C4 Differentiation Exam Questions
Section A: Implicit Differentiation and/or Exponential Functions
[C4 June 2014(R) Q3]
1.
x2 + y2 + 10x + 2y – 4xy = 10
dy
(a) Find
in terms of x and y, fully simplifying your answer.
dx
dy
 0.
(b) Find the values of y for which
dx
(5)
(5)
[C4 June 2013 Q7]
2.
A curve is described by the equation
x2 + 4xy + y2 + 27 = 0
(a) Find
dy
in terms of x and y.
dx
(5)
A point Q lies on the curve.
The tangent to the curve at Q is parallel to the y-axis.
Given that the x-coordinate of Q is negative,
(b) use your answer to part (a) to find the coordinates of Q.
(7)
[C4 June 2013(R) Q2]
3.
The curve C has equation
3x–1 + xy –y2 +5 = 0
dy
1
3
at the point (1, 3) on the curve C can be written in the form ln(  e ) , where λ

dx
and μ are integers to be found.
(7)
Show that
[C4 June 2012 Q5]
4.
The curve C has equation
16y3 + 9x2y − 54x = 0.
(a) Find
dy
in terms of x and y.
dx
(b) Find the coordinates of the points on C where
(5)
dy
= 0.
dx
(7)
[C4 Jan 2012 Q1]
5.
The curve C has the equation 2x + 3y2 + 3x2 y = 4x2.
The point P on the curve has coordinates (–1, 1).
(a) Find the gradient of the curve at P.
(5)
(b) Hence find the equation of the normal to C at P, giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
(3)
[C4 June 2011 Q5]
6.
Find the gradient of the curve with equation
ln y = 2x ln x,
x > 0, y > 0,
at the point on the curve where x = 2. Give your answer as an exact value.
(7)
[C4 Jan 2011 Q2]
7.
The current, I amps, in an electric circuit at time t seconds is given by
I = 16 – 16(0.5)t,
t  0.
dI
Use differentiation to find the value of
when t = 3 .
dt
Give your answer in the form ln a, where a is a constant.
(5)
[C4 June 2010 Q3]
8.
A curve C has equation
2x + y2 = 2xy.
Find the exact value of
dy
at the point on C with coordinates (3, 2).
dx
(7)
[C4 Jan 2010 Q3]
9.
The curve C has equation
cos 2x + cos 3y = 1,
–



 x  , 0 y  .
4
4
6
dy
in terms of x and y.
dx

The point P lies on C where x = .
6
(b) Find the value of y at P.
(c) Find the equation of the tangent to C at P, giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
(a) Find
(3)
(3)
(3)
[C4 June 2009 Q4]
10. The curve C has the equation ye–2x = 2x + y2.
dy
(a) Find
in terms of x and y.
(5)
dx
The point P on C has coordinates (0, 1).
(b) Find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0,
where a, b and c are integers.
(4)
[C4 Jan 2009 Q1]
11. A curve C has the equation y2 – 3y = x3 + 8.
dy
(a) Find
in terms of x and y.
dx
(b) Hence find the gradient of C at the point where y = 3.
(4)
(3)
[C4 June 2008 Q4]
12. A curve has equation 3x2 – y2 + xy = 4. The points P and Q lie on the curve. The gradient of
the tangent to the curve is 83 at P and at Q.
(a) Use implicit differentiation to show that y – 2x = 0 at P and at Q.
(6)
(b) Find the coordinates of P and Q.
(3)
[C4 Jan 2008 Q5]
13. A curve is described by the equation
x3  4y2 = 12xy.
(a) Find the coordinates of the two points on the curve where x = –8.
(b) Find the gradient of the curve at each of these points.
[C4 Jan 2007 Q5]
14. A set of curves is given by the equation sin x + cos y = 0.5.
dy
(a) Use implicit differentiation to find an expression for
.
dx
For – < x <  and – < y < ,
dy
(b) find the coordinates of the points where
= 0.
dx
(3)
(6)
(2)
(5)
[C4 Jan 2007 Q6]
15.
dy
= 2x ln 2. (2)
dx
at the point with coordinates (2, 16).
(4)
(a) Given that y = 2x, and using the result 2x = ex ln 2, or otherwise, show that
(b) Find the gradient of the curve with equation y = 2 ( x
2
)
[C4 June 2006 Q1]
16. A curve C is described by the equation
3x2 – 2y2 + 2x – 3y + 5 = 0.
Find an equation of the normal to C at the point (0, 1), giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
(7)
[C4 Jan 2006 Q1]
17. A curve C is described by the equation
3x2 + 4y2 – 2x + 6xy – 5 = 0.
Find an equation of the tangent to C at the point (1, –2), giving your answer in the form ax + by
+ c = 0, where a, b and c are integers.
(7)
[C4 June 2005 Q2]
18. A curve has equation
x2 + 2xy – 3y2 + 16 = 0.
Find the coordinates of the points on the curve where
dy
= 0.
dx
(7)
[C4 Jan 2014(I) Q3]
19. The number of bacteria, N, present in a liquid culture at time t hours after the start of a scientific
study is modelled by the equation
N = 5000(1.04)t,
t≥0
where N is a continuous function of t.
(a) Find the number of bacteria present at the start of the scientific study.
(b) Find the percentage increase in the number of bacteria present from t = 0 to t = 2.
Given that N = 15 000 when t = T,
dN
(c) find the value of
when t = T, giving your answer to 3 significant figures.
dt
(1)
(2)
(4)
Section B: Parametric Differentiation
[C4 June 2014(R) Q8]
1.
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,
(b) find the gradient of the curve at the point A.
(2)
(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 Jan 2014(I) Q7]
2.
The curve C has parametric equations
x = 2cos t, y =
3 cos 2t ,
0≤t≤π
where t is a parameter.
dy
in terms of t.
dx
2
The point P lies on C where t 
.
3
The line l is a normal to C at P.
(b) Show that an equation for l is
2x  2 3 y 1  0
(a) Find an expression for
(2)
(5)
The line l intersects the curve C again at the point Q.
(c) Find the exact coordinates of Q.
You must show clearly how you obtained your answers.
[C4 June 2013 Q4]
3.
A curve C has parametric equations
x = 2sin t,
y = 1 – cos 2t,

