MATHEMATICS AS/P2/M18 AS PAPER 2

Transcription

1 Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks MATHEMATICS AS PAPER 2 March Mock Exam (Edexcel Version) CM Time allowed: 1 hour and 15 minutes Instructions to candidates: In the es 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 round brackets. There are 9 questions in this question paper. The total mark for this paper is 60. 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. AS/P2/M crashmaths Ltd.

2 2 Section A: Statistics 1 Callum collects some data about the average fuel consumption (f miles per gallon) of his car at various different speeds (s miles per hour) between mph. The regression line for f on s using Callum s data is f = s. (a) Give an interpretation of the gradient of Callum s regression line. (1) (b) Use the regression line to estimate the fuel consumption of Callum s car at 85 mph. (1) (c) Comment on the reliability of your estimate in part (b). (1)

3 3 Question 1 continued TOTAL 3 MARKS

4 4 2 Joshua compares the amount of rain in 2015 between Heathrow and the city X on the continent of Asia using the Large Data Set. (a) Write down the name of the city X that Joshua compares with Heathrow. (1) At random, he selects 8 data points about the daily total rainfall, in mm, in May 2015 for the two cities. These 8 data points are shown below. Heathrow: tr City X: (b) Explain what is meant by the reading tr. (1) (c) State one (i) advantage (1) (ii) disadvantage (1) of Joshua using 8 data points from the large data set for his comparisons. Figure 1 below shows a -plot for the data collected by Joshua on the rainfall in the city X in May City X Figure 1 daily total rainfall (mm) A copy of Figure 1 is provided on Page 5. (d) On this copy on Page 5, draw another -plot to represent the data collected by Joshua for Heathrow. In your data processing, take tr to mean 0.0 mm of rainfall and ignore outliers. (2) (e) Compare the amount of rainfall in May 2015 between Heathrow and the city X. (1)

5 5 Question 2 continued City X daily total rainfall (mm) Copy of Figure 1 TOTAL 7 MARKS

6 6 3 Jasmine records the speed of cars, in miles per hour (mph), on a stretch of a UK motorway. Her results are given in the table below. Speed (s mph) Frequency (f) 40 s < s < s < s < s < 85 A histogram has been drawn to represent these data. The bar representing the speed 55 s < 65 has width 2 cm and height 4 cm Speed midpoint (x) (a) Justify the use of a histogram to represent these data. (1) (b) Calculate the width and height of the bar representing the speed 65 s < 70. (2) (c) (i) Show that an estimate of the mean speed of the cars on the motorway is 70 mph. (1) (ii) Find an estimate for the standard deviation of the speeds of the car on the motorway. (2) Jasmine thinks she has miscounted and left out a number of cars that were travelling at 70 mph from the table. She wants to include these missing data points into the calculation of the mean and standard deviation. (d) Without further calculations, state, giving a reason, what effect including these data points will have on your estimate of the standard deviation. (1) (You may use x.) 2 f =

7 7 Question 3 continued

8 8 Question 3 continued

9 9 Question 3 continued TOTAL 7 MARKS

10 10 4 (a) State two conditions for a random variable X to have a binomial distribution. (1) In a regular deck of cards, the probability of taking a clubs card is a quarter. Jeremy takes a deck of cards. Jeremy takes, at random, 32 cards from the deck. He writes down whether the card is a clubs card or not. After each individual card selection, the card is replaced and the deck is shuffled before the next card is picked. Jeremy suspects that the deck is not a regular deck of cards. He uses a hypothesis test to test his claim. (b) Write down the hypotheses Jeremy should use to test his claim. (1) (c) Using a 10% level of significance, find the critical region for a two-tailed test to answer Jeremy s suspicion. You should state the probability of rejection in each tail, which should be less than (3) Jeremy found that 14 of the 32 cards randomly selected from the deck were clubs cards. (d) Comment on Jeremy s suspicion in the light of this observation. (1) It turns out that on a number of occassions Jeremy forgot to shuffle the deck after replacing the card he previously selected. (e) Comment on the validity of the model used to obtain the answer to part (d), giving a reason for your answer. (1)

