Mathematics AS/P2/D17 AS PAPER 2

Transcription

1 Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks Mathematics AS PAPER 2 December Mock Exam (AQA Version) CM 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 16 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. AS/P2/D crashmaths Ltd.

2 2 Section A Answer all questions in the spaces provided. 1 Find the coordinates of the minimum point on the curve y = 2x 2 8x. Circle your answer. ( 2, 8) (2, 4) (2, 8) ( 2, 4) 2 Which of the following trigonometric identities is correct? Circle your answer. cos 2 x 1+ sin 2 x cos x tan x 1 sin x tan x cos x sin x sin 2 ( 3x) + cos 2 3x ( ) 1

3 3 3 Three vectors p, q and r are defined such that p = 12i aj q = 6i + (9 5a)j r = q p where i and j are perpendicular unit vectors. 3 (a) Given that p and q are parallel vectors, find the value of the constant a. [3 marks] 3 (b) Find r. [3 marks] Turn over

4 4 4 The curve C has the equation y = f(x), where f(x) = px 18 x, x 0, and p is a constant. Figure 1 shows a sketch of the curve C. y 4 y = f(x) x R Figure 1 The region R, shown shaded in Figure 1, is bounded by the curve, the x axis and the line x = 4. 4 (a) Given that the area of the region R is 48 units 2, find the value of p. [5 marks]

5 5 4 (a) [Extra space] 4 (b) Use calculus to find the coordinates of the minimum point on the curve C. [4 marks] 4 (c) Using further differentiation, verify that the point found in (b) is a minimum. [3 marks] Turn over

6 6 5 The curve C has the equation y = f(x), where f(x) = 2x 3 + 9x 2 x 12 5 (a) Show that the curve C crosses the x axis when x = 4. 5 (b) Express f(x) as a product of three linear factors. [3 marks]

7 7 5 (c) Sketch the curve with equation y = f(x). [2 marks] 5 (d) Find all the solutions to the equation 2( x 4) 3 + 9( 4 x) 2 ( x 4) 12 = 0 [2 marks] Turn over

8 8 6 (a) In descending powers of x, find the first four terms in the binomial expansion of 2 1 x 8 [5 marks]

9 9 6 (b) Bernoulli s inequality states that ( 1+ x) r 1+ rx for all integers r 0 and every real number x 1. ( ) r x > 0 6 (b) (i) By using the binomial theorem on 1+ x, prove Bernoulli s inequality for. [2 marks] 6 (b) (ii) Verify Bernoulli s inequality for the case x = 0. 6 (b) (iii) Use a counter-example to show that Bernoulli s inequality is not valid for x < 1. [2 marks] Turn over

10 10 7 The straight line l is perpendicular to the line qx = 2y + 4, where q is a constant. 7 (a) Find, in terms of q, the gradient of the line l. [2 marks] 7 (b) The curve C has the equation y = 1, where p is a constant and x is positive. x + 3 x 2 p The tangent to the curve C at x = 1 is parallel to l. Express p in terms of q. [5 marks]

11 11 7 (b) [Extra space]

12 12 8 (a) Prove, from first principles, that d ( ) dx x3 = 3x 2 [3 marks]

13 13 8 (b) By considering derivatives, or otherwise, evaluate (x + h) 3 x 3 lim h 0 x + h x [2 marks] END OF SECTION A TURN OVER FOR SECTION B

14 14 Section B Answer all questions in the spaces provided. 9 The discrete random variable X has probability function: P( Y = y) = ky 2 y = 2, 1, 0, otherwise Find the value of k. Circle your answer

15 15 10 Jessie has a copy of the Large Data Set for the household purchases in England between (a) Explain why Jessie cannot use the Large Data Set to compare purchases of school milk. 10 (b) Jessie wants to use the large data set to obtain the mean amount of welfare milk purchased per person in England each year. Explain how Jessie can calculate this value using the Large Data Set. [2 marks] Turn over

16 16 11 The total marks, m marks, scored by n pupils in an exam are summarised as follows ( m 20) = 350 m = 45 Find the value of n. [3 marks]

17 17 12 A university wants to select 200 students to interview about their experiences at university. All of the students at the university are listed alphabetically on the university s computer system. The list is enumerated from 1 to and 200 five digit random numbers are generated. The university uses each number, where possible, as the number of the member in the sample. 12 (a) State the name given to the sampling method being used by the university. 12 (b) Suggest why the university s method may not generate a sample of size (c) The first 10 random numbers generated are: How many distinct individuals do these random numbers select for the university to interview?

18 18 13 Chris investigates the price of petrol, p pence, per litre at fuel stations r miles from his house. He collects data from his local petrol stations and summarises his data in the scatter graph in Figure price of petrol per litre (p pence) Distance (r miles) Figure 2 13 (a) Which of the following values is most likely to be the product moment correlation coefficient for these data? Explain your answer. Most likely value Explanation

19 19 13 (b) A new petrol station opens 10 miles from Chris house. Explain why the data obtained by Chris cannot be used to make reliable predictions about the price of petrol at this new petrol station. Turn over for the next question

20 20 14 A fair four-sided die has faces numbered 1, 2, 3 and 4. A coin is biased so that the 1 probability of tossing heads is. The die is thrown once and the number n that it 5 lands on is recorded. The biased coin is then thrown (n + 2) times. So, for example, if the die lands on 3, the coin is thrown 5 times. 14 (a) Find the probability that the die lands on 4 and the coin shows heads 4 times. [3 marks] 14 (b) Find the probability that the number the die lands on is the same as the number of times the coin shows heads. [3 marks]

21 21 BLANK PAGE DO NOT WRITE ON THIS PAGE TURN OVER FOR THE NEXT QUESTION

22 22 15 Emma collects information on the number of hours it took individuals to pass their driving test. To collect her data, Emma uses an opportunity sample. She samples 53 individuals and obtained 50 data points. All of Emma s data is summarised by the histogram in Figure 3. frequency density Number of hours taken to pass Figure 3 15 (a) Suggest why Emma did not obtain 53 data points. 15 (b) Find the number of individuals in Emma s sample that took between hours to pass their driving test. [4 marks]

23 23 15 (b) [Extra space] 15 (c) Calculate an estimate for the median of these data. [2 marks]

24 24 16 (a) Explain briefly what you understand by the critical region of a test statistic. 16 (b) A commercial airline reports that two in every thirty of its passengers do not turn up to their flight, and therefore the airline routinely overbooks their flights. An investigative journalist disputes the airline s claims, believing that the proportion is much lower. The journalist picks 50 scheduled passengers at random and finds that one passenger failed to turn up their flight. Investigate the journalist s claims at the 5% level of significance. [4 marks]

25 25 16 (b) [Extra space] END OF QUESTIONS Copyright 2017 crashmaths Ltd.