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

4 hours32 questions
1
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3 marks

A 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 2 t cubed plus 2 t

Use parametric differentiation to find an expression for fraction numerator d y over denominator d x end fraction in terms of t.

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2
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5 marks

A sketch of the curve with parametric equations

x equals 8 t space space space space space space space space space space y equals t squared plus 1

is shown below.

q3a-9-2-further-parametric-equations-easy-a-level-maths-pure-screenshot
  • The point t subscript 1 has x-coordinate 8

  • The point t subscript 2 has x-coordinate 16

(i) Show that the area of the shaded region is given by

integral subscript 1 superscript 2 open parentheses 8 t squared plus 8 close parentheses d t

(ii) Hence find, by algebraic integration, the exact area of the shaded region.

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3a
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2 marks

The curve C with parametric equations

x equals 5 sin space theta space space space space space space space space space space y equals theta squared space space space space space space space space space space minus pi less or equal than theta less or equal than pi

is shown in the figure below.

q5a-9-2-further-parametric-equations-easy-a-level-maths-pure-screenshot

Find the exact coordinates of the point A.

3b
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2 marks

(i) Write down the value of fraction numerator d y over denominator d theta end fraction at the origin.

(ii) Write down the value(s) of  fraction numerator d x over denominator d theta end fraction  at the points where space x equals negative 5 spaceandspace x equals 5.

3c
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4 marks

Find the exact gradient of the point on the curve where  theta equals pi over 3

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4a
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3 marks

A curve C has parametric equations

x equals 5 t squared minus 1 space space space space space space space space space space y equals 3 t space space space space space space space space space space t greater than 0

Find an expression for fraction numerator d y over denominator d x end fraction in terms of t.

4b
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3 marks

Find the equation of the tangent to C at the point open parentheses 4 comma space 3 close parentheses.

Given your answer in the form a x plus b y plus c equals 0 where a, b and c are integers to be found.

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5a
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3 marks

A curve C has parametric equations

x equals 2 t cubed space space space space space space space space space space y equals 4 t minus 1 space space space space space space space space space space t greater than 0

Find an expression for fraction numerator d y over denominator d x end fraction in terms of t.

5b
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5 marks

Find the equation of the normal to C at the point open parentheses 16 comma space 7 close parentheses.

Given your answer in the form a x plus b y plus c equals 0 where a, b and c are integers to be found.

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6
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4 marks

A sketch of the curve with parametric equations

x equals 3 plus 2 cos space t space space space space space space space space space space y equals negative 3 sin space t space space space space space space space space space space pi less or equal than t less or equal than 2 pi

is shown below, where x and y are measured in centimetres.

q8a-9-2-further-parametric-equations-easy-a-level-maths-pure-screenshot

(i) Find an expression for fraction numerator d x over denominator d t end fraction in terms of t

(ii) Show that the shaded area is given by

6 integral subscript straight pi superscript 2 straight pi end superscript sin to the power of 2 space end exponent t space d t

(iii) Hence using your calculator, or otherwise, find the exact area.

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7a
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2 marks

A curve C has parametric equations

x equals t minus 1 space space space space space space space space space space y equals 2 ln space t

Find the Cartesian equation of C.

7b
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3 marks

(i) Find  fraction numerator d y over denominator d x end fraction  in terms of x

(ii) Find the gradient of C at the point where t equals 1

7c
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2 marks

Hence find the equation of the tangent to C at the point where t equals 1

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1a3 marks

The curve C has parametric equations

table row cell x equals sin space 2 theta end cell blank cell y equals cosec cubed theta end cell blank cell 0 less than theta less than pi over 2 end cell end table

Find an expression for fraction numerator straight d y over denominator straight d x end fraction in terms of theta

1b
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3 marks

Hence find the exact value of the gradient of the tangent to C at the point where y equals 8

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2
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5 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 with 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

The line l is the normal to C at the point P where t equals pi over 4

Using parametric differentiation, show that an equation for l is

y equals negative 1 half x plus 17 over 2

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3a
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1 mark

A particle travels along a curve with parametric equations

x equals 6 t space space space space space space space space space space y equals 8 t squared minus 8 t plus 3 space space space space space space space space space space 0 less or equal than t less or equal than 1

where the coordinates open parentheses x comma space y close parentheses is the position of the particle after time t seconds.

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

3b
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of t.

3c
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3 marks

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

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4a
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3 marks

Find an expression for fraction numerator d y over denominator d x end fraction in terms of t for the curve with parametric equations

x equals straight e to the power of 2 t end exponent space space space space space space space space space space y equals 3 t squared plus 1

4b
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3 marks

The graph of y against x passes through the point P with coordinates open parentheses 1 comma space 1 close parentheses.

Show that P is a stationary point.

