AS MATHEMATICS...Practice paper – set 1 Time allowed: 1 hour 30 minutes Materials • You must have the AQA Formulae for A-level Mathematics booklet. • You should have a graphical

PRACTICE PAPER SET 1

AS MATHEMATICS Paper 1

Practice paper – set 1 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 Consider the two statements, P and Q, below:

P: θ =2 3sin

4

Q: θ =1

cos2

Which of the relationships below is correct?

Circle your answer. [1 mark]

⇔Q P ⇒P Q ⇐P Q ⇐Q P

2 The lines ay x= − 43

and by x= −34

are perpendicular.

Find the value of ab.

Circle your answer. [1 mark]

34

–12 −43

12

AS Mathematics Paper 1 Practice paper – Set 1 Version 1.0

3

3

The curve, C, has equation y x x k= + +22 5 . The minimum value of C is −34

3 (a) Find the value of k.

[4 marks]

3 (b) The curve C is translated by d

1to obtain the curve ′C

The curve ′C touches the x-axis.

State the value of d.

[1 mark]

AS Mathematics Paper 1 Practice paper – Set 1

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4

4

The graph of y = f(x) is shown below for x≤ ≤0 6

4 (a) Evaluate ( )f dx x∫6

0

[2 marks]


5

4 (b) Deduce values for each of the following, giving reasons for your answers.

4 (b) (i) ( )f dx x−∫7

11

[3 marks]

4 (b) (ii) ( )( )f dx x−∫6

01

[3 marks]


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6

5 (a)

Show that n(n – 1)(n + 1) + 6n ≡ n3 + 5n [1 mark]

5 (b)

Given that n is an integer, prove that 6 is a factor of n3 + 5n [4 marks]


7

6

f ( ) xxx−

=9

5

1 2 for x > 0

Prove that ( )f x is a decreasing function.

[6 marks]


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8

7

A curve has equation x y x+ + =2 2 12 64

A line has equation y mx= +10

7 (a) (i) In the case that the line intersects the curve at two distinct points, show that

( ) ( )m m+ − + >2 220 12 144 1 0

[4 marks]

7 (a) (ii)

Hence find the possible values of m.

[2 marks]


9

7 (b) (i)

On the same diagram, sketch the curve and the line in the case when m = 0 [4 marks]

7 (b) (ii)

State the relationship between the curve and the line. [1 mark]


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10

8

(x – 3) is a common factor of ( )f x and ( )g x where:

( ) ( )( )

f

g

x x x p x q

x x x px q

= − + − +

= − + +

3 2

3 2

2 11 15

2 17 2

8 (a) (i)

Show that 3p + q = 90 and 3p + 2q = 99

Fully justify your answer.

[4 marks]

8 (a) (ii)

Hence find the values of p and q.

[1 mark]


11

8 (b)

( ) ( ) ( )h f gx x x= +

Using your values of p and q, fully factorise ( )h x

[4 marks]

Turn over for the next question


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12

9

Martin tried to find all the solutions of θ θ θ− =2 2 24sin cos cos 0 for ° ≤ ≤ °0 360θ

His working is shown below:

θ θ θ

θ θ θ

θ

θ

θ

θ

− =

⇒ =

⇒ =

⇒ =

⇒ =

⇒ = ° °

2 2 2

2 2 2

2

2

4sin cos cos 0

4sin cos cos

4sin 11

sin41

sin2

30 ,150

Martin did not find all the correct solutions because he made two errors.

9 (a) Identify the two errors and explain the consequence of each error. [4 marks]


13

9 (b) Find all the solutions that Martin did not find. [3 marks]

END OF SECTION A TURN OVER FOR SECTION B


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Section B

Answer all questions in the spaces provided.

10 A block is at rest on a horizontal playground. The normal reaction force acting on the block has magnitude 400 N.

Find the approximate mass of the block.

Circle your answer.

[1 mark]

4 kg 40 kg 400 kg 4000 kg

11 A car travels 2.4 km in 6 minutes.

Find the average speed of the car.

Circle your answer.

[1 mark]

0.4 m s–1 6.67 m s–1 24 m s–1 400 m s–1


15

12

The position vector of point A is 7i + 9j

The position vector of the midpoint of the line joining point A to point B is 3i + 6j

12 (a) Find the position vector of the point B.

[2 marks]

12 (b) Find

AB

[2 marks]


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13

A toy train travels on a straight track, of length 11 metres.

It is initially at rest with the back of the train at one end of the track.

It accelerates uniformly for 8 seconds and reaches a speed of 2 m s–1.

It then travels at this speed until it reaches the end of the track.

13 (a)

Sketch a velocity-time graph for the train.

[2 marks]


17

13 (b)

Find the time it takes the train to reach the end of the track.

[2 marks]

13 (c) (i)

Describe how the model that you have used could be refined.

[1 mark]

13 (c) (ii)

Explain how your refinement would affect your answer to part (b). [1 mark]


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18

14

A car, of mass 1200 kg, tows a trailer, of mass 300 kg. They move with a constant acceleration in a straight line on a horizontal road. The trailer is connected to the car by a horizontal tow bar.

A resistance force of magnitude 40 N acts on the trailer.

A resistance force of magnitude 200 N acts on the car.

The speed of the car and trailer increases from 4 m s–1 to 6 m s–1 as they travel 50 metres.

A tractive force of magnitude P newtons acts on the car.

14 (a)

Find the acceleration of the car and trailer.

[2 marks]


19

14 (b)

Find the tension in the tow bar.

[2 marks]

14 (c)

Find P.

[2 marks]

Turn over for the next question


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20

15

In this question use g = 9.8 m s–2.

A crate of mass 19 kg is attached to a cable, which is used to lift the crate vertically.

The tension in the cable is 204 N.

15 (a)

Assuming that there is no air resistance on the crate, find the acceleration of the crate.

[3 marks]

15 (b)

In reality, there is air resistance on the crate.

State what can be deduced about the acceleration when the effect of air resistance is considered.

[1 mark]


21

16 A particle moves in a straight line so that at time, t seconds, its velocity is v m s–1 where

( )v t t= − +213 21 2

5

Find the speed of the particle when the acceleration is zero.

[5 marks]

END OF QUESTIONS


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AS MATHEMATICS...Practice paper – set 1 Time allowed: 1 hour 30 minutes Materials • You must have the AQA Formulae for A-level Mathematics booklet. • You should have a graphical