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MathematicsAS PAPER 1
December Mock Exam (AQA Version) Time allowed: 1 hour and 30 minutes
Instructions to candidates:
• In the boxes above, write your centre number, candidate number, your surname, other names
and signature.
• Answer ALL of the questions.
• You must write your answer for each question in the spaces provided.
• You may use a calculator.
Information to candidates:
• Full marks may only be obtained for answers to ALL of the questions.
• The marks for individual questions and parts of the questions are shown in square brackets.
• There are 19 questions in this question paper. The total mark for this paper is 80.
Advice to candidates:
• You should ensure your answers to parts of the question are clearly labelled.
• You should show sufficient working to make your workings clear to the Examiner.
• Answers without working may not gain full credit.
CM
AS/P1/D17© 2017 crashMATHS Ltd.
1 2 3 3 2 2 1 1 8 D 1 7 5
Surname
Other Names
Candidate Signature
Centre Number Candidate Number
Examiner Comments Total Marks
2
1
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Find the value of p such that .
Circle your answer.
[1 mark]
Answer all questions in the spaces provided.
2 The circle C has the equation .
Which of the following options correctly describes the circle C?
Circle your answer.
[1 mark]
2p = 4 × 14
32
−1 − 12
3
x − 2( )2 + y + 3( )2 = 4
Option Centre of C Radius of C
A (2, –3)
B (–2, 3)
C (2, –3)
D (–2, 3)
4
4
2
2
3 Here are two statements.
Statement A:
Statement B:
Choose the most appropriate option below.
Circle your answer.
[1 mark]
x + 2 = 4
x = 2
A ⇒B A ⇐B A ⇔BThere is no connection
between A and B
Section A
3
1 2 3 3 2 2 1 1 8 D 1 7 5
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4 The equation has two equal roots.
Find the possible values of the constant k.
[3 marks]
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kx2 + (3 − k)x − 4 = 0
5 Solve the equation .
[3 marks]
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a12 + 4a = 3
4
6
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Figure 1 shows a sketch of the curve with equation . y = f(x)
y
x
Figure 1
−1−4
−2
6 (i) On the axes below, sketch the curve with equation .
[3 marks]y = 1
2f(x)
5
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Turn over ►
6 (ii) On the axes below, sketch the curve with equation .
[3 marks]y = f(−x)
Turn over for the next question
6
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7 (a) Find the length of AD.
[2 marks]
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7
The shape ABCDA, as shown in Figure 2, consists of a triangle BCD joined to a sector ABD of a circle with centre D.
Angle DBC = 53o, angle BCD = 50o and BC = 10 cm.
Figure 2
A
B
CD
10 cmR
53°
50°
7
1 2 3 3 2 2 1 1 8 D 1 7 5
Turn over ►
7 (b) Find the area of the shaded region R.
[3 marks]
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7 (c) Calculate the perimeter of the shape ABCDA.
Give your answer to one decimal place.
[4 marks]
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8
8 (i)
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The function f is defined such that , where a and b are constants.
Given that the curve with equation y = f(x) passes through the points (4, 5) and
(8, 12), find the values of the constants a and b.
[4 marks]
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f(x) = axb
8 (ii) The table below shows the atomic number n and the melting point (y degrees Celsius) for some alkali metals.
Metal
n
y
Lithium Sodium Potassium Rubidium Caesium
3 11 19 37 55
180.5 97.8 63.7 38.9 28.5
9
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8 (ii) A graph of ln(y) against ln(n) is produced using these data. A line of best fit is then
drawn for these data and it passes through the points (5, 2.79) and (45, – 22.77).
8 (ii) (a) Express y in terms of n.
[4 marks]
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8 (ii) (b) Francium is also a metal. The atomic number of Francium is 87. Using your answer to (a), estimate the melting point of Francium.
[2 marks]
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8 (ii) (c) Comment on the reliability of your estimate to (b).
