CIE A Level Maths: Pure 3

Topic Questions

4.3 Differentiation of Parametric Equations

1a
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2 marks

Given

   space x equals e t space  and y equals 2 t cubed plus 3 t

find  fraction numerator dx space over denominator dt end fraction and  fraction numerator dy space over denominator dt end fraction.

1b
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2 marks

Hence, or otherwise, find fraction numerator dy over denominator dx space end fraction in terms of t.

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2a
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2 marks

Find the Cartesian equation of the curve C, defined by the parametric equations

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

2b
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3 marks
i)
Find  fraction numerator dy over denominator dx space end fraction in terms of x.
ii)
Find the gradient of C at the point where space t equals 1.

2c
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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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3a
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2 marks

A particle travels along a path defined by the parametric equations

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

where  left parenthesis x space comma space y right parenthesis  are the coordinates of the particle at time t seconds.

Find the coordinates of the particle after 0.2 seconds.

3b
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3 marks
i)
Find  dx over dt  and  dy over dt.
ii)
Hence find  dy over dxin terms of t.
3c
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2 marks

Find the coordinates of the particle when it is at its minimum point.

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4a
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2 marks

The graph of the curve C shown below is defined by the parametric equations

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

q5-easy-4-3-differentiation-of-parametric-equatioon-cie-maths-pure-

Find the exact coordinates of point A.

4b
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2 marks
i)
Write down the value of  dy over dθ at the origin.
ii)
Write down the value of  dx over dθat the points wherespace x equals negative 5 and x equals 5.
4c
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4 marks
i)
Find space fraction numerator d x over denominator dθ end fraction space and  space fraction numerator d y over denominator dθ end fraction.
ii)
Hence find  space fraction numerator d y over denominator d x end fraction  in terms of theta.
iii)
Find the gradient at the point where  theta equals space straight pi over 3.

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5a
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3 marks

The curve C has parametric equations

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

i)
Find fraction numerator space dx space space over denominator dt end fraction and  fraction numerator space d y space space over denominator dt end fraction.
ii)
Hence find  fraction numerator space d y space space over denominator d x end fractionin terms of t.
5b
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3 marks
i)
Find the gradient of the tangent to C at the point (4 , 3).
ii)
Hence find the equation of the tangent to C at the point (4 , 3).

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6a
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3 marks

The curve C has parametric equations

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

i)
Find  fraction numerator d x over denominator d t end fractionand  fraction numerator d y over denominator d t end fraction.
ii)
Hence find  fraction numerator d y over denominator d x end fractionin terms of t.
6b
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5 marks
i)
Find the gradient of the tangent to C at the point (16 , 7).
ii)
Hence find the gradient of the normal to C at the point (16 , 7).
iii)
Find the equation of the normal to C at the point (16 , 7).

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1a
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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 t for the parametric equations

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

1b
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2 marks

Verify that the graph of x against y passes through the point (0 , 1) and find the gradient at that point.

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2a
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3 marks

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

x equals 8 t minus 4 space    y equals 16 t squared minus 16 t plus 5    0 less or equal than t less or equal than 1

as shown in the diagram below.

q4-9-2-further-parametric-equations-hard-a-level-maths-pure

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball, at timespace t equals 0.

Find a Cartesian equation of the curve in the form  y equals straight f open parentheses x close parentheses comma  and state the domain of space straight f open parentheses x close parentheses.

2b
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2 marks

Find the difference between the maximum and minimum heights of the wrecking ball during its motion.

2c
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3 marks

The crane is positioned such that point A is 7 m horizontally from the wall the wrecking ball is to destroy.
Find the height at which the wrecking ball will strike the wall.

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3a
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3 marks

The graph of the curve C shown below is defined by the parametric equations

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

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

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

3b
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4 marks
(i)
Show that the gradient of the tangent to C, at the point where  theta equals space pi over 4,  is  negative 5 over 6 .

(ii)
Hence find the equation of the tangent to C at the point where  theta equals space pi over 4 .

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4a
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3 marks

The curve C has parametric equations

x equals space fraction numerator 1 space over denominator t squared end fraction    space y equals t plus space 1 over t space   space t greater than 0

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

4b
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5 marks
(i)
Find the gradient of the tangent to C at the point where  space t equals space fraction numerator 1 space over denominator 2 end fraction.

