Edexcel A Level Maths

Paper 2: Pure Mathematics 2 (Set A)

Friday 18 April 2025

Morning (Time: 2 hours 0 minutes)

Instructions
Try to complete this mock exam paper in one sitting, under exam conditions. Use all the time available and
check your answers to each question at the end before submitting.
Remember this is PRACTICE. Mistakes are fine and will help you improve in time for the real exam - just do
your best.
Candidates may use any calculator allowed by Pearson regulations.
You should show sufficient working to make your methods clear. Answers without working may not gain
full credit.
Inexact answers should be given to three significant figures unless otherwise stated.

Materials
Mathematical Formulae and Statistical Tables

Scan here to mark your mock exam

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1 The 4th and 8th terms of an arithmetic sequence are 20 and 64 respectively. Find the first
term and the common difference.

(3 marks)

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2 (a) The functions f(x) and g(x) are defined as follows

1
f x 4x 3 x
2

gx 0 5x 0 75 x

Find

(i)

fg x

(ii)

gf x

(3 marks)

(b) Write down f−1(x) and state its domain and range.

(3 marks)

3 Solve the equation

log3 x 4 4 2 log3 x

giving your answers correct to 3 significant figures.

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 (3 marks)

4 Given that θ is small, write an approximation in terms of θ for

2 sin θ cos θ tan2 θ

(3 marks)

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5 (a) A curve has the equation y x 3 12x 7 .

dy d2y
Find expressions for and .
dx dx 2

(3 marks)

(b) Determine the coordinates of the local minimum of the curve.

(3 marks)

6 The diagram below shows the sector of a circle with centre O . The radii OA and OB
r cm, and the angle at the centre,AOB , is equal to θ radians.
are each equal to The
line DC is perpendicular to the line OB .

Given that BC : CO 2:3 , show that the area of the shaded shape ABCD is given by

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 1 2
r 25θ 9 tan θ cm2.
50

(6 marks)

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 4
7 (a) The curve C has equation y 3x 2 6 . The point P 1 1 lies on C.
x

dy
Find an expression for .
dx

(2 marks)

(b) Show that an equation of the normal to C at point P is x 2y 3.

(3 marks)

8 The diagram below shows part of the graph of y 2x 3x 2 2x 3

Find the area of the shaded region labelled R.

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 (4 marks)

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9 (a) Show that the derivative function of the curve given by

ln y 2xy 3 8

is given by

dy 2y 4
dx 1 6xy 3

(5 marks)

(b) Find the equation of the normal to the curve given in part (a) at the point where y 1,
giving your answer in the form ax by c 0 where a b and c are integers to be
found.

(3 marks)

10 A geometric series is given by 1 2x 4x 2

(i)

Write down the common ratio, r , of the series.

(ii)

Given that the series is convergent, and that 2x n 1 19 , calculate the value of x .
n 1

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 (4 marks)

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11 (a) An exponential model of the form

D Ae kt

is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s

The graph below shows values of D plotted against t

Using the points markedP and Q , find an equation for the line of best fit, giving your
answer in the form ln D mt ln c , where m and c are constants to be found.

(2 marks)

(b) Hence find estimates for the constants A and k

(2 marks)

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(c) The patient is allowed a second injection of the drug once the amount of drug in the
bloodstream falls below 1% of the initial dose. Find, to the nearest minute, how long until
the patient is allowed a second injection of the drug.

(2 marks)

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12 (a) A sketch of the graph with equation y f x where f x 10x x 2 16 is shown
below.

Points A and B are the x -axis intercepts and point C is the maximum point on the
graph.

1 1
On the diagram above, sketch the graph of y 4
f 2
x labelling the image of the
points A B and C with A B and C .

(3 marks)

(b) Show that the area of ABC is twice the area of triangle A BC .

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 (4 marks)

13 Use the substitution u 2 ln x to show that

1 1
3
dx 2
c
x 2 ln x 2 ln x 2

where c is the constant of integration.

(5 marks)

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14 (a) The curve C has parametric equations

x t 2 4 y 3t

dy
Show that at the point (0 , 6), t 2 and find the value of at this point.
dx

(4 marks)

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

(3 marks)

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15 (a) A large block of ice used by sculptors is in the shape of a cuboid with dimensions x m by
2x m by 5x m. The block melts uniformly with its surface area decreasing at a constant
rate of k m2 s-1. You may assume that as the block melts, the shape remains
mathematically similar to the original cuboid.

Show that the rate of melting, by volume, is given by

15kx 3 1
m s
34

(5 marks)

(b) In the case when k 0 2 , the block of ice remains solid enough to be sculpted as long as
the rate of melting, by volume, does not exceed 0 05 m3 s 1 .

Find the value of x for the largest block of ice that can be used for ice sculpting under
such conditions, giving your answer as a fraction in its lowest terms.

(3 marks)

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16 (a) The number of daylight hours, h , in the UK, d days after the spring equinox (the day in
spring when the number of daylight hours is 12) is modelled using the function

2
h A B sin d
365

where A are B constants.

(i)

Write down the value of A .

(ii)

Given that the maximum number of daylight hours is 16.5, write down the value of B.

(2 marks)

(b) For how many days of the year does the number of daylight hours remain below 10?
Give your answer as a whole number of days.

(2 marks)

(c) If the spring equinox falls on the 21st March, find the dates throughout the year when
there are 16 hours of daylight.

(3 marks)

(d) The model needs to be adjusted every four years. Suggest a reason why.

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 (1 mark)

17 In the parallelogram ABCD, AB is parallel to CD, and BC is parallel to AD.

Given that AB 3 2 and AD 7 4 , find the area of the parallelogram.

(8 marks)

18 Prove that the square of an odd number is always odd.

(3 marks)

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