Parametric Equations - Core 4 Revision
1.
A curve is defined by the parametric equations
x = 4sin3, y = 9sin2 – 6sin ,
where 0 <  <
(a)

The tangent to the curve at the point P has gradient 1.
2
dy
in terms of  and hence show that the value of  at the point P satisfies the
dx
equation
Find
2sin2  – 3sin  + 1 = 0.
(5)
(b)
Find the coordinates of P.
(4)
(Total 9 marks)
2.
A curve is given by the parametric equations
x = 1 – t2 ,
(a)
Find
y = 2t.
dy
in terms of t.
dx
(2)
(b)
Hence find the equation of the normal to the curve at the point where t = 3.
(4)
(Total 6 marks)
3.
(a)
Sketch the curve C with parametric equations
x = 2cos t
y = sin t.
Indicate the coordinates of the points where C crosses the axes.
(2)
(b)
The region bounded by C and the positive x and y axes is denoted by R. Given that the
area of R is

2
 2 sin ²t dt ,
0
show, by evaluating this integral, that the area of R is

.
2
(3)
(Total 5 marks)
South Wolds Comprehensive School
1
4.
A curve is defined by the parametric equations
x = t², y = 4 + t³, t > 0.
Show that the equation of the tangent to the curve at the point P(p2, 4+ p3) is
2y = 3px – p3 + 8.
(Total 5 marks)
5.
A curve is defined by the parametric equations
x = 3 sin t
(a)
Show that, at the point P where t =
and
y = cos t.
π
1
, the gradient of the curve is – .
4
3
(3)
(b)
Find the equation of the tangent to the curve at the point P, giving your answer in the
form y = mx + c.
(4)
(Total 7 marks)
6.
A curve is given by the parametric equations
x = 3t2,
(a)
(i)
Find
y = 6t
dy
in terms of t.
dx
(2)
(ii)
Find the gradient of the curve at the point where t =
1
.
2
(1)
(b)
(i)
Find the equation of the curve in the form x = f(y).
(2)
(ii)
Find
dx
in terms of y and hence verify your answer to part (a)(ii).
dy
(4)
(Total 9 marks)
7.
A curve is given by the parametric equations
x = 3t – 1,
(a)
Find
l
y= .
t
dy
in terms of t.
dx
(2)
(b)
Hence find the equation of the normal to the curve at the point where t = 1.
(4)
(Total 6 marks)
South Wolds Comprehensive School
2
8.
(a)
Sketch the curve given by the parametric equations
for t  0.
x = 4t2, y = 8t
(1)
(b)
Find
dy
in terms of t.
dx
(2)
(c)
Hence find the equation of the tangent at the point where t = 0.5.
(3)
(Total 6 marks)
9.
A curve C is defined parametrically by the equations
x=
(a)
Determine
t
,
1 t
y=
2
,
1 t
where t  1, t  –1.
dy
dy
as a function of t and show that
< 0 at all points of C.
dx
dx
(4)
(b)
Express t in terms of x and hence show that the cartesian equation of C may be written in
the form
y 1
M
,
Nx  1
for constants M and N whose values are to be determined.
(4)
(c)
Use your answer to part (b) to state the equations of the asymptotes of C.
(2)
(d)
Find, in terms of , the volume generated when the region of the plane bounded by C and
the lines y = 1, x = 1 and x = 4 is rotated through 2 radians about the line y = 1.
(4)
No credit will be given for a numerical approximation or for a numerical answer from a
calculator without supporting working.
(Total 14 marks)
10.
A curve is given by the parametric equations
x = 2t + 3 ,
y=
2
.
t
Find the equation of the normal at the point on the curve where t = 2.
(Total 6 marks)
11.
A curve has parametric equations
x = t2 – 9,
y = 2 + 2t.
(a)
The curve crosses the x-axis at the point P. Find the coordinates of P.
South Wolds Comprehensive School
(1)
3
(b)
The curve crosses the y-axis at the points Q and R. Find the length QR.
(2)
(Total 3 marks)
12.
The parametric equations of a curve are
x = t2, y = 2t,
where the parameter t takes all positive values.
Find the equation of the normal to the curve at the point P(p², 2p).
(Total 5 marks)
13.
The diagram shows a sketch of part of the curve C which is defined parametrically by
x = t2, y = sin t, t  0.
y
P( 2,0)
O
x
Q
The curve cuts the positive x-axis for the first time at the point P (2, 0).
The normal to the curve C at P intersects the y-axis at the point Q.
(a)
Show that the equation of the normal PQ is
y = 2x – 2 3.
(5)
(b)
(i)
Find  t sin dt.
(3)
(ii)
Find, in terms of , the area of the shaded region bounded by the
curve C, the normal PQ and the y-axis.
(5)
(Total 13 marks)
South Wolds Comprehensive School
4
14.
The parametric equations of a curve are
x = 1 – cos 2t,
(a)
(i)
Show that
y = 2t – sin 2t,
(0 < t <

).
2
dy
= tan t.
dx
(4)
(ii)
The tangent to the curve at the point where t =

cuts the x-axis at X.
4
Determine the exact value of the x-coordinate of the point X.
(3)
(b)
The line y = x cuts the curve at P.
(i)
Show that the value of t at P satisfies the equation
t=
1
(1+ sin 2t – cos 2t).
2
(1)
(ii)
1
(1+ sin 2tn – cos 2tn) with t1 = 1.2 to find the
2
value of t3, giving your answer correct to five decimal places.
Use the iterative formula tn+1 =
(2)
(iii)
Hence find an approximation for the x-coordinate of P, giving the value correct to
three decimal places.
(1)
(Total 11 marks)
15.
A general point p on an ellipse has coordinates (cos, 3 sin ). The point X on the x-axis is
such that angle OXP is a right angle.
y
P(cos ,  sin)
O
(a)
X
x
Find the exact value of the acute angle , such that OP = 2.
(4)
(b)
Express cos  + 3 sin  in the form r sin( + ), where the exact values of r and 
should be stated. Hence find the exact value of the acute angle , such that OX + XP = 2.
(5)
(Total 9 marks)
South Wolds Comprehensive School
5