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5a
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2 marks

The graph shows the curve with parametric equations

x equals t cubed space space space space space space space space space space y equals 2 t squared minus 1

q2a-9-2-further-parametric-equations-medium-a-level-maths-pure
  • The point where t equals t subscript 1 space end subscripthas coordinates open parentheses 1 comma space 1 close parentheses

  • The point where t equals t subscript 2 has coordinates open parentheses 8 comma space 7 close parentheses

Find the values of t subscript 1 and t subscript 2.

5b
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5 marks

Hence find the exact area of the shaded region.

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6a
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2 marks

The graph of the curve C with parametric equations

 x equals 3 sin space 3 theta space space space space space space space space space space y equals 6 cos space 2 theta space space space space space space space space space space minus pi over 2 less or equal than theta less or equal than pi over 2

is shown in the figure below.

q4a-9-2-medium-a-level-maths

(i) Write down the value of  fraction numerator straight d y over denominator straight d theta end fraction  at the point open parentheses 0 comma space 6 close parentheses

(ii) Write down the value(s) of fraction numerator straight d x over denominator straight d theta end fraction at the points open parentheses negative 3 comma space 3 close parentheses and open parentheses 3 comma space 3 close parentheses

6b
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3 marks

Find an expression for  fraction numerator straight d y over denominator straight d x end fraction  in terms of theta.

6c
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4 marks

Hence show that the equation of the tangent to C at the point where theta equals space pi over 12 is

2 square root of 2 x plus 3 y minus open parentheses 9 square root of 3 plus 6 close parentheses equals 0

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7a
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3 marks

The curve C has parametric equations

 x equals 6 t squared plus 2 space space space space space space space space space space y equals 1 over t space space space space space space space space space space t greater than 0

Find an expression for fraction numerator straight d y over denominator straight d x end fraction in terms of t.

7b
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5 marks

Hence find the equation of the normal to C at the point with coordinate open parentheses 8 comma space 1 close parentheses.

Give your answer in the form y equals m x plus c.

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8a
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3 marks

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

q7a-9-2-further-parametric-equations-easy-a-level-maths-pure-screenshot

The curve has parametric equations

 x equals 3 cos space t space space space space space space space space space space y equals sin space 2 t space space space space space space space space space minus pi less or equal than space t less or equal than pi

where x and y are measured in metres.

(i) Find the values of xat the points where t equals negative pi and t equals negative pi over 2

(ii) Find the coordinates of the point on the curve where t equals negative fraction numerator 3 pi over denominator 4 end fraction

8b
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7 marks

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

4 integral subscript negative straight pi end subscript superscript negative straight pi over 2 end superscript open parentheses negative 6 cos space t space sin squared space t close parentheses space straight d t

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

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9a
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3 marks

The curve C has parametric equations

x equals sin space 2 t space space space space space space space space space space y equals straight e to the power of t

Find an expression for fraction numerator straight d y over denominator straight d x end fraction in terms of t.

9b
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3 marks

Show that the graph of C goes through the point open parentheses 0 comma space 1 close parentheses and find the gradient at this point.

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10
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6 marks

The graph of the curve C with parametric equations

 x equals 5 t minus 1 space space space space space space space space space space y equals square root of t space space space space space space space space space space t greater or equal than 0

is shown in the figure below.

q2a-9-2-further-parametric-equations-hard-a-level-maths-pure

The shaded region is the area bounded by C and the x-axis between x equals negative 1 and x equals 4.

Use algebraic integration to find the exact area of the shaded region.

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1
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4 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

The curve C cuts the y-axis at the point P.

Show that the equation of the tangent to C at P can be written in the form

a x plus b y equals c ln 5

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

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2a
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3 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

The region R, shown shaded in Figure 3, is bounded by the curve and the x-axis.

Show that the area of R is given by integral subscript 0 superscript pi over 2 end superscript 60 sin t cos squared t space straight d t

2b
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3 marks

Hence show, by algebraic integration, that the area of R is exactly 20

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3
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6 marks

The curve C has parametric equations

 x equals t squared space space space space space space space space space space y equals 2 sin space t space space space space space space space space space space 0 less or equal than t less than 2 pi

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

2 square root of pi to the power of 4 plus 4 end root

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4
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6 marks

The graph of the curve C with parametric equations

 x equals 2 cos space 3 theta space space space space space space space space space space y equals 5 sin space theta space space space space space space space space space space 0 less or equal than theta less than 2 pi

is shown in the figure below.

usbvpK9r_q4-9-2-further-parametric-equations-hard-a-level-maths-pure

Find the equation of the tangent to C at the point where  theta equals space pi over 4 .

Give your answer in the form y equals m x plus c.

