Edexcel A Level Maths: Pure

Topic Questions

9.1 Parametric Equations (A Level only)

1a
Sme Calculator
2 marks

The curve C has parametric equations x equals 4 t and  y equals t squared.

Complete the table below.

t -2 -1 0 1 2
x          
y          

1b
Sme Calculator
2 marks

Plot the graph of C on the axes below.

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

Did this page help you?

2a
Sme Calculator
1 mark

A company’s logo is created using an arc of a circle as shown in the diagram below.

q2a-9-1-parametric-equations-easy-a-level-maths-pure-screenshot

When the end points of the arc are joined to the origin, they form the minor sector of a circle with angle θ  radians at the centre.

The arc is formed using the parametric equations  space x equals sin space t space  and  space y equals cos space t space, with domain pi over 4 less or equal than t less or equal than fraction numerator 3 pi over denominator 4 end fraction.

Write down the size of the angle θ.

2b
Sme Calculator
3 marks
i)
By finding both x squared and y squared show that  x squared plus y squared equals 1.

ii)
Hence write down the radius of the sector.
2c
Sme Calculator
1 mark

Use the formula “l = r θ to find the length of the arc.

Did this page help you?

3
Sme Calculator
3 marks

A curve C has parametric equations  x equals t minus 3  and  y equals t squared minus 4.

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

Did this page help you?

4a
Sme Calculator
4 marks

A curve is defined parametrically by the equations space x equals 2 sin space theta   and  y equals 2 cos space theta .

(i)
Find expressions, in terms of θ, for space x squared spaceand y squared.
(ii)
Hence find the Cartesian equation for the curve.
4b
Sme Calculator
2 marks

Describe the shape of the curve.

Did this page help you?

5a
Sme Calculator
2 marks

A sketch of the graph with parametric equations  x equals 4 t minus 8 space and  y equals t squared minus 1 is shown below.

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

Find the value of t when x equals 0 and hence write down the coordinates of the y-axis intercept.

5b
Sme Calculator
3 marks

Find the values of t when y equals 0 and hence write down the coordinates of the x-axis intercepts.

Did this page help you?

6a
Sme Calculator
2 marks

A curve C is defined by the parametric equations  x equals t plus 1  and space y equals t squared minus 1.

Show that the equation of the curve can be given in the Cartesian form space y equals straight f open parentheses x close parentheses, with space straight f open parentheses x close parentheses equals x squared minus 2 x.

6b
Sme Calculator
3 marks

Sketch the graph of  y equals straight f left parenthesis x right parenthesis, labelling any points where the graph intercepts the coordinate axes.

6c
Sme Calculator
4 marks
(i)
Find  fraction numerator d y over denominator d x end fraction and  fraction numerator d squared y over denominator d x squared end fraction.

(ii)
Hence show the coordinates of the minimum point on the graph of  y equals straight f left parenthesis x right parenthesis  are (1 , -1).
6d
Sme Calculator
2 marks

Given t element of straight real numbers, write down the domain and range of straight f left parenthesis x right parenthesis.

Did this page help you?

7
Sme Calculator
4 marks

The curve defined parametrically by  x equals p t cubed  and  y equals 2 t plus 1, where p is a non-zero constant, passes through the point Q (16 , 5).

(i)
Show that at point Q,space t equals 2.

(ii)
Hence show that p equals 2.

Did this page help you?

8a
Sme Calculator
2 marks

A curve C is defined by the parametric equations  x equals e to the power of t  and space y equals vertical line t vertical line, where space t space element of space straight real numbers.

Show thatspace y equals straight f left parenthesis x right parenthesis, where straight f open parentheses x close parentheses equals open vertical bar ln space x close vertical bar .

8b
Sme Calculator
2 marks

Sketch the graph of C, labelling any points where the graph intercepts the coordinate axes.

8c
Sme Calculator
2 marks

Determine the domain and range for the function straight f left parenthesis x right parenthesis.

Did this page help you?

