AS MATHEMATICS. Paper 1 PRACTICE PAPER SET 1

Transcription

1 PRACTICE PAPER SET 1 Please write clearly, in block capitals. Centre number Candidate number Surname Forename(s) Candidate signature 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. Version 1.0

2 2 Section A Answer all questions in the spaces provided. 1 Consider the two statements, P and Q, below: 2 3 P: sin θ = 4 Q: cosθ = 1 2 Which of the relationships below is correct? Circle your answer. [1 mark] Q P P Q P Q Q P 2 The lines a y = x 4 3 and b y = 3 x are perpendicular. 4 Find the value of ab. Circle your answer. [1 mark]

3 3 2 3 The curve, C, has equation y = 2x + 5 x+ k. The minimum value of C is (a) Find the value of k. [4 marks] 1 3 (b) The curve C is translated by to obtain the curve C d The curve C touches the x-axis. State the value of d. [1 mark] Practice paper Set 1 Turn over

4 4 4 The graph of y = f(x) is shown below for x (a) Evaluate f( x) dx [2 marks]

5 5 4 (b) Deduce values for each of the following, giving reasons for your answers (b) (i) f( x 1 ) dx [3 marks] 6 0 ( 1 ) 4 (b) (ii) f( x) dx [3 marks] Practice paper Set 1 Turn over

6 6 5 (a) Show that n(n 1)(n + 1) + 6n n 3 + 5n [1 mark] 5 (b) Given that n is an integer, prove that 6 is a factor of n 3 + 5n [4 marks]

7 7 1 2x 6 f( x) = 5 x 9 for x > 0 Prove that f ( x ) is a decreasing function. [6 marks] Practice paper Set 1 Turn over

8 A curve has equation x + y + 12x= 64 A line has equation y = mx (a) (i) In the case that the line intersects the curve at two distinct points, show that 2 2 ( 20m+ 12) 144( m + 1) > 0 [4 marks] 7 (a) (ii) Hence find the possible values of m. [2 marks]

9 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] Practice paper Set 1 Turn over

10 10 8 (x 3) is a common factor of f ( x) and g ( x) where: f g 3 2 ( ) 2 11 ( 15) 3 2 ( x) = 2x 17x + px + 2q x = x x + p x+ q 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 11 8 (b) h( x) = f( x) + g( x) Using your values of p and q, fully factorise h ( x ) [4 marks] Turn over for the next question Practice paper Set 1 Turn over

12 Martin tried to find all the solutions of 4 sin θ cos θ cos θ = 0 for 0 θ 360 His working is shown below: sin θ cos θ cos θ = sin θ cos θ = cos θ 2 4 sin θ = sin θ = 4 1 sinθ = 2 θ = 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 13 9 (b) Find all the solutions that Martin did not find. [3 marks] END OF SECTION A TURN OVER FOR SECTION B Practice paper Set 1 Turn over

14 14 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 m s 1 24 m s m s 1

15 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] Practice paper Set 1 Turn over

16 16 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 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] Practice paper Set 1 Turn over

18 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 19 14 (b) Find the tension in the tow bar. [2 marks] 14 (c) Find P. [2 marks] Turn over for the next question Practice paper Set 1 Turn over

20 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 21 16 A particle moves in a straight line so that at time, t seconds, its velocity is v m s 1 where 1 v= ( 3 2 t 21 t+ 2 ) 5 Find the speed of the particle when the acceleration is zero. [5 marks] END OF QUESTIONS

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