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5
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7 marks

The curve C has parametric equations

x equals 1 over t squared space space space space space space space space space space y equals t plus 1 over t space space space space space space space space space space t greater than 0

Find the equation of the normal to C at the point where t equals space 1 half.

Given your answer in the form y equals m x plus c.

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6a
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4 marks

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

 x equals 10 t space space space space space space space space space space y equals 4.9 t squared minus 4.9 t plus 2 space space space space space space space space space space 0 less or equal than t less or equal than 1

as shown in the figure below.

xBS~8dH5_q1a-10-1-solving-equations-easy-a-level-maths-pure

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

The wrecking ball is initially released from the point A.

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

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

6b
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4 marks

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

Find the horizontal distance from A to the building.

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7
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7 marks

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

x equals 2 cos open parentheses theta plus pi over 3 close parentheses space space space space space space space space space space y equals 4 sin space theta space space space space space space space space space space minus pi less than theta less or equal than pi

q3-9-2-further-parametric-equations-very-hard-a-level-maths-pure

Find the equation of the tangent to E at the point where theta equals negative pi over 6.

Given your answer in the form y equals a minus b x, where a and b are exact real numbers to be found.

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1a
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5 marks
Graph showing a shaded region, R, under a curve, C, bounded by the x-axis, vertical line at x=4, and curve C. Axes are labelled x and y.
Figure 6

Figure 6 shows a sketch of the curve C with parametric equations

x equals 8 sin squared space t space space space space space space space space space space space y equals 2 sin space 2 t plus 3 sin space t space space space space space space space space space space space 0 space less or equal than t less or equal than pi over 2

The region R, shown shaded in Figure 6, is bounded by C, the x-axis and the line with equation x equals 4

Show that the area of R is given by

integral subscript 0 superscript a open parentheses 8 minus 8 cos space 4 t plus 48 sin squared space t cos space t close parentheses d t

where a is a constant to be found.

1b
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4 marks

Hence, using algebraic integration, find the exact area of R.

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2
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9 marks

The curve C has parametric equations

x equals 9 minus t squared space space space space space space space space space space y equals 5 minus t

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

q7-9-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots

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

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3
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7 marks

The curve C has parametric equations

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

The tangent at the point open parentheses 0 comma space 6 close parentheses on C is parallel to the normal at the point P on C.

Find the exact coordinates of the point P.

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4
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8 marks

The curve C has parametric equations

 x equals 3 t space space space space space space space space space space y equals t plus 1 over t space space space space space space space space space space t greater than 0

Find the equation of the normal to C at the point where C intersects the straight line y equals x.

Give your answer in the form y equals m x plus c.

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5
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6 marks

The graph of the curve with parametric equations

x equals straight e to the power of 2 t end exponent space space space space space space space space space space y equals straight e to the power of negative 3 t end exponent

is shown in the figure below.

q5-9-2-further-parametric-equations-very-hard-a-level-maths-pure

(i) Show that the graph passes through the point with coordinates open parentheses 1 comma space 1 close parentheses.

(ii) Prove that the straight line with equation y equals xis not the normal to the curve at the point open parentheses 1 comma space 1 close parentheses.

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6a
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2 marks

The graph of the curve C with parametric equations

 x equals 3 cos space t space space space space space space space space space space y equals 5 sin space 2 t space space space space space space space space space space 0 less or equal than t less than 2 pi

is shown in the figure below.

q6-9-2-further-parametric-equations-very-hard-a-level-maths-pure

(i) Write down the equations of any horizontal tangents to C.

(ii) Write down the equations of any vertical tangents to C.

6b
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8 marks

The tangents in part (a) create a rectangle around C, as shown below.

The shaded region is the area enclosed by C.

Graph with shaded lobes in the shape of a butterfly, centred on the origin, enclosed within a square. X and Y axes intersect at the centre.

Find the percentage of the area of the rectangle that is shaded.

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7a
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5 marks

The graph of the curve C with parametric equations

 x equals 4 t space space space space space space space space space space y equals straight e to the power of t squared end exponent

is shown in the figure below.

q7-9-2-further-parametric-equations-very-hard-a-level-maths-pure

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

Find the values of t at A and B.

7b
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3 marks

Hence show that the area of triangle O A B is 

2 square root of 2 straight e to the power of 1 half end exponent

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8a
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3 marks

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

x equals cos space t space space space space space space space space space space y equals sin space 3 t space space space space space space space space space 0 less or equal than space t less or equal than 20 pi

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

Graph of two overlapping curves on a grid with the shaded intersection forming an ellipse-like shape in the centre, axes labelled x and y.

 (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?

8b
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6 marks

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.

Determine if 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

integral sin space t space sin space 3 t space straight d t equals cos space t space sin cubed space t space plus c

where c is a constant.

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