assuming there are no restrictions on the values of t, find the domain and range of .
Sketch the graph of .
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The curve defined parametrically by
where p is a non-zero constant, passes through the point (3 , -3).
Show that at point ,.
Hence, or otherwise, find the value of p.
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A curve C is defined by the parametric equations
Show that.
Determine the ranges of x and y in the given domain of t.
Sketch the graph of C.
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The curve C has parametric equations
and
t | -3 | -2 | -1 | 0 | 1 | 2 |
x | ||||||
y |
Plot the graph of C on the axes below.
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A company’s logo is created using an arc of a circle and a straight line as shown in the diagram below.
The arc part of the logo is formed using the parametric equations
with domain -
Find the equation of the line passing through points A and B, stating values to three significant figures where appropriate.
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Find a Cartesian equation for the curve that has parametric equations
giving your answer in the form
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A curve is defined by the parametric equations
By changing these equations into Cartesian form, show that this is the equation of a circle, and determine its centre and radius.
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A sketch of the graph with parametric equations
is shown below.
Find the coordinates of all points where the graph intersects the coordinate axes.
Find the coordinates of the points where the graph intersects the line with equation .
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A curve C is defined by the parametric equations
Show that the Cartesian equation of can be written in the form , where is a function of y.
Sketch the curve C.
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The curve defined parametrically by
where is a constant, passes through the point (139 , 43).
Find the value of .
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A curve C is defined by the parametric equations
Sketch the graph of C.
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The curve C has parametric equations
and
t | -3 | -2 | -1 | -0.5 | 0 | 0.5 | 1 | 2 | 3 |
x | |||||||||
y |
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A company’s logo is a major arc of an ellipse as shown in the diagram below.
When the end points of the arc are connected to the origin, they form an elliptic sector with angle radians at the centre.
The logo is formed using the parametric equations
with domain
Find the angle , giving your answer to three significant figures.
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Find a Cartesian equation for the curve that has parametric equations
giving your answer in the form .
For the curve in part (a), explain why the domain must be restricted to.
How would your answer to part (a) change if the domain were changed to
?
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A curve is defined by the parametric equations
where a and b are non-zero constants, and p and q are constants.
In the case where and , the curve is used to represent the position of a robot at time t.
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A sketch of the graph with parametric equations
is shown below.
Find the coordinates of the points where the graph crosses the coordinate axes.
Find the coordinates of the points where the graph intersects the square with vertices at (-1 , 1), (1 , 1), (1 , -1), (-1 , -1).
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A curve C has parametric equations
Determine the ranges of x and y in the given domain of t.
Show that the Cartesian equation of C can be written in the form
Sketch the graph of C.
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The curve defined parametrically by
where is a constant, passes through the point (-22 , 0).
Find the possible values of .
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Show that can be written as .
A curve C is defined by the parametric equations
Sketch the graph of C, showing clearly the beginning and end points of the curve, as well as the points of intersection with the coordinate axes and any minimum and maximum points.
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