11 11 Question 4 continued

12 12 Question 4 continued

13 13 Question 4 continued TOTAL 7 MARKS

14 14 5 The events A and B satisfy P(A or B) = 0.63 P(A) = P(B) = 4P(A and B) (a) Show that P(A and B) = (1) (b) Find (i) P(not A), (1) (ii) P(A and not B), (1) (iii) P(A or not B). (1) (c) Show that the events A and B are not statistically independent. (2)

15 15 Question 5 continued TOTAL 6 MARKS TOTAL FOR SECTION A IS 30 MARKS

16 16 Section B: Mechanics Unless otherwise indicated, whenever a numerical value of g is required, take g = 9.8 m s 2 and give your answer to either 2 significant figures or 2 significant figures. 6 A ball, with mass 0.5 kg, is thrown vertically upwards from a point P at 22 m s 1. The point P is 10 m above a large water reservoir. The ball is modelled as a particle that moves freely under the influence of gravity until it reaches the reservoir. (a) Find the maximum height reached by the ball above the reservoir. (3) (b) Find the speed of the ball as it hits the reservoir. (2) After the ball hits the reservoir, it decelerates uniformly and comes to rest in 3 s. (c) Calculate the deceleration of the ball in the reservoir. (1) (d) Hence, find the magnitude of the resistive forces acting on the ball in the reservoir. (2)

17 17 Question 6 continued TOTAL 8 MARKS

18 18 7 A particle P moves on the x axis. At time t s, P is moving with a velocity v m s 1, where v = a bt 2 0 t 5 0 t > 5 and a and b are positive constants. The initial velocity of the particle is 4 m s 1. The magnitude of the acceleration of P at t = 2 is 4 m s 2. (a) Find the values of the constants a and b. (3) (b) Sketch the velocity-time graph for the particle P. On your sketch, you should show clearly the coordinates of any points where the graph crosses or meets the coordinate axes. (2) (c) Find the total distance travelled by the particle P. (4)

19 19 Question 7 continued

20 20 Question 7 continued

21 21 Question 7 continued TOTAL 9 MARKS

22 22 8 [In this question, i and j are perpendicular unit vectors.] A particle P has a position vector (xi + yj) m relative to a fixed origin O. Two variable forces, F 1 N and F 2 N, act on the particle as it moves, where F 1 = (4ysin 2 x + x)i + e y j F 2 = (4ycos 2 x)i 6j The particle passes through the point Q, which has position vector (ai + bj) m relative to O. When the particle passes through Q, it is moving at constant speed. Find the exact value of a and b. (4)

23 23 Question 8 continued TOTAL 4 MARKS

24 24 9 A P B Figure 2 The mass A is held at rest on a rough horizontal table and is attached to one end of a string. The mass of A is 2 kg. The string passes over a pulley P, which is fixed at the edge of the table. The other end of the string is attached to the mass B, which has mass 4.5 kg and hangs freely, vertically below P. The magnitude of the frictional force between A and the table is modelled as having the constant value 0.4R N, where R is the magnitude of the normal reaction force exerted by the table on A. The system is released from rest, with the string taut, as shown in Figure 2. The masses are modelled as particles, the string is modelled as light and inextensible, the pulley is modelled as small and the acceleration due to gravity, g, is modelled as being 9.8 m s 2. The pulley is not modelled as a smooth pulley and the difference in the tension between the two sides, T N, is modelled as T = α + βa where α = 3 N, β = 0.3 kg and a is the acceleration of the masses. Given that the tension in the string at B is greater than the tension in the string at A, (a) calculate the magnitude of the acceleration of the masses. (4) (b) Find the magnitude of the resultant force acting on the pulley. (3) (c) Suggest two improvements that can be made to the model. (2)

25 25 Question 9 continued

26 26 Question 9 continued

27 27 Question 9 continued TOTAL 9 MARKS TOTAL FOR SECTION B IS 30 MARKS TOTAL FOR PAPER IS 60 MARKS Copyright 2018 crashmaths Ltd.