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

Exam code: 9MA0

3 hours34 questions
Easy
Medium
Hard
Very Hard
1a2 marks

The curve C has parametric equations

x equals 4 t space space space space space space space space space space y equals t squared

Complete the table below.

t

-2

-1

0

1

2

x

 

 

 

 

 

y

 

 

 

 

 

How did you do?

1b2 marks

Plot the graph of C on the axes below.

q1b-9-1-parametric-equations-easy-a-level-maths-pure-screenshot

How did you do?

22 marks

A curve C has parametric equations

x equals t minus 3 space space space space space space space space space space y equals t squared minus 4

Find the Cartesian equation for the curve C in the form 

y equals a x squared plus b x plus c

where a, b and c are integers to be found.

How did you do?

3a2 marks

The curve C has parametric equations

x equals t cubed space space space space space space space space space space y equals t squared

Complete the table below.

t

-2

-1

0

1

2

x

 

 

 

 

 

y

 

 

 

 

 

How did you do?

3b2 marks

Plot the graph of C on the axes below.

q1b-9-1-parametric-equations-easy-a-level-maths-pure-screenshot

How did you do?

4a2 marks

A sketch of part of the curve with parametric equations

x equals 4 t minus 8 space space space space space space space space space space y equals t squared minus 1

is shown in the figure below.

q5a-9-1-parametric-equations-easy-a-level-maths-pure-screenshot

Find the coordinates of the point at which the curve crosses the y-axis.

How did you do?

4b3 marks

Find the coordinates of the points at which the curve crosses the x-axis.

How did you do?

5a3 marks

A curve has parametric equations

x equals 2 sin theta space space space space space space space space space space y equals 2 cos theta space space space space space space space space space space 0 less or equal than theta less than 2 pi

Find a Cartesian equation for the curve.

How did you do?

5b2 marks

Describe the shape of the curve.

How did you do?

6a2 marks

The curve C has parametric equations

x equals t plus 1 space space space space space space space space space space y equals t squared minus 1

Find the Cartesian equation for the curve C in the form 

y equals a x squared plus b x

where a and b are integers to be found.

How did you do?

6b2 marks

Hence sketch the curve C, showing clearly the value(s) at which the curve meets the coordinate axes.

How did you do?

73 marks

The curve C has parametric equations

x equals p t cubed space space space space space space space space space space y equals 2 t plus 1

where p is a non-zero constant.

Given that C passes through the point open parentheses 16 comma space 5 close parentheses, find the value of p.

How did you do?

83 marks

A curve C has parametric equations

x equals 2 t minus 1 space space space space space space space space space space y equals 4 t squared plus 3

Find the Cartesian equation for the curve C in the form 

y equals a x squared plus b x plus c

where a, b and c are integers to be found.

How did you do?

13 marks

A curve C has parametric equations

x equals fraction numerator t squared plus 5 over denominator t squared plus 1 end fraction space space space space y equals fraction numerator 4 t over denominator t squared plus 1 end fraction space space space space space t element of straight real numbers

Show that all points on C satisfy

open parentheses x minus 3 close parentheses squared plus y squared equals 4

How did you do?

23 marks

A sketch of the curve with parametric equations

x equals t squared minus 4 space space space space space space space space space y equals 3 t

is shown below.

q5-9-1-parametric-equations-medium-a-level-maths-pure-screenshot

Find the coordinates of the point(s) at which the curve intersects the line x equals 12.

How did you do?

3
Sme Calculator4 marks

The curve C has parametric equations

x equals t squared space space space space space space space space space space y equals straight e to the power of t

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

t

-3

-2

-1

0

1

2

x

 

 

 

 

 

 

y

 

 

 

 

 

 

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

q1a-9-1-parametric-equations-hard-a-level-maths-pure-screenshot

How did you do?

44 marks

A curve C has parametric equations

x equals 4 cos theta space space space space space space space space space space y equals 4 sin theta space space space space space space space space space space 0 less or equal than theta less than 2 pi

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

How did you do?

5a1 mark

The graph of C has parametric equations

x equals t squared plus 1 space space space space space space space space space space y equals t squared minus 1

Show that all points on C satisfy

y equals x minus 2

How did you do?

5b2 marks

Given that t element of straight real numbers, explain why all points on C must satisfy

x greater or equal than 1

y greater or equal than negative 1

How did you do?

5c2 marks

Sketch C.

How did you do?

63 marks

The curve C has parametric equations

x equals p t cubed space space space space space space space space space space y equals p t squared

where p is a non-zero constant.

Given that C passes through the point open parentheses 3 comma space minus 3 close parentheses, find the value of p.

How did you do?

