Calculus & Modelling with Parametric Equations (AQA A Level Maths: Pure): Exam Questions

A particle travels along a curve with parametric equations

where the coordinates  give the position of the particle after time seconds.

Find the coordinates of the position of the particle after 0.2 seconds.

Find an expression for  in terms of .

Find the coordinates of the position of the particle when it is at the minimum point on the curve.

Find an expression for  in terms of for the curve with parametric equations

The graph of  against  passes through the point with coordinates .

Show that is a stationary point.

The graph shows the curve with parametric equations

• The point where has coordinates

• The point where has coordinates

Find the values of and .

Hence find the exact area of the shaded region.

The graph of the curve with parametric equations

is shown in the figure below.

(i) Write down the value of    at the point

(ii) Write down the value(s) of at the points and

Find an expression for   in terms of .

Hence show that the equation of the tangent to at the point where  is

The curve has parametric equations

Find an expression for  in terms of .

Hence find the equation of the normal to  at the point with coordinate .

Give your answer in the form .

A company logo, in the shape of the symbol for infinity (), is printed on a flag, as shown below.

The curve has parametric equations

where and are measured in metres.

(i) Find the values of at the points where  and

(ii) Find the coordinates of the point on the curve where

(i) Show that the total area of the logo is given by

(ii) Hence find the total area of the logo.

The curve has parametric equations

Show that the distance between the maximum point and the minimum point on is 

The graph of the curve with parametric equations

is shown in the figure below.

Find the equation of the tangent to  at the point where  .

Give your answer in the form .

The curve has parametric equations

Find the equation of the normal to  at the point where .

Given your answer in the form .

A crane swings a wrecking ball along a two-dimensional path modelled by the parametric equations

as shown in the figure below.

The variables and are the horizontal and vertical displacements, in metres, from the origin, , and is the time in seconds.

The wrecking ball is initially released from the point .

(i) Find the vertical height of the wrecking ball when it is at the point .

(ii) Find the shortest distance between the wrecking ball and the horizontal ground during its motion.

The crane is positioned such that the wrecking ball hits a building at a vertical height of 1.4 metres above the ground, on the upwards part of the swing.

Find the horizontal distance from to the building.

The ellipse , shown in the figure below, has parametric equations

Find the equation of the tangent to at the point where .

Given your answer in the form , where and are exact real numbers to be found.

The curve has parametric equations

The tangents to  at the points and meet at the point , as shown in the figure below.

Given that the -coordinate of both points and is 5, find the area of the triangle .

The curve has parametric equations

The tangent at the point on is parallel to the normal at the point on .

Find the exact coordinates of the point .

The curve has parametric equations

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

Give your answer in the form .

The graph of the curve with parametric equations

is shown in the figure below.

(i) Show that the graph passes through the point with coordinates .

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

The graph of the curve with parametric equations

is shown in the figure below.

The two tangents to that pass through the origin, , touch at the points and (not shown on the diagram).

Find the values of at and .

Hence show that the area of triangle is 

A model car travels around a track that follows the curve with parametric equations

where and are the horizontal and vertical displacements, in metres, from the origin , at time t seconds.

 (i) Write down the coordinates of the starting position of the model car.

(ii) Indicate on the graph the direction in which the model car travels.

(iii) How many laps of the track does the model car complete?

A second track is to be constructed within the central area of the original track, indicated by the shaded region.

The design for the second track requires a minimum area of 1.25 m².

Use algebraic integration to determine whether there is sufficient room for the second track to be built within the central area of the original track.

In your calculations, you may use without proof the result that

where  is a constant.