1
Sme Calculator
4 marks

The curve C has parametric equations

x equals t squared   and   y equals e to the power of t

(i)
Complete the table below.
Give values to three 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

Did this page help you?

2
Sme Calculator
3 marks

A company’s logo is created using an arc of a circle and a straight line as shown in the diagram below.

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

The arc part of the logo is formed using the parametric equations

 space x equals sin space ϕ    y equals cos space ϕ

with domain  -1 less or equal than ϕ less or equal than 3.

Find the equation of the line passing through points A and B, stating values to three significant figures where appropriate.

Did this page help you?

3
Sme Calculator
3 marks

Find a Cartesian equation for the curve that has parametric equations

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

giving your answer in the form y equals straight f left parenthesis x right parenthesis.

Did this page help you?

4a
Sme Calculator
5 marks

A curve is defined by the parametric equations

x equals 2 cos space 3 theta space plus 3 space    y equals 2 sin space 3 theta minus 2 space.

By changing these equations into Cartesian form, show that this is the equation of a circle, and determine its centre and radius.

4b
Sme Calculator
3 marks
(i)
Given that θ is non-negative, write down a minimum possible domain for θ that will produce a complete circle.

(ii)
Give a restricted domain for θ that would create a semicircle.

Did this page help you?

5a
Sme Calculator
5 marks

A sketch of the graph with parametric equations

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

is shown below.

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

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

5b
Sme Calculator
4 marks

Find the coordinates of the points where the graph intersects the line with equation x minus 5 y plus 19 equals 0.

Did this page help you?

6a
Sme Calculator
3 marks

A curve C is defined by the parametric equations

 space x equals t to the power of 4 minus 4 t squared space    y equals 4 t    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, where straight f open parentheses y close parentheses is a function of y.

6b
Sme Calculator
4 marks

Sketch the curve C.

Did this page help you?

7
Sme Calculator
4 marks

The curve defined parametrically by

  x equals p t cubed plus 4     y equals p t squared minus 2

where p is a constant, passes through the point (139 , 43).

Find the value of p.

Did this page help you?

8a
Sme Calculator
5 marks

A curve C is defined by the parametric equations

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

(i)
Determine the ranges of x and y in the given domain of t.

 

(ii)
Show that y equals sin space x.
8b
Sme Calculator
3 marks

Sketch the graph of C.

Did this page help you?

1a
Sme Calculator
2 marks

The curve C has parametric equations

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

Complete the table below.

t -2 -1 0 1 2
x          
y          

1b
Sme Calculator
2 marks

Plot the graph of C on the axes below.

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

Did this page help you?

2
Sme Calculator
6 marks

A company’s logo is created using an arc of a circle as shown in the diagram below.

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

When the end points of the arc are joined to the origin, they form the major sector of a circle with angle straight theta radians at the centre.

The arc is formed using the parametric equations

space space x equals sin space t space              y equals cos space t

with domain  negative 2 less or equal than t less or equal than 2.

Find the size of the angle theta, as well as the length of the arc used in the logo.

Did this page help you?

3
Sme Calculator
3 marks

A curve C has parametric equations

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

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

Did this page help you?

4
Sme Calculator
4 marks
(i)
Find the Cartesian equation for the curve defined by the parametric equations

  space x equals space 4 cos space theta   space y equals 4 sin space theta

(ii)
Describe the shape of the curve.

Did this page help you?

5a
Sme Calculator
2 marks

A sketch of the graph with parametric equations

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

is shown below.

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

Find the value of t whenspace y equals 0 spaceand hence write down the coordinates of the x-axis intercept.

5b
Sme Calculator
3 marks

Find the values of t whenspace x equals 0 spaceand hence write down the coordinates of the y-axis intercepts.

5c
Sme Calculator
3 marks

Find the coordinates of the points where the graph intersects the line with equation x equals 12.

Did this page help you?

6a
Sme Calculator
2 marks

A curve C is defined by the parametric equations

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

Show that the equation of the curve can be given in the Cartesian form y equals straight f open parentheses x close parentheses,  with  straight f open parentheses x close parentheses equals x minus 2.