7a2 marks

The curve C has parametric equations

x equals cos squared theta space space space space space space space space space space y equals sin squared theta space space space space space space space space space space minus pi less or equal than theta less or equal than pi

Show that all points on C satisfy

y equals 1 minus x

How did you do?

7b2 marks

Determine the range of values that xand y can take for all points on the curve C.

How did you do?

7c2 marks

Sketch the graph of C.

How did you do?

8a2 marks

The curve C has parametric equations

x equals straight e to the power of t space space space space space space space space space space y equals vertical line t vertical line space space space space space space space space space space t element of straight real numbers

Find the Cartesian equation for the curve C in the formspace y equals straight f left parenthesis x right parenthesis.

How did you do?

8b2 marks

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

How did you do?

8c2 marks

For the function straight f left parenthesis x right parenthesis, determine

(i) the domain

(ii) the range

How did you do?

93 marks

The curve C has parametric equations

x equals 4 t squared plus 1 space space space space space space space space space space y equals ln open parentheses 2 t close parentheses space space space space space space space space space space t greater than 0

Find the Cartesian equation for the curve C in the form y equals straight f open parentheses x close parentheses.

How did you do?

10a4 marks

The curve C has parametric equations

x equals 3 plus 2 cos 3 theta space space space space space space space space space space space space y equals 2 open parentheses negative 1 plus sin 3 theta close parentheses

By finding a Cartesian equation for C, describe fully the shape of C when theta element of straight real numbers.

How did you do?

10b
Sme Calculator2 marks

Given that theta is restricted to 0 less or equal than theta less or equal than k where k is a positive constant, write down the minimum value of k that gives

(i) a full circle

(ii) a semicircle

How did you do?

1
Sme Calculator5 marks
Graph displaying a shaded region R under a curved line in the first quadrant of the xy-plane. The axes are labelled x and y, and the origin is labelled O.
Figure 3

The curve shown in Figure 3 has parametric equations

x equals 6 sin t space space space space space space space space space space space space y equals 5 sin 2 t space space space space space space space space space space space space 0 less or equal than t less or equal than pi over 2

Graph with axes labelled x and y. A shaded area is inside the curve from part (a), under a line segment MN, and to the left of the vertical line from N to the x-axis. The height of MN is marked as 4.2.
Figure 4

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

  • x and y are in metres

  • the vertical wall of the dam is 4.2 metres high

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

calculate the width of the walkway.

How did you do?

23 marks

The curve C has parametric equations

x equals t squared plus 6 t minus 16 space space space space space space space space y equals 6 ln open parentheses t plus 3 close parentheses space space space space space space space space t greater than negative 3

Show that a Cartesian equation for C is

y equals A ln open parentheses x plus B close parentheses space space space space space space space space x greater than negative B

where A and B are integers to be found.

How did you do?

34 marks

A sketch of the curve with parametric equations

x equals t squared plus 4 t space space space space space space space space space space space space y equals t squared minus 1

is shown below.

q5a-9-1-parametric-equations-hard-a-level-maths-pure-screenshot

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

x minus 5 y plus 19 equals 0

How did you do?

4a3 marks

The graph of C has parametric equations

x equals t to the power of 4 minus 4 t squared space space space space space space space space space space y equals 4 t space space space space space space space space space space t element of straight real numbers

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

x equals space 1 over 256 space straight f open parentheses y close parentheses

How did you do?

4b4 marks

Sketch the curve C.

How did you do?

5
Sme Calculator4 marks

The curve C has parametric equations

x equals t ln open parentheses t squared plus 0.75 close parentheses space space space space space space space space space space y equals t

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

t

-3

-2

-1

-0.5

0

0.5

1

2

3

x

 

 

 

 

 

 

 

 

 

y

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

q1a-9-1-parametric-equations-vh-a-level-maths-pure-screenshot

How did you do?

6a4 marks

A curve has parametric equations

x equals p plus a cos 2 t space space space space space space space space space space y equals q plus b sin 2 t space space space space space space space space space space t greater or equal than 0

where a, b, p and q are constants.

In the case where vertical line a vertical line equals vertical line b vertical line, find a Cartesian equation for C and hence describe fully the shape of the curve.

How did you do?

6b4 marks

In the case where  a equals negative 6 space and  b equals 6, the curve is used to represent the position of a robot at time t.

(i) Find the total distance travelled by the robot between times t equals 0and t equals 2 pi.

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

How did you do?

74 marks

A curve has parametric equations

x equals m t cubed minus 6 space space space space space space space space space y equals m t plus t squared

where m is a constant.

If the curve passes through the point with coordinates open parentheses negative 22 comma space 0 close parentheses, find the possible values of m.

How did you do?

