Edexcel past paper
questions
Core Mathematics 4
Parametric Equations
Edited by: K V Kumaran
Email: [email protected]
C4 Maths Parametric equations
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Co-ordinate Geometry
A parametric equation of a curve is one which does not give the relationship
between x and y directly but rather uses a third variable, typically t, to do so. The
third variable is known as the parameter. A simple example of a pair of parametric
equations:
x = 5t + 3
y = t2 + 2t
Converting to Cartesian
You need to be able to find the Cartesian equation of the curve from parametric
equations, that is the equation that relates x and y directly. To do this you need to
eliminate the parameter. The easiest way to do this is to rearrange on parametric
equation to get the parameter as the subject and then substitute this into the
other equation.
A circle with an origin (a, b) has the parametric equation:
𝑥 = 𝑎 + 𝑟𝑐𝑜𝑠𝜃
𝑦 = 𝑏 + 𝑟𝑠𝑖𝑛𝜃
You can use the result sin2 𝜃 + cos 2 𝜃 = 1 to derive these. As before, θ is the
parameter instead of t in the equations. You need to be able to recognise these as
parametric equations of circles in the exam.
C4 Maths Parametric equations
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C4 Maths Parametric equations
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Example
The curve C is described by the parametric equations
x = 5cost, y = cos2t, 0 ≤ t ≤ π
a)
Find a Cartesian equation for the curve C.
b)
Draw a sketch of the curve C.
a)
From C3 again you should remember that:
cos2t  cos2t – sin2t  2cos2t – 1
Therefore:
y = 2cos2t – 1
If x = 5cost then cost =
x
5
So finally
2x2
-1
25
y=
b)
This is simply a quadratic that is symmetrical about the y axis and intercepts
with the y axis at -1.
-1
C4 Maths Parametric equations
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Example
C4 Maths Parametric equations
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The parametric equations of a curve are
x = 14 sin t, y = 14t cos t
dy
π
. Find
in terms of t, and hence show that the gradient of the
dx
2
1
curve is zero where tan t 
t
dy
Since the curve is given parametrically we can use the chain rule to find
dx
dy dy dt


dx dt dx
where 0 < t <
So by differentiating the parametric:
x = 14 sin t
y = 14t cos t (a Product)
dx
= 14 cos t
dt
dy
= 14 cos – 14t sin t
dt
dy dy dt


dx dt dx
dy 14 cost - 14t sin t

 1  t tan t
dx
14cost
When the gradient is zero
1  t tan t  0
 tan t 
1
t
This equation could be solved by iterative methods (C3).
C4 Maths Parametric equations
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Example 4
A curve is given by the parametric equations
x = 7 sin3 t,
Show that
y = 6 cos 2t,
0<t<
dx
7
  sin t
dy
8
By chain rule

.
4
dx dx dt


dy dt dy
dx
 21sin2 tcos t
dt
dy
 12sin2t
dt
By C3 trig identities
sin 2t = 2 sin t cos t
don't forget the 2.
dx dx dt 21sin2 t cos t



dy dt dy 24 sin t cos t
dx 7 sin t

dy
8
The final example in this section deals with tangents and normals to curves.
Example 5
The curve C is described by the parametric equations
x = tan t
y = sin 2t


t
2
2
a)
Find the gradient of the curve at the point P where t=
b)
Find the equation of the normal to the curve at P.
a)
π
3
Find the gradient of the curve at the point P where t=
C4 Maths Parametric equations

3
Page 7
Using chain rule:
dy
 2 cos 2t
dt
dx
 sec 2 t
dt
dx dx dt


dy dt dy
Let t=
dy 2 cos 2t

 2 cos 2 t cos 2t
2
dx
sec t
π
3
2

2
Grad = 2 cos  cos  0.25
3
3

b)
Find the equation of the normal to the curve at P.
We are asked for the equation of the normal therefore the gradient will be 4
(why?).
Using t=