(ii)
Hence find the equation of the normal to C at the point where  space t equals space fraction numerator 1 space over denominator 2 end fraction .

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5a
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4 marks

The curvespace C space has parametric equations

x equals t squared minus 4   space y equals 3 t

Show that at the point (0 , 6),space t equals 2 spaceand find the value of  fraction numerator straight d y over denominator straight d x end fraction  at this point.

5b
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3 marks

The tangent at the point (0 , 6) is parallel to the normal at the point P.
Find the exact coordinates of point P

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6
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10 marks

A curve C has parametric equations

space space x equals 9 minus t squared space space space space space space 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 diagram 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 RST.

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1a
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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 parametric equations

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

1b
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3 marks

The graph of  y  against  x  passes through the point P (1 , 1).

(i)
Find the value of t at the point P.

(ii)
Find the gradient at the point P.

(iii)
What does the value of the gradient tell you about point P?

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2a
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2 marks

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

space space x equals 12 t space space   space y equals 9 t squared minus 9 t plus 4 space    0 less or equal than t less or equal than 1

as shown in the diagram below.

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

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball, at time t equals 0.

Find the height of the wrecking ball after 0.3 seconds.

2b
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3 marks

Find the minimum height of the wrecking ball during its motion.

2c
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4 marks

Find the horizontal distances from point A at the times when the wrecking ball is at a height of 2.9 m, giving your answers accurate to 1 decimal place.

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3a
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2 marks

The graph of the curve C shown below is defined by the parametric equations

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

q4a-9-2-further-parametric-equations-medium-a-level-maths-pure

(i)
Write down the value of  fraction numerator straight d y over denominator straight d theta end fraction  at the point (0 , 6).

(ii)
Write down the value of  fraction numerator straight d x over denominator straight d theta end fraction at the points (-3 , 3) and (3 , 3).
3b
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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.

3c
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4 marks
(i)
Find the values of x, y and  fraction numerator straight d y over denominator straight d x end fraction  at the point where  theta equals space pi over 12.

(ii)
Hence show the equation of the tangent to C at the point where space theta equals space pi over 12 space spaceis
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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4a
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3 marks

The curve C has parametric equations

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

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

4b
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5 marks
(i)
Find the gradient of the tangent to C at the point (8 , 1).

(ii)
Hence write down the gradient of the normal to C at the point (8 , 1).

(iii)
Find the equation of the normal to C at the point (8 , 1).

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5a
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3 marks

The curve C has parametric equations

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

Show that, in terms of t

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator cos space t over denominator t end fraction

5b
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4 marks

Show that the distance between the maximum and minimum points on C is  2 square root of straight pi to the power of 4 plus 4 end root square units.

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1a
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4 marks

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

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

as shown in the diagram below.

q2-9-2-further-parametric-equations-very-hard-a-level-maths-pure

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball.

(i)
Write down the height of the wrecking ball when it is at point A.

 

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

1b
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4 marks

The destruction of a building requires the wrecking ball to strike it at a height of 1.4 m whilst on the upward part of its path.
Find the horizontal distance from point A at which the ball hits the building.

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2a
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3 marks

The graph of the ellipse E shown below is defined by the parametric equations

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

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

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

2b
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4 marks

Find the equation of the tangent to E, at the point where  theta equals negative space pi over 6 , giving your answer in the form y equals a minus b x, where a and b are real numbers that should be given in exact form.

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3
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9 marks

The curve C has parametric equations

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

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

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4
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6 marks

The graph of the curve defined by the parametric equations

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

is shown below.

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

(i)

Verify that the graph passes through the point (1 , 1).

(ii)
Prove that the line with equation space y equals x space spaceis not the normal to the curve at the point (1 , 1).

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5a
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5 marks

The diagram below shows a sketch of the curve defined by the parametric equations

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

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

The tangents to the curve that pass through the origin meet the curve at points A and B

Show that the values of t at points A  and B are  t equals negative space fraction numerator square root of 2 over denominator 2 end fraction  and space t equals space fraction numerator square root of 2 over denominator 2 end fraction space .

5b
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3 marks

Hence, or otherwise, show that the area of the triangle OAB is  2 square root of 2 e to the power of begin inline style 1 half end style end exponent square units.

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