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Differentiation
Question Paper
Course CIE IGCSE Additional Maths
Section Calculus
Topic Differentiation
Difficulty Hard
Time Allowed 110
Score /84
Percentage /100
Page 1 of 11
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Question 1
5
A curve has equation y = ln(5 – 3x ) where x < . The normal to the curve at the point where x = – 5 , cuts the x -
3
axis, at the point P .
Find the equation of the normal and the x -coordinate of P .
[7 marks]
Question 2
x
Variables x and y are such that y = e2 + x cos 2x , where x is in radians. Use differentiation to find the approximate
change in y as x increases from 1 to 1 + h , where h is small.
[6 marks]
Page 2 of 11
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Question 3
x3
The tangent to the curve y = ln(3x 2 − 4) − , at the point where x = 2 , meets the y -axis at the point P . Find the
6
exact coordinates of P .
[6 marks]
Page 3 of 11
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Question 4
A container is a circular cylinder, open at one end, with a base radius of r cm and a height of h cm. The volume of the
container is 1000 cm3. Given that r and h can vary and that the total outer surface area of the container has a minimum
value, find this value.
[8 marks]
Page 4 of 11
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Question 5a
Find the x -coordinates of the stationary points of the curve y = e 3x (2x + 3) 6 .
[6 marks]
Question 5b
A curve has equation y = f (x ) and has exactly two stationary points. Given that f " (x ) = 4x − 7 ,
f '(0 . 5) = 0 and f'(3) = 0 , use the second derivative test to determine the nature of each of the stationary points of
this curve.
[2 marks]
Page 5 of 11
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Question 5c
In this question all lengths are in centimetres.
The diagram shows a solid cuboid with height h and a rectangular base measuring 4x by x . The volume of the cuboid is
40 cm3 . Given that x and h can vary and that the surface area of the cuboid has a minimum value, find this value.
[5 marks]
Page 6 of 11
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Question 6a
π
Find the equation of the tangent to the curve 2y = tan 2x + 7 at the point where x = .
8
π
Give your answer in the form ax − y = + c , where a , b and c are integers.
b
[5 marks]
Question 6b
This tangent intersects the x -axis at P and the y -axis at Q . Find the length of PQ .
[2 marks]
Question 7a
dy Ax + B
y = x x + 2 Given that , show that = , where A and B are constants.
dx 2 x + 2
[5 marks]
Page 7 of 11
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Question 7b
Find the exact coordinates of the stationary point of the curve y = x x + 2 .
[3 marks]
Question 7c
Determine the nature of this stationary point.
[2 marks]
Question 8a
Differentiate y = tan(x + 4) − 3 sin x with respect to x .
[2 marks]
Question 8b
ln(2x + 5)
Variables x and y are such that y = . Use differentiation to find the approximate change in y as x increases
2e 3x
from 1 to 1 + h , where h is small.
[6 marks]
Page 8 of 11
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Question 9a
tan 3x
It is given that y = .
sin x
dy π
Find the exact value of when x = .
dx 3
[4 marks]
Question 9b
π π
Hence find the approximate change in y as x increases from to + h , where h is small.
3 3
[1 mark]
Question 9c
π
Given that x is increasing at the rate of 3 units per second, find the corresponding rate of change in y when x = , giving
3
your answer in its simplest surd form.
[2 marks]
Page 9 of 11
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Question 10a
π
It is given that y = ln(sin x + 3 cos x ) for 0 < x < .
2
dy
Find .
dx
[3 marks]
Question 10b
dy 1
Find the value of x for which =− .
dx 2
[3 marks]
Question 11a
e2x − 3 dy
Given that y = , find .
x2 + 1 dx
[3 marks]
Page 10 of 11
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Question 11b
Hence, given that y is increasing at the rate of 2 units per second, find the exact rate of change of x when x = 2 .
[3 marks]
Page 11 of 11
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