8a5 marks

The curve C has parametric equations

x equals arccos t space space space space space space space space space space y equals square root of 1 minus t squared end root space space space space space space space space space space minus 1 less or equal than t less or equal than 1

(i) Determine the range of values that xand y can take for all points on C

(ii) Show that all points on C satisfy

y equals sin space x

How did you do?

8b2 marks

Sketch the graph of C.

How did you do?

9a2 marks

A curve C has parametric equations

x equals t squared minus 4 space space space space space space space space y equals t cubed minus 4 t space space space space space space space space t element of straight real numbers

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

How did you do?

9b3 marks

Show that all points on C satisfy

y squared equals x squared open parentheses x plus k close parentheses

where k is a constant to be found.

How did you do?

1a2 marks

The curve C is defined by the parametric equations

x equals 2 tan t plus 1 space space space space space space space space space space space space space y equals 2 sec squared t plus 3 space space space space space space space space space space space space space minus pi over 4 less or equal than t less or equal than pi over 3

Show that all points on C satisfy the equation

y equals 1 half open parentheses x minus 1 close parentheses squared plus 5

How did you do?

1b5 marks
Graph of a curve C intersecting a line l at point P in the first quadrant, on an xy-plane with origin O. The curve goes down and then back up with a minimum point in the first quadrant. The line has a negative gradient and is perpendicular to the curve at the point of intersection.
Figure 6

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

y equals negative 1 half x plus 17 over 2

The line l intersects the curve C at exactly one point.

A different straight line with equation

y equals negative 1 half x plus k space space space space space space space space space space space space space where k is a constant

intersects C at two distinct points.

Find the range of possible values for k.

How did you do?

2
Sme Calculator6 marks
Graph with x and y axes showing intersecting curves C1 and C2, intersecting at point O, with point S marked on the x-axis near C1.
Figure 2

The curve C subscript 1 with parametric equations

x equals 10 cos t comma space space space space y equals 4 square root of 2 sin t comma space space space space 0 less or equal than t less than 2 pi

meets the circle C subscript 2 with equation

x squared plus y squared equals 66

at four distinct points as shown in Figure 2.

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

How did you do?

3a2 marks

A curve C has parametric equations

x equals 3 plus 2 sin t comma space space space space space y equals 4 plus 2 cos 2 t comma space space space space space space 0 less or equal than t less than 2 pi

Show that all points on C satisfy y equals 6 minus open parentheses x minus 3 close parentheses squared.

How did you do?

3b3 marks

(i) Sketch the curve C.

(ii) Explain briefly why C does not include all points of y equals 6 minus open parentheses x minus 3 close parentheses squared comma space space space space x element of straight real numbers.

How did you do?

3c
Sme Calculator5 marks

The line with equation x plus y equals k, where k is a constant, intersects C at two distinct points.

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

How did you do?

4
Sme Calculator4 marks

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

q2-9-1-parametric-equations-vh-a-level-maths-pure-screenshot

The arc is formed using the parametric equations

x equals 2 sin ϕ space space space space space space space space space space y equals cos ϕ space space space space space space space space space space minus 2 less or equal than ϕ less or equal than 2

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

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

How did you do?

5a4 marks

The curve C has parametric equations

x equals open parentheses ln 2 t close parentheses squared space space space space space space space space space space y equals t space sin 2 t

In the case where t greater or equal than 1 half, find the Cartesian equation for the curve C in the formspace y equals straight f left parenthesis x right parenthesis.

How did you do?

5b3 marks

(i) If the restriction of t greater or equal than 1 half is changed, explain whyspace t greater than 0 must be satisfied.

(ii) In the case where 0 less than t less than space 1 half, find the Cartesian equation for the curve C.

How did you do?

6a4 marks

A sketch of the curve with parametric equations

x equals cos 2 ϕ space space space space space space space space space y equals sin 3 ϕ space space space space space space space space space pi over 2 less or equal than ϕ less or equal than fraction numerator 3 pi over denominator 2 end fraction

is shown below.

q5-9-1-parametric-equations-vh-a-level-maths-pure-screenshot

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

How did you do?

6b4 marks

A square has vertices open parentheses negative 1 comma space minus 1 close parentheses, open parentheses 1 comma space minus 1 close parentheses, open parentheses 1 comma space 1 close parentheses and open parentheses negative 1 comma space 1 close parentheses.

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

How did you do?

77 marks

A curve C has parametric equations

x equals theta plus pi over 3 space space space space space space space space y equals cos theta minus square root of 3 sin theta space space space space space space space space space 0 less or equal than theta less or equal than 2 pi

Sketch the graph of C, showing clearly the coordinates of any

  • start and end points on the curve

  • points of intersection with the coordinate axes

  • minimum and maximum points

How did you do?