AS MATHEMATICS

PRACTICE PAPER SET 2

AS MATHEMATICS Paper 1

Practice paper – Set 2 Time allowed: 1 hour 30 minutes Materials • You must have the AQA Formulae for A-level Mathematics booklet. • You should have a graphical or scientific calculator that meets the

requirements of the specification. Instructions • Use black ink or black ball-point pen. Pencil should be used for drawing. • Answer all questions. • You must answer each question in the space provided for that question.

If you require extra space, use an AQA supplementary answer book; do not use the space provided for a different question.

• Show all necessary working; otherwise marks for method may be lost. • Do all rough work in this book. Cross through any work that you do

not want to be marked. Information • The marks for questions are shown in brackets. • The maximum mark for this paper is 80. Advice • Unless stated otherwise, you may quote formulae, without proof, from the booklet. • 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

Version 1.0

2

Section A

Answer all questions in the spaces provided.

1 The graph shows a cubic curve.

Only one of the equations below is the equation of this curve. Which equation is it?

Circle your answer. [1 mark]

y = x2(x + k) y = x(x – k)2 y = 2x2(x + k) y = 3x2(x – k)

2 Solve (5 – x)(2x + 1) > 0


( , ) ( , )−∞ − ∪ ∞1

52

−

15

2,

( ),− −5 1 ( , ) ( , )−∞ − ∪ − ∞5 1

AS Maths Paper 1 Practice papers – Set 2 Version 1.0

3

3

Given that a ≠12

, fully simplify

a a

a a

−

−

−

−

3 12 2

1 12 2

4

2

[4 marks]

AS Maths Paper 1 Practice papers – Set 2

4

4 Stephen is differentiating y xx

= −22

13

2

His working is shown below.

dddd

y xx

y x xy x xxy xx x

−

−

= −

⇒ = −

⇒ = +

⇒ = +

22

2 2

1

13

23 2

6 4

46

4 (a)

Identify the two errors that Stephen has made, explaining what he has done wrong. [4 marks]


5

4 (b)

Find a correct expression for ddyx

[1 mark]

Turn over for the next question


6

5 The depth, h metres, of water in a harbour on a particular day can be modelled by the formula

h = 5 + 3 sin (30t) where t is the time in hours after midday.

5 (a)

Find the maximum depth of water in the harbour, using this model. [1 mark]

5 (b)

The ship, Aqamarine, can enter the harbour when the depth of the water is at least 6.5 metres.

Find the times after mid-day during which Aqamarine can enter the harbour.

[4 marks]


7

6 Prove that

n – 1 is divisible by 3 ⇒ n3 – 1 is divisible by 9 [5 marks]


8

7

The function ( )f x is such that ( )f – –x x x kx= + +3 22 10

The graph of ( )= fy x crosses the x-axis at the points with coordinates (a, 0), (2, 0) and (b, 0) where a < b

7 (a)

Show that k = 5

[2 marks]

7 (b)

Find the exact value of a and the exact value of b

[3 marks]


9

7 (c)

The functions ( )g x and ( )h x are such that

( )( )

g – –

h – –

x x x x

x x x x

= +

= + +

3 2

3 2

2 5 10

8 8 10 10

7 (c) (i) Explain how the graph of ( )fy x= can be transformed into the graph of ( )= gy x

Fully justify your answer. [2 marks]

7 (c) (ii) Explain how the graph of ( )fy x= can be transformed into the graph of ( )= hy x

Fully justify your answer. [2 marks]


10

8 Prove that the equations

x xyx y k+ =+ =

23 2 10

have two distinct real pairs of solutions for all values of k [6 marks]


11



12

9

The point P lies on the curve y = e2x where x = 0

The normal to the curve at P cuts the x-axis at the point Q

The tangent to the curve at P cuts the x-axis at the point R

Find the area of triangle PQR

[7 marks]


13


10 Stephanie is making a wire sculpture. She bends a piece of thin wire of length 48 cm into a pentagon ABCDE, as shown in the diagram.

Angle AED = Angle CDE = 90º, AB = BC = ED and AE = CD

Stephanie wants to maximise the area of the pentagon.

Use calculus to find the maximum area of the pentagon.

Fully justify your answer.

[10 marks]

14

END OF SECTION A TURN OVER FOR SECTION B


15

Section B

Answer all questions in the spaces provided.

11 A particle, of mass 3 kg, accelerates vertically upwards at 2.5 m s–2

Find the magnitude of the resultant force on the particle.


7.5 N 21.9 N 29.4 N 36.9 N

12 The velocity, v m s–1, of a particle at time, t seconds, is given by

v t t= − +212 6 9

Find an expression for the acceleration in m s–2


t t t− +3 24 3 9 t −24 6 t −12 6 t t t− +3 24 6 9


16

13

The graph below shows how the velocity of a lift varies during a 40-second period.

13 (a) Find the total distance travelled by the lift in the 40-second period.

[3 marks]


17

13 (b)

Find the average velocity of the lift during the 40-second period.

[3 marks]


18

14

A tractor, of mass 3000 kg, is used to drag a skip, of mass 1500 kg, across a rough horizontal surface.

The skip is connected to the tractor by a horizontal rope.

A resistive force of magnitude 9000 N acts on the skip.

The resistive force on the tractor is negligible.

The skip and tractor accelerate at 0.2 m s–2.

14 (a) Find the tension in the rope.

[3 marks]

14 (b)

Find the magnitude of the forward driving force acting on the tractor.

[2 marks]


19

15 In this question use g = 9.8 m s–2 A crate, of mass 30 kg, is on the floor of a lift.

The lift has mass 250 kg and is attached to a single cable.

The lift accelerates upwards at 0.6 m s–2

15 (a)

Draw a diagram to show the forces acting on the crate. [1 mark]

15 (b)

Draw a diagram to show the forces acting on the lift. [1 mark]


20

15 (c)

Find the magnitude of the normal reaction force acting on the crate. [3 marks]



21

16 A particle moves in a straight line so that its acceleration, a, at time t is given by

a = pt – q where p and q are constants.

When t = 0, the velocity of the particle is U

When a = 0, the velocity of the particle is U2

16 (a)

Show that qUp

=2

[5 marks]


22

16 (b)

Find the distance travelled by the particle as its velocity changes from U to U2

Give your answer in terms of p and q

[4 marks]

END OF QUESTIONS


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