A-level
MATHEMATICS
Paper 3
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 3
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 State the range of values of x for which the binomial expansion of
ffiffiffiffiffiffiffiffiffiffiffi
1 x
4
r
is valid.
Circle your answer.
[1 mark]
jxj <
1
4 jxj < 1 jxj < 2 jxj < 4
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(02)
3
2 The shaded region, shown in the diagram below, is defined by
x2 7x þ 7 y 7 2x
O 5 x
y
Identify which of the following gives the area of the shaded region.
Tick (3) one box.
[1 mark]
ð
(7 2x) dx
ð
(x2 7x þ 7) dx
ð5
0
(x2 5x) dx
ð5
0
(5x x2) dx
ð5
0
(x2 9x þ 14) dx
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(03)
4
3 The function f is defined by
f (x) ¼ 2x þ 1
Solve the equation
f (x) ¼ f 1ðx)
Circle your answer.
[1 mark]
x ¼ 1 x ¼ 0 x ¼ 1 x ¼ 2
4 Find
ð
x2 þ x
1
2
dx
[2 marks]
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(04)
5
5 (a) Sketch the graph of
y ¼ sin 2x
for 0 x 360
O x
y
90° 180° 270° 360°
[2 marks]
5 (b) The equation
sin 2x ¼ A
has exactly two solutions for 0 x 360
State the possible values of A.
[1 mark]
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(05)
6
6 A design for a surfboard is shown in Figure 1.
Figure 1
length
width
The curve of the top half of the surfboard can be modelled by the parametric
equations
x ¼ 2t 2
y ¼ 9t 0:7t2
for 0 t 9:5 as shown in Figure 2, where x and y are measured in centimetres.
Figure 2
O
y
x
6 (a) Find the length of the surfboard.
[2 marks]
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(06)
7
6 (b) (i) Find an expression for dy
dx
in terms of t.
[3 marks]
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6 (b) (ii) Hence, show that the width of the surfboard is approximately one third of its length.
[4 marks]
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(07)
8
7 A planet takes T days to complete one orbit of the Sun.
T is known to be related to the planet’s average distance d, in millions of kilometres,
from the Sun.
A graph of log10 T against log10 d is shown with data for Mercury and Uranus labelled.
log10 T
log10 d
Uranus
(3.46, 4.49)
Mercury
(1.76, 1.94)
7 (a) (i) Find the equation of the straight line in the form
log10 T ¼ a þ b log10 d
where a and b are constants to be found.
[3 marks]
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(08)
9
7 (a) (ii) Show that
T ¼ K d n
where K and n are constants to be found.
[2 marks]
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7 (b) Neptune takes approximately 60 000 days to complete one orbit of the Sun.
Use your answer to 7(a)(ii) to find an estimate for the average distance of Neptune
from the Sun.
[2 marks]
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(09)
10
8 Water is poured into an empty cone at a constant rate of 8 cm3/s
After t seconds the depth of the water in the inverted cone is h cm, as shown in the
diagram below.
h
When the depth of the water in the inverted cone is h cm, the volume, Vcm3, is given
by
V ¼ ph3
12
8 (a) Show that when t ¼ 3
dV
dh ¼ 6 ffiffiffiffiffiffi
6p p3
[4 marks]
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(10)
11
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8 (b) Hence, find the rate at which the depth is increasing when t ¼ 3
Give your answer to three significant figures.
[3 marks]
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(11)
12
9 Assume that a and b are integers such that
a2 4b 2 ¼ 0
9 (a) Prove that a is even.
[2 marks]
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9 (b) Hence, prove that 2b þ 1 is even and explain why this is a contradiction.
[3 marks]
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(12)
13
9 (c) Explain what can be deduced about the solutions of the equation
a2 4b 2 ¼ 0
[1 mark]
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(13)
14
10 The function f is defined by
f (x) ¼ x2 þ 10
2x þ 5
where f has its maximum possible domain.
The curve y ¼ f (x) intersects the line y ¼ x at the points P and Q as shown below.
x
y = f (x)
y = x
O
Q
P
y
10 (a) State the value of x which is not in the domain of f.
[1 mark]
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(14)
15
10 (b) Explain how you know that the function f is many-to-one.
[2 marks]
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10 (c) (i) Show that the x-coordinates of P and Q satisfy the equation
x2 þ 5x 10 ¼ 0
[2 marks]
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10 (c) (ii) Hence, find the exact x-coordinate of P and the exact x-coordinate of Q.
[1 mark]
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(15)
16
10 (d) Show that P and Q are stationary points of the curve.
Fully justify your answer.
[5 marks]
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10 (e) Using set notation, state the range of f
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