[1 mark]
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10
9
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The curve C has the equation y = g(x), where
Given that the curve passes through the point (2, –12), find the values of a, b and c
such that .
[7 marks]
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g x( ) = a x − b( )2 + c
dydx
= 16x3 − 9x
x(3 − 4x), x >1
11
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9 [Extra space]
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Turn over for the next question
12
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10 (a) (i) Prove that .
[1 mark]
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a3 − b3 = a − b( ) a2 + ab + b2( )
10 (a) (ii) Hence, show that
[3 marks]
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1+ sin xcos xcos3 x − sin3 x
+ 1cos x + sin x
≡ 2cos xcos2 x − sin2 x
13
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10 (a) (iii) Deduce that
[3 marks]
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10 (b) Use a suitable counter-example to show that .
[1 mark]
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1+ sin xcos xcos3 x − sin3 x
+ 1cos x + sin x
+ sin2 x − 2cos x −1cos2 x − sin2 x
≡ 1tan2 x −1
cos2 x − sin2 x ≡ 1
END OF SECTION ATURN OVER FOR SECTION B
14
15
1 2 3 3 2 2 1 1 8 D 1 7 5
A particle is projected vertically upwards at 16 m/s from a point 5 m above the
ground.
Find the maximum height of the particle above the ground.
Circle your answer.
[1 mark]
Answer all questions in the spaces provided.
Section B
18.1 m 13.1 m 31.1 m 18.0 m
15
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16 A particle of mass m is attached to one end of a light inextensible string and the
other end of the string is fixed to the ceiling. The mass is displaced at an angle,
with the string taut, and released.
16 (a) (i) State one assumption made by modelling the mass as a particle.
[1 mark]
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16 (a) (ii) State one assumption made by modelling the mass as a particle.
[1 mark]
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16 (b) Suggest one assumption it may be useful to make about the environment.
[1 mark]
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16
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17 A particle of mass 3 kg is moving on a smooth horizontal surface. At time
t = 0, the particle passes through the point A and is moving at a constant speed
of 15 m s–1. After 8 s, the surface becomes rough and the particle is subject to a
constant frictional force of magnitude 18 N. The particle subsequently comes to rest
at the point B on the surface.
17 (i) Find the total time taken for the particle to come to rest.
[4 marks]
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17
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17 (ii) Find the distance between the points A and B.
[3 marks]
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Turn over for the next question
18
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18 [In this question, i and j are unit vectors directed due east and due north respectively.]
Two forces (8i + 3j) N and (xi + yj) N act on a particle.
The resultant force acting on the particle acts at a bearing of 315o.
18 (i) Show that x + y + 11 = 0.
[2 marks]
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18 (ii) The magnitude of the resultant force acting on the particle is .
Find the value of x and the value of y.
[6 marks]
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28 2
19
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18 (ii) [Extra space]
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Turn over for the next question
20
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19 A light lift L is attached to a vertical light inextensible string. The lift carries two
masses A and B and the mass A rests on top of B, as shown in Figure 3. The
mass of A is 300 g and the mass of B is 750 g.
The lift is raised vertically using the string at 2.5 m s–2. 19 (a) Find the tension in the string.
[2 marks]
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Figure 3
300 g
750 g
A
B
21
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19 (b) Find the force exerted on the mass B by the mass A.
[3 marks]
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Turn over for the rest of the question
22
1 2 3 3 2 2 1 1 8 D 1 7 5
The lift L and a particle P, of mass m kg, are then attached to the ends of a light
inextensible string. The string passes over a small smooth fixed pulley. The lift and
the particle hang freely with the string taut, as shown in Figure 4. The mass of P is
chosen such that the lift rises vertically with acceleration 2.5 m s–2, as before.
19 (c) Find the value of m.
[3 marks]
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Figure 4
mP
L
AB
23
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19 (d) Calculate the magnitude and direction of the force exerted by the string on the
pulley.
[2 marks]
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END OF QUESTIONS
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