6b
Sme Calculator
2 marks

Assuming there are no restrictions on the values of t, find the domain and range of straight f left parenthesis x right parenthesis.

6c
Sme Calculator
3 marks

Sketch the graph of y equals straight f left parenthesis x right parenthesis.

Did this page help you?

7a
Sme Calculator
3 marks

The curve defined parametrically by

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

where p is a non-zero constant, passes through the point Q (3 , -3).

Show that at point Q,space t equals negative 1.

7b
Sme Calculator
2 marks

Hence, or otherwise, find the value of p.

Did this page help you?

8a
Sme Calculator
2 marks

A curve C is defined by the parametric equations

  x equals cos squared t    y equals sin squared t    negative pi less or equal than t less or equal than pi

Show thatspace y equals 1 minus x.

8b
Sme Calculator
2 marks

Determine the ranges of x and y in the given domain of t.

8c
Sme Calculator
3 marks

Sketch the graph of C.

Did this page help you?

1
Sme Calculator
5 marks

The curve C has parametric equations

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

(i)
Complete the table below.
Give values to three 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

Did this page help you?

2
Sme Calculator
4 marks

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

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

When the end points of the arc are connected to the origin, they form an elliptic sector with angle theta radians at the centre.

The logo is formed using the parametric equations

space space x equals 2 sin space ϕ    y equals cos space ϕ

with domain negative 2 less or equal than ϕ less or equal than 2.

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

Did this page help you?

3a
Sme Calculator
4 marks

Find a Cartesian equation for the curve that has parametric equations

  x equals left parenthesis ln space 2 t right parenthesis squared   space y equals t sin space 2 t space   space t greater or equal than space 1 half

giving your answer in the form y equals straight f left parenthesis x right parenthesis.

3b
Sme Calculator
3 marks
(i)

For the curve in part (a), explain why the domain must be restricted tospace t greater than 0.

(ii)

How would your answer to part (a) change if the domain were changed to      

0 less than t less than space 1 half ?

Did this page help you?

4a
Sme Calculator
6 marks

A curve is defined by the parametric equations

 space x equals a cos space 2 t space plus p space    y equals b sin space 2 t space plus q

where a and b are non-zero constants, and p and q are constants.

(i)
Show that if a=b{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} then this is the equation of a circle, giving the equation of the circle in Cartesian form.

(ii)
State the centre and radius of the circle.
4b
Sme Calculator
4 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 space t equals 0 spaceand t equals 2 pi.

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

Did this page help you?

5a
Sme Calculator
6 marks

A sketch of the graph with parametric equations

x equals cos space 2 ϕ space y equals sin space 3 ϕ italic space pi over 2 space less or equal than ϕ less or equal than space 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 the points where the graph crosses the coordinate axes.

5b
Sme Calculator
6 marks

Find the coordinates of the points where the graph intersects the square with vertices at (-1 , 1), (1 , 1), (1 , -1), (-1 , -1).

Did this page help you?

6a
Sme Calculator
2 marks

A curve C has parametric equations

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

Determine the ranges of x and y in the given domain of t.

6b
Sme Calculator
3 marks

Show that the Cartesian equation of C can be written in the form  y squared equals x squared left parenthesis x plus 4 right parenthesis.

6c
Sme Calculator
4 marks

Sketch the graph of  C.

Did this page help you?

7
Sme Calculator
4 marks

The curve defined parametrically by

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

where m is a constant, passes through the point (-22 , 0).

Find the possible values of m.

Did this page help you?

8a
Sme Calculator
3 marks

Show that cos space theta minus square root of 3 space sin space theta italic space can be written as 2 cos space open parentheses theta plus pi over 3 close parentheses .

8b
Sme Calculator
6 marks

A curve C is defined by the parametric equations


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


Sketch the graph of C, showing clearly the beginning and end points of the curve, as well as the points of intersection with the coordinate axes and any minimum and maximum points.

Did this page help you?