Further Parametric Equations (A Level only) (OCR A Level Maths: Pure)

Find an expression for  in terms of for the parametric equations

The graph of  y  against  x  passes through the point P (1 , 1).

The graph defined by the parametric equations

is shown below.

The point where has coordinates (1 , 1).

The point where has coordinates (8 , 7).

Find the values of and .

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.

The graph of the curve C shown below is defined by the parametric equations

Write down the value of    at the point (0 , 6).

Write down the value of  at the points (-3 , 3) and (3 , 3).

Find an expression for   in terms of .

The curve C has parametric equations

Find an expression, in terms of t, for  .

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.

A company logo is in the shape of the symbol for infinity () as shown on the graph below.

The company wishes to produce a sign of its logo and requires it to be painted, as indicated by the shading in the diagram.

The graph of the logo is defined by the parametric equations

where x and y are measured in metres.

Show that when ,   ,  and that when  ,   .

Find the coordinates of the point on the graph corresponding to .

A model car travels on a model track along the path of the curve shown in the diagram below.  The curve is defined by the parametric equations

where x and y are, respectively, the horizontal and vertical displacements in metres from the origin O, and t is the time in seconds.

Verify that the starting position of the model car is at the origin, and find the position of the car at the times ,  and  seconds.

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.

The graph defined by the parametric equations

is shown below.

The point where has coordinates (-1 , 0).

The point where has coordinates (4 , 1).

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.

The graph of the curve C shown below is defined by the parametric equations

Find an expression for    in terms of  .

The curve C has parametric equations

Find an expression, in terms of  t, for  .

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

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.

A model car travels on a model track along the path of the curve shown in the diagram below.  The curve is defined by the parametric equations

       0

where x and y are, respectively, the horizontal and vertical displacements in metres from the origin O, and t is the time in seconds.

Find the times during the first lap at which the model car is at a “crossroads” – indicated by points P and Q on the graph.

Find the speed of the model car at the start of the final lap.

The shaded area in the diagram below is bounded on three of its sides by the -axis, the -axis, and the line  .  On the remaining side, the boundary is defined by the parametric equations

Show that the shaded area is not a trapezium.

In your work, you may use without proof the result

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.

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.

The curve C has parametric equations

Find the equation of the normal to C at the point where C intersects the line  .

The graph of the curve defined by the parametric equations

is shown below.

Prove that the line with equation is not the normal to the curve at the point (1 , 1).

The diagram below shows a sketch of the curve defined by the parametric equations

The four tangents from part (a) create a rectangle around the curve as shown below.

Find the percentage of the area of the rectangle enclosed by the curve

(the shaded area on the diagram).

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.

A model car travels on a model track along the path of the curve shown in the diagram below.  The curve is defined by the parametric equations

where x and y are, respectively, the horizontal and vertical displacements in metres from the origin O, and t is the time in seconds.

A second track is to be constructed within the central area of the original track, indicated by shading on the graph above.
The design for the second track requires an area of at least 1.25 m².

Determine if there is sufficient room for the second track to be built within the central area of the original track.

In your work, you may use without proof the result 

where  c  is a constant of integration.