dy

at the point where t = .
dx
6
(b) Find a cartesian equation for C in the form
(6)


≤t≤
2
2
(a) Find
y = f(x),
(4)
–k ≤ x ≤ k,
stating the value of the constant k.
(c) Write down the range of f(x).
(3)
(2)
[C4 June 2012 Q6]
4.
Figure 2 shows a sketch of the curve C with parametric equations
x = 3 sin 2t,
y = 4 cos2 t,
0  t  .
dy
(a) Show that
= 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.
(c) Find a cartesian equation of C.
(4)
(3)
[C4 Jan 2012 Q5]
5.
Figure 2
Figure 2 shows a sketch of the curve C with parametric equations
 
x = 4 sin  t   ,
y = 3 cos 2t, 0  t < 2.
 6
dy
(a) Find an expression for
in terms of t.
dx
dy
(b) Find the coordinates of all the points on C where
= 0.
dx
(3)
(5)
[C4 June 2010 Q4]
6.
A curve C has parametric equations
x = sin2 t, y = 2 tan t , 0 ≤ t <
(a) Find
dy
in terms of t.
dx
The tangent to C at the point where t =
(b) Find the x-coordinate of P.

.
2
(4)

cuts the x-axis at the point P.
3
(6)
[C4 June 2009 Q5]
7.
Figure 2 shows a sketch of the curve with parametric equations
x = 2 cos 2t,
y = 6 sin t,
(a) Find the gradient of the curve at the point where t =
0t

.
3

.
2
(4)
(b) Find a cartesian equation of the curve in the form
y = f(x), –k  x  k,
(c)
stating the value of the constant k.
Write down the range of f(x).
(4)
(2)
[C4 Jan 2009 Q7]
8.
The curve C shown in Figure 3 has parametric equations
x = t 3 – 8t, y = t 2
(c)
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.
Find the coordinates of B.
(6)
[C4 June 2007 Q6]
9.
A curve has parametric equations
x = tan2 t,
y = sin t,
0<t<

.
2
dy
in terms of t. You need not simplify your answer.
dx

(b) Find an equation of the tangent to the curve at the point where t = .
4
(a) Find an expression for
(3)
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 Jan 2007 Q3]
10. A curve has parametric equations
x = 7 cos t – cos 7t,
y = sin t – sin 7t,