the x and y coordinates are
3
3 and
3
respectively.
2
Using y = mx + c
3
=4 3 +c
2
c=
7 3
2
Therefore the equation of the normal is
y  4x 
7 3
2
Differentiating ax
C4 Maths Parametric equations
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This function describes growth and decay, and its derivative gives a measure of
the rate of change of this growth/decay.
Since 𝑦 = 𝑎 𝑥 , taking logs of both sides gives ln 𝑦 = ln 𝑎 𝑥 = 𝑥 ln 𝑎. Using implicit
differentiation to differentiate ln 𝑦:
1 𝑑𝑦
= ln 𝑎
𝑦 𝑑𝑥
𝑑𝑦
= 𝑦 ln 𝑎 = 𝑎 𝑥 ln 𝑎
𝑑𝑥
This result needs to be learn, and is not given in the formula sheet.
C4 Parametric differentiation past paper questions
C4 Maths Parametric equations
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1.
A curve has parametric equations
x = 2 cot t, y = 2 sin2 t, 0 < t 
(a) Find an expression for

.
2
dy
in terms of the parameter t.
dx
(4)
(b) Find an equation of the tangent to the curve at the point where t =

. (4)
4
(c) Find a cartesian equation of the curve in the form y = f(x). State the domain on
which the curve is defined.
(4)
(C4 June 2005, Q6.)
2.
Figure 2
y
0.5
–0.5
–1
O
1
0.5
x
The curve shown in Figure 2 has parametric equations

x = sin t, y = sin  t   ,

6


2
<t<

.
2
(a) Find an equation of the tangent to the curve at the point where t =

.
6
(6)
(b) Show that a cartesian equation of the curve is
C4 Maths Parametric equations
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y=
1
3
x + (1 – x2),
2
2
–1 < x < 1.
(3)
(C4 June 2006, Q4.)
3.
A curve has parametric equations
x = 7 cos t – cos 7t,
(a) Find an expression for
y = 7 sin t – sin 7t,


<t< .
8
3
dy
in terms of t. You need not simplify your answer.
dx
(3)
(b) Find an equation of the normal to the curve at the point where t =

.
6
Give your answer in its simplest exact form.
(6)
(C4 Jan 2007, Q3.)
4.
A curve has parametric equations
x = tan2 t,
(a) Find an expression for
y = sin t,
0<t<

.
2
dy
in terms of t. You need not simplify your answer.
dx
(3)
(b) Find an equation of the tangent to the curve at the point where t =

.
4
Give your answer in the form y = ax + b , where a and b are constants to be
determined.
(5)
(c) Find a cartesian equation of the curve in the form y2 = f(x).
(4)
(C4 June 2007, Q6.)
5.
C4 Maths Parametric equations
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Figure 3
The curve C shown in Figure 3 has parametric equations
x = t 3 – 8t, y = t 2
where t is a parameter. Given that the point A has parameter t = –1,
(a) find the coordinates of A.
(1)
The line l is the tangent to C at A.
(b) Show that an equation for l is 2x – 5y – 9 = 0.
(5)
The line l also intersects the curve at the point B.
(c) Find the coordinates of B.
(6)
(C4 Jan 2009, Q7.)
C4 Maths Parametric equations
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6.
Figure 2
Figure 2 shows a sketch of the curve with parametric equations
x = 2 cos 2t,
y = 6 sin t,
0t
(a) Find the gradient of the curve at the point where t =

.
2

.
3
(4)
(b) Find a Cartesian equation of the curve in the form
y = f(x), –k  x  k,
Stating the value of the constant k.
(4)
(c) Write down the range of f(x).
(2)
(C4 June 2009, Q5.)
C4 Maths Parametric equations
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7.
A curve C has parametric equations
x = sin2 t, y = 2 tan t , 0 ≤ t <
(a) Find

.
2
dy
in terms of t.
dx
(4)
The tangent to C at the point where t =

cuts the x-axis at the point P.
3
(b) Find the x-coordinate of P.
(6)
(C4 June 2010, Q4.)
8.
The curve C has parametric equations
x = ln t,
y = t2 −2,
t > 0.
Find
(a) An equation of the normal to C at the point where t = 3,
(6)
(b) A Cartesian equation of C.
(3)
(C4 Jan 2011, Q6.)
C4 Maths Parametric equations
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9.
Figure 3
Figure 3 shows part of the curve C with parametric equations
x = tan ,
y = sin ,
The point P lies on C and has coordinates   3,

