Given
and
find and .
Hence, or otherwise, find in terms of .
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Given
and
find and .
Hence, or otherwise, find in terms of .
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Find the Cartesian equation of the curve , defined by the parametric equations
and
Hence find the equation of the tangent to at the point where .
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A particle travels along a path defined by the parametric equations
and
where are the coordinates of the particle at time seconds.
Find the coordinates of the particle after 0.2 seconds.
Find the coordinates of the particle when it is at its minimum point.
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The graph of the curve C shown below is defined by the parametric equations
and
Find the exact coordinates of point .
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The curve has parametric equations
and
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The curve has parametric equations
and
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Find an expression for in terms of for the parametric equations
The graph of y against x passes through the point P (1 , 1).
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A crane swings a wrecking ball along a two-dimensional path defined by the parametric equations
as shown in the diagram below.
x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds. Point A indicates the initial position of the wrecking ball, at time .
Find the height of the wrecking ball after 0.3 seconds.
Find the minimum height of the wrecking ball during its motion.
Find the horizontal distances from point A at the times when the wrecking ball is at a height of 2.9 m, giving your answers accurate to 1 decimal place.
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The graph of the curve C shown below is defined by the parametric equations
Find an expression for in terms of .
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The curve C has parametric equations
Find an expression, in terms of t, for .
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The curve C has parametric equations
Show that, in terms of ,
Show that the distance between the maximum and minimum points on C is square units.
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Find an expression for in terms of t for the parametric equations
Verify that the graph of x against y passes through the point (0 , 1) and find the gradient at that point.
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A crane swings a wrecking ball along a two-dimensional path defined by the parametric equations
as shown in the diagram below.
x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds. Point A indicates the initial position of the wrecking ball, at time.
Find a Cartesian equation of the curve in the form and state the domain of .
Find the difference between the maximum and minimum heights of the wrecking ball during its motion.
The crane is positioned such that point A is 7 m horizontally from the wall the wrecking ball is to destroy.
Find the height at which the wrecking ball will strike the wall.
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The graph of the curve C shown below is defined by the parametric equations
Find an expression for in terms of .
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The curve C has parametric equations
Find an expression, in terms of t, for .
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The curve has parametric equations
Show that at the point (0 , 6),and find the value of at this point.
The tangent at the point (0 , 6) is parallel to the normal at the point P.
Find the exact coordinates of point P
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A curve C has parametric equations
The tangents to C at the points R and S meet at the point T, as shown in the diagram below.
Given that the x-coordinate of both points R and S is 5, find the area of the triangle RST.
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A crane swings a wrecking ball along a two-dimensional path defined by the parametric equations
as shown in the diagram below.
x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds. Point A indicates the initial position of the wrecking ball.
The destruction of a building requires the wrecking ball to strike it at a height of 1.4 m whilst on the upward part of its path.
Find the horizontal distance from point A at which the ball hits the building.
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The graph of the ellipse E shown below is defined by the parametric equations
Find an expression for in terms of θ.
Find the equation of the tangent to E, at the point where , giving your answer in the form , where a and b are real numbers that should be given in exact form.
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The curve C has parametric equations
Find the equation of the normal to C at the point where C intersects the line .
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The graph of the curve defined by the parametric equations
is shown below.
Verify that the graph passes through the point (1 , 1).
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The diagram below shows a sketch of the curve defined by the parametric equations
The tangents to the curve that pass through the origin meet the curve at points A and B
Show that the values of t at points A and B are and .
Hence, or otherwise, show that the area of the triangle OAB is square units.
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