<t< .
8
3
dy
in terms of t. You need not simplify your answer.
dx

(b) Find an equation of the normal to the curve at the point where t = .
6
(a) Find an expression for
Give your answer in its simplest exact form.
[C4 June 2006 Q4]
11.
(3)
(6)
Figure 2
y
0.5
–1
–0.5
O
0.5
The curve shown in Figure 2 has parametric equations
 
x = sin t, y = sin  t   ,
 6



<t< .
2
2
1
x
(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
y=
1
3
x + (1 – x2),
2
2
–1 < x < 1.
(3)
[C4 June 2005 Q6]
12.
A curve has parametric equations
(a) Find an expression for
x = 2 cot t, y = 2 sin2 t, 0 < t 
dy
in terms of the parameter t.
dx
(b) Find an equation of the tangent to the curve at the point where t =

.
2
(4)

.
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)
Section C: Rates of Change
[C4 June 2014(R) Q5]
1.
At time t seconds the radius of a sphere is r cm, its volume is V cm3 and its surface area is
S cm2.
4
[You are given that V = πr3 and that S = 4πr2]
3
The volume of the sphere is increasing uniformly at a constant rate of 3 cm3 s–1.
dr
(a) Find
when the radius of the sphere is 4 cm, giving your answer to 3 significant figures.
dt
(4)
(b) Find the rate at which the surface area of the sphere is increasing when the radius is
4 cm.
(2)
[C4 Jan 2014(I) Q6]
2.
Oil is leaking from a storage container onto a flat section of concrete at a rate of 0.48 cm3 s–1.
The leaking oil spreads to form a pool with an increasing circular cross-section. The pool has
a constant uniform thickness of 3 mm.
Find the rate at which the radius r of the pool of oil is increasing at the instant when r = 5 cm.
Give your answer, in cm s–1, to 3 significant figures.
(5)
[C4 June 2012 Q2]
3.
Figure 1 shows a metal cube which is expanding uniformly as it is heated.
At time t seconds, the length of each edge of the cube is x cm, and the volume of the cube
is V cm3.
dV
(a) Show that
= 3x2.
(1)
dx
Given that the volume, V cm3, increases at a constant rate of 0.048 cm3 s–1,
dx
(b) find
when x = 8,
(2)
dt
(c) find the rate of increase of the total surface area of the cube, in cm2 s–1, when x = 8. (3)
[C4 June 2011 Q3]
4.
A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl.
When the depth of the water is h m, the volume V m3 is given by
1
V=
 h2(3 – 4h),
0  h  0.25.
12
dV
when h = 0.1.
dh

Water flows into the bowl at a rate of
m3 s–1.
800
(b) Find the rate of change of h, in m s–1, when h = 0.1.
(a) Find, in terms of ,
[C4 Jan 2010 Q6]
(4)
(2)
5.
The area A of a circle is increasing at a constant rate of 1.5 cm2 s–1. Find, to 3 significant figures,
the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm2.
(5)
[C4 Jan 2009 Q5]
6.
A container is made in the shape of a hollow inverted right circular cone. The height of the
container is 24 cm and the radius is 16 cm, as shown in Figure 2. Water is flowing into the
container. When the height of water is h cm, the surface of the water has radius r cm and the
volume of water is V cm3.
4h 3
(a) Show that V =
.
(2)
27
[The volume V of a right circular cone with vertical height h and base radius r is given by the
1
formula V =  r 2h .]
3
Water flows into the container at a rate of 8 cm3 s–1.
(b) Find, in terms of π, the rate of change of h when h = 12.
(5)
[C4 June 2008 Q3]
7.
Figure 2 shows a right circular cylindrical metal rod which is expanding as it is heated. After
t seconds the radius of the rod is x cm and the length of the rod is 5x cm.
The cross-sectional area of the rod is increasing at the constant rate of 0.032 cm2 s–1.
(a) Find
dx
when the radius of the rod is 2 cm, giving your answer to 3 significant figures.
dt
(4)
(b) Find the rate of increase of the volume of the rod when x = 2.
(4)
Solutions
Section A
Question A1
Question A2
Question A3
Question A4
Question A5
Question A6
Question A7
Question A8
Question A9
Question A10
Question A11
Question A12
Question A13
Question A14
Question A15
Question A16
Question A17
Question A18
Question A19
Section B
Question B1
Question B2
Question B3
Question B4
Question B5
Question B6
Question B7
Question B8
Question B9
Question B10
Question B11
Question B12
Section C
Question C1
Question C2
Question C3
Question C4
Question C5
Question C6
Question C7