0 <

.
2
1 
 3 .
2 
(a) Find the value of  at the point P.
(2)
The line l is a normal to C at P. The normal cuts the x-axis at the point Q.
(b) Show that Q has coordinates (k3, 0), giving the value of the constant k.
(6)
(C4 June 2011, Q7.)
C4 Maths Parametric equations
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10.
Figure 2
Figure 2 shows a sketch of the curve C with parametric equations

x = 4 sin  t   ,

(a) Find an expression for
6
y = 3 cos 2t,
0  t < 2.
dy
in terms of t.
dx
(3)
(b) Find the coordinates of all the points on C where
dy
= 0.
dx
(5)
(C4 Jan 2012, Q5.)
C4 Maths Parametric equations
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11.
Figure 2
Figure 2 shows a sketch of the curve C with parametric equations
x = 3 sin 2t,
(a) Show that
y = 4 cos2 t,
0  t  .
dy
= k3 tan 2t, where k is a constant to be determined.
dx
(5)
(b) Find an equation of the tangent to C at the point where t =

.
3
Give your answer in the form y = ax + b, where a and b are constants.
(4)
(c) Find a Cartesian equation of C.
(3)
(C4 June 2012, Q6.)
C4 Maths Parametric equations
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12.
Figure 2
Figure 2 shows a sketch of part of the curve C with parametric equations
x=1–
1
t,
2
y = 2t – 1.
The curve crosses the y-axis at the point A and crosses the x-axis at the point B.
(a) Show that A has coordinates (0, 3).
(2)
(b) Find the x-coordinate of the point B.
(2)
(c) Find an equation of the normal to C at the point A.
(5)
(C4 Jan 2013, part of Q5.)
C4 Maths Parametric equations
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13. A curve C has parametric equations
x = 2sin t,
(a) Find

y = 1 – cos 2t,


≤t≤
2
2
dy

at the point where t = .
dx
6
(4)
(b) Find a cartesian equation for C in the form
y = f(x),
–k ≤ x ≤ k,
stating the value of the constant k.
(3)
(c) Write down the range of f(x).
(2)
(C4 June 2013, Q4)
14.
Figure 2
Figure 2 shows a sketch of the curve C with parametric equations
x  27sec3 t , y  3tan t ,
C4 Maths Parametric equations
0≤t≤

3
Page 19
(a) Find the gradient of the curve C at the point where t =

.
6
(4)
(b) Show that the cartesian equation of C may be written in the form
y  ( x 3  9) 2 ,
2
1
a≤x≤b
stating values of a and b.
(3)
(C4 June 2013_R, part of Q7)
15.
Figure 3
Figure 3 shows a sketch of the curve C with parametric equations
 
y = 2sin t,
0 ≤ t ≤ 2π
x  4cos  t   ,
 6
(a) Show that
x + y = 2√3 cos t
(3)
(b) Show that a cartesian equation of C is
(x + y)2 + ay2 = b
where a and b are integers to be determined.
(2)
(C4 June 2014, Q5)
C4 Maths Parametric equations
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16.
Figure 3
The curve shown in Figure 3 has parametric equations
x = t – 4 sin t, y = 1 – 2 cos t,

2
2
t 
3
3
The point A, with coordinates (k, 1), lies on the curve.
Given that k > 0
(a) find the exact value of k,
(2)
(b) find the gradient of the curve at the point A.
(4)
1
There is one point on the curve where the gradient is equal to  .
2
(c) Find the value of t at this point, showing each step in your working and giving your answer to
4 decimal places.
[Solutions based entirely on graphical or numerical methods are not acceptable.]
(6)
(C4 June 2014_R, Q8)
C4 Maths Parametric equations
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17.
A curve C has parametric equations
x = 4t + 3,
(a) Find the value of
y = 4t + 8 +
5
,
2t
t  0.
dy
at the point on C where t = 2, giving your answer as a fraction in its
dx
simplest form.
(3)
(b) Show that the Cartesian equation of the curve C can be written in the form
y=
x 2  ax  b
,
x3
x  3,
where a and b are integers to be determined.
(3)
(C4 June 2015, Q5)
C4 Maths Parametric equations
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