Parametric Equations (Edexcel A Level Maths: Pure): Exam Questions

A curve has parametric equations

Show that all points on satisfy

A sketch of the curve with parametric equations

is shown below.

Find the coordinates of the point(s) at which the curve intersects the line .

The curve has parametric equations

(i) Complete the table below, giving values to 3 significant figures where appropriate.

(ii) Plot the graph of  on the axes below.

A curve has parametric equations

By finding a Cartesian equation for , describe the shape of .

The graph of has parametric equations

Show that all points on satisfy

Given that , explain why all points on must satisfy

Sketch .

The curve has parametric equations

where is a non-zero constant.

Given that passes through the point , find the value of .

Hence, or otherwise, find the value of p.

The curve has parametric equations

Show that all points on satisfy

Determine the range of values that and can take for all points on the curve .

Sketch the graph of .

The curve has parametric equations

Find the Cartesian equation for the curve  in the form.

Sketch the curve , showing clearly the value(s) at which the curve meets the coordinate axes.

For the function , determine

(i) the domain

(ii) the range

The curve has parametric equations

Find the Cartesian equation for the curve  in the form .

The curve has parametric equations

By finding a Cartesian equation for , describe fully the shape of when .

Given that is restricted to where is a positive constant, write down the minimum value of that gives

(i) a full circle

(ii) a semicircle

The curve shown in Figure 3 has parametric equations

Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4.

Using the model and given that

• and are in metres

• the vertical wall of the dam is 4.2 metres high

• there is a horizontal walkway of width along the top of the dam

calculate the width of the walkway.

The curve has parametric equations

Show that a Cartesian equation for is

where and are integers to be found.

A sketch of the curve with parametric equations

is shown below.

Find the coordinates of any points at which the curve intersects the line with equation

The graph of has parametric equations

Show that the Cartesian equation of can be written in the form

Sketch the curve .

The curve has parametric equations

(i) Complete the table below, giving values to 3 significant figures where appropriate.

(ii) Plot the graph of on the axes below.

A curve has parametric equations

where , , and are constants.

In the case where , find a Cartesian equation for and hence describe fully the shape of the curve.

In the case where  and  , the curve is used to represent the position of a robot at time .

(i) Find the total distance travelled by the robot between times and .

(ii) In which direction (clockwise or anticlockwise) does the robot move along the curve?

A curve has parametric equations

where is a constant.

If the curve passes through the point with coordinates , find the possible values of .

The curve has parametric equations

(i) Determine the range of values that and can take for all points on

(ii) Show that all points on satisfy

Sketch the graph of .

A curve has parametric equations

Determine the range of values that and can take for all points on the curve .

Show that all points on satisfy

where is a constant to be found.

The curve is defined by the parametric equations

Show that all points on satisfy the equation

Figure 6 shows a sketch of the curve and the line which has equation

The line intersects the curve at exactly one point.

A different straight line with equation

where is a constant

intersects at two distinct points.

Find the range of possible values for .

The curve with parametric equations

meets the circle with equation

at four distinct points as shown in Figure 2.

Given that one of these points, , lies in the 4th quadrant, find the Cartesian coordinates of .

A curve has parametric equations

Show that all points on satisfy .

(i) Sketch the curve .

(ii) Explain briefly why does not include all points of .

The line with equation , where is a constant, intersects at two distinct points.

State the range of values of , writing your answer in set notation.

A company’s logo is a major arc of an ellipse, as shown in the figure below.

The arc is formed using the parametric equations

The end points of the arc are connected to the origin, forming an angle  radians at the centre, as shown.

Find the angle , giving your answer to 3 significant figures.

The curve has parametric equations

In the case where , find the Cartesian equation for the curve  in the form.

(i) If the restriction of is changed, explain why must be satisfied.

(ii) In the case where , find the Cartesian equation for the curve .

A sketch of the curve with parametric equations

is shown below.

Find the coordinates of all points where the graph crosses the coordinate axes.

A square has vertices , , and .

Find the coordinates of all points where the curve meets the square.

A curve has parametric equations

Sketch the graph of , showing clearly the coordinates of any

• start and end points on the curve

• points of intersection with the coordinate axes

• minimum and maximum points