A-level
MATHEMATICS
Paper 2
Time allowed: 2 hours
Materials
l You must have the AQA Formulae for A‑level Mathematics booklet.
l You should have a graphical or scientific calculator that meets the
requirements of
...
A-level
MATHEMATICS
Paper 2
Time allowed: 2 hours
Materials
l You must have the AQA Formulae for A‑level Mathematics booklet.
l You should have a graphical or scientific calculator that meets the
requirements of the specification.
Instructions
l Use black ink or black ball-point pen. Pencil should only be used for drawing.
l Fill in the boxes at the top of this page.
l Answer all questions.
l You must answer each question in the space provided for that question.
If you need extra space for your answer(s), use the lined pages at the end
of this book. Write the question number against your answer(s).
l Do not write outside the box around each page or on blank pages.
l Show all necessary working; otherwise marks for method may be lost.
l Do all rough work in this book. Cross through any work that you do not want
to be marked.
Information
l The marks for questions are shown in brackets.
l The maximum mark for this paper is 100.
Advice
l Unless stated otherwise, you may quote formulae, without proof, from the
booklet.
l You do not necessarily need to use all the space provided.
Please write clearly in block capitals.
Centre number Candidate number
Surname ________________________________________________________________________
Forename(s) ________________________________________________________________________
Candidate signature ________________________________________________________________________
For Examiner’s Use
Question Mark
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
TOTAL
I declare this is my own work.
2
Section A
Answer all questions in the spaces provided.
1 A circle has centre (4, 5) and radius 6
Find the equation of the circle.
Tick (3) one box.
[1 mark]
(x 4)2 þ (y þ 5)2 ¼ 6
(x þ 4)2 þ (y 5)2 ¼ 6
(x 4)2 þ (y þ 5)2 ¼ 36
(x þ 4)2 þ (y 5)2 ¼ 36
2 State the value of
lim
h!0
sin (p þ h) sin p
h
Circle your answer.
[1 mark]
cos h 10 1
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outside the
box
(02)
3
3 The function f is concave and is represented by one of the graphs below.
Identify the graph which represents f.
Tick (3) one box.
[1 mark]
x
y
O
x
y
O
x
y
O
x
y
O
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outside the
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Turn over s
(03)
4
4
8.7 cm
6.1 cm
A
B C
38°
The diagram shows a triangle ABC.
AB is the shortest side. The lengths of AC and BC are 6.1 cm and 8.7 cm
respectively.
The size of angle ABC is 38
Find the size of the largest angle.
Give your answer to the nearest degree.
[3 marks]
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(04)
DO NOT WRITE ON THIS PAGE
ANSWER IN THE SPACES PROVIDED
5
Turn over for the next question
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(05)
6
5 The binomial expansion of (2 þ 5x)
4 is given by
(2 þ 5x)
4 ¼ A þ 160x þ Bx2 þ 1000x3 þ 625x4
5 (a) Find the value of A and the value of B.
[2 marks]
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5 (b) Show that
(2 þ 5x)
4 (2 5x)
4 ¼ Cx þ Dx3
where C and D are constants to be found.
[2 marks]
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(06)
7
5 (c) Hence, or otherwise, find
ð
(2 þ 5x)
4 (2 5x)
4
dx
[2 marks]
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(07)
8
6 (a) Asif notices that 242 ¼ 576 and 2 þ 4 ¼ 6 gives the last digit of 576
He checks two more examples:
272 ¼ 729 292 ¼ 841
2 þ 7 ¼ 9 2 þ 9 ¼ 11
Last digit 9 Last digit 1
Asif concludes that he can find the last digit of any square number greater than 100
by adding the digits of the number being squared.
Give a counter example to show that Asif’s conclusion is not correct.
[2 marks]
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6 (b) Claire tells Asif that he should look only at the last digit of the number being squared.
272 ¼ 729 242 ¼ 576
72 ¼ 49 42 ¼ 16
Last digit 9 Last digit 6
Using Claire’s method determine the last digit of 234567892
[1 mark]
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(08)
9
6 (c) Given Claire’s method is correct, use proof by exhaustion to show that no square
number has a last digit of 8
[2 marks]
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(09)
10
7 The curve y ¼ 15 x2 and the isosceles triangle OPQ are shown on the diagram
below.
q x
y
O
P Q
Vertices P and Q lie on the curve such that Q lies vertically above some point (q, 0)
The line PQ is parallel to the x-axis.
7 (a) Show that the area, A, of the triangle OPQ is given by
A ¼ 15q q3 for 0 < q < c
where c is a constant to be found.
[3 marks]
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(10)
11
7 (b) Find the exact maximum area of triangle OPQ.
Fully justify your answer.
[6 marks]
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(11)
12
8 (a) Sketch the graph of y ¼ 1
x2
[2 marks]
x
y
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(12)
13
8 (b) The graph of y ¼ 1
x2 can be transformed onto the graph of y ¼ 9
x2 using a stretch in
one direction.
Beth thinks the stretch should be in the y-direction.
Paul thinks the stretch should be in the x-direction.
State, giving reasons for your answer, whether Beth is correct, Paul is correct, both
are correct or neither is correct.
[3 marks]
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(13)
14
9 Given that
log2 x3 log2 y2 ¼ 9
show that
x ¼ Ay p
where A is an integer and p is a rational number.
[4 marks]
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(14)
DO NOT WRITE ON THIS PAGE
ANSWER IN THE SPACES PROVIDED
15
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(15)
16
10 A gardener has a greenhouse containing 900 tomato plants.
The gardener notices that some of the tomato plants are damaged by insects.
Initially there are 25 damaged tomato plants.
The number of tomato plants damaged by insects is increasing by 32% each day.
10 (a) The total number of plants damaged by insects, x, is modelled by
x ¼ A Bt
where A and B are constants and t is the number of days after the gardener first
noticed the damaged plants.
10 (a) (i) Use this model to find the total number of plants damaged by insects 5 days after the
gardener noticed the damaged plants.
[3 marks]
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10 (a) (ii) Explain why this model is not realistic in the long term.
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