FP0060 1 (Continued) UNIVERSITY OF WARWICK File-based Open Book Assessment (2 hours) Warwick Foundation Studies FP0060 STATISTICS AND FURTHER MATHEMATICS Student Instructions: 1. Read these instructions fully and before you start the assessment. Read through the paper at least once before you start writing 2. There are 10 questions. Candidates should attempt every question. 3. You should not submit answers to more than the required number of questions. 4. The maximum mark for this paper is 100. 5. Answers must be handwritten, either on paper or using an electronic tablet, and uploaded to the AEP within a single PDF file. 6. You should write legibly, preferably in dark blue or black ink. If you use a pencil ensure that it is not too faint to be captured by a scan or photograph. 7. When uploading photographs or scans of your work please check for legibility before uploading. It is your responsibility to ensure that your work can be read. 8. Start each question on a new page, and only write on one side of each piece of paper. Label each question (number and part) clearly. 9. Write your student ID number in the top corner of each page (but not your name). 10. You are allowed to access module materials, notes and resources on Moodle during the assessment. You must not use any other websites or resources. 11. You may use a scientific calculator and the formula book provided to help you answer the questions. 12. You must not communicate with any other candidate during the assessment period. You may be contacted for a video call to discuss some of your answers. You should therefore keep the original copy of your work and be available via email/Teams in case you are contacted. 13. By starting this assessment you are declaring yourself fit to undertake it. You are expected to make a reasonable attempt at the assessment by answering the questions in the paper. 14. You have 2 hours to complete the assessment. You should upload your assignment to the AEP within the required timeframe (see below). FP0060 2 (Continued) Please note that: - We strongly recommend you use Google Chrome or Mozilla Firefox to access the Alternative Exams Portal. - You must have completed and uploaded the assessment within the fixed time available. - You have an additional 45 minutes beyond the stated duration of this assessment to allow for downloading and uploading the assessment, your files, and technical delays. This time should not be used to continue writing your answers. - Late submissions are not accepted. Once started, you must complete the assessment within 2 hours and then upload your files within 45 minutes. - If you are unable to submit your assessment, you can record Mitigating Circumstances. Your case will be considered and you will be notified of the outcome. - If you have pre-approved Alternative Exam Arrangements (Reasonable Adjustments) that permit extra time and/or rest breaks this time will be added on to the stated duration. FP0060 3 (Continued) Support During the Assessment Operational Support Use the AEP to seek advice immediately if during the assessment period: • you cannot access the online assessment; • you believe you have been given access to the wrong online assessment; Notify the invigilator (using the ‘Contact an Invigilator’ tool in the AEP) if you have an academic query about the assessment. Operational support will be available between 09:00 and 17:00 (UK Time). Technical Support • If you experience any technical difficulties with the Alternative Exam Portal please contact helpdesk@warwick.ac.uk • If you experience technical difficulties with Moodle please contact moodle@warwick.ac.uk Technical support will be available between 09:00 and 17:00 GMT (UK Time). Academic Support If you have an academic query, contact the invigilator (using the ‘Contact an Invigilator’ tool in AEP) to raise your issue. Please be aware that two-way communication in AEP is not currently possible. Academic support will be provided for the duration of the full examination period (i.e. 2 hours (+45 min)). Academic support beyond this time is at the discretion of the department. Other Support Write to your module convenor/department immediately if you cannot complete your assessment for the following reasons: • you lose your internet connection; • your device fails; • you become unwell and are unable to continue; • you are affected by circumstances beyond your control (e.g. fire alarm). Your assessment starts below. FP0060 1. The events A and B are independent with P (A) = 1 6 and P (A ∪B) = 4 9 . (a) Find P (B). Give your answer as a fraction. (3 marks) (b) Draw a Venn diagram to illustrate the events A and B and the probabilities for each region. (3 marks) (c) Find P ((A′ ∩B) ∪ (A ∩B′)). Give your answer as a fraction. (2 marks) (d) Find P (B′ |A). Give your answer as a fraction. (2 marks) 2. The probability distribution of the discrete random variable X is given by: x 1 3 5 P (X = x) 0.2− a 2a 0.8− a where a is a constant. (a) State the range of possible values of a. (2 marks) (b) Show that E(X) is independent of a. (2 marks) (c) Given that Var(X) = 1.6, calculate the value of a. (3 marks) Two independent observations of X, denoted X1 and X2, are considered. (d) Calculate P (X1 +X2 = 6). Give your answer as a fraction. (3 marks) 4 (Continued) FP0060 3. The length of time, in minutes, that patients spend waiting at a doctor’s surgery is modelled by the continuous random variable, T , with the following cumulative distribution function: F (t) =  0 t < 0 1 56000 (−t3 + 45t2 + 1200t) 0 ≤ t ≤ 40 1 t > 40 (a) Find the probability that a patient has to wait more than 20 minutes. Give your answer as a fraction. (3 marks) (b) Define fully the probability density function f(t) of T . (3 marks) (c) Find the modal waiting time in minutes of a patient at the doctor’s surgery. (3 marks) (d) Give one reason why this model may need to be refined. (1 mark) 4. A manufacturing company produces batteries. The probability that a battery is faulty is 0.25. A random sample of 10 batteries is selected. (a) Write down a suitable distribution to model the number of faulty batteries in the sample. (1 mark) (b) Write down two assumptions you have made for your model in part (a). (2 marks) (c) Find the probability that there is exactly 2 faulty batteries in the sample. Give your answer to four decimal places. (2 marks) (d) Find the probability that there are at least 3 faulty batteries in the sample. Give your answer to four decimal places. (2 marks) The batteries are sold in multipacks of 10. A customer buys 4 multipacks. (e) Find the probability that exactly 2 of these multipacks contain at least 3 faulty batteries. Give your answer to four decimal places. (3 marks) 5 (Continued) FP0060 5. In a population, 10% of the people are left-handed. A random sample of 50 people is taken from this population. (a) Using a suitable Binomial distribution, calculate the probability that in this sample there are more than 10 people in the sample who are left-handed. Give your answer to four decimal places. (2 marks) A second random sample of 400 people is taken from the population. (b) Using a suitable Normal approximation, estimate the probability that in this sample there are between 30 and 50 (inclusive) people who are left-handed. Give your answer to four decimal places. (8 marks) 6. Use the method of Lagrange Multipliers to optimise the function f(x, y, z) = x+ y + z subject to the constraint g(x, y, z) = x2 + y2 + z2 = 3 (10 marks) 7. The plane Π1 has vector equation: r ·  2-5 1  = 4 (a) Find the perpendicular distance from the point (7, 3, 13) to the plane Π1. (2 marks) The plane Π2 passes through the point (-2, -3, 2) and is perpendicular to the vector  31 -1 . (b) Show that the Cartesian equation of Π2 is 3x+ y − z = −11. (2 marks) (c) Show that Π1 and Π2 are perpendicular. (2 marks) (d) Find an equation for the line of intersection of Π1 and Π2. Give your answer in the form r = a + tb, where a and b are constant vectors and t is a scalar parameter. (4 marks) 6 (Continued) FP0060 8. The matrix A is given by A = 1 0 20 1 -1 0 0 2  (a) Calculate the determinant of A. (2 marks) The plane Π1 has vector equation: r =  2-2 4 + λ 10 1 + µ -22 1  The plane Π1 is transformed to the plane Π2 by the transformation matrix represented by the matrix A. (b) Find an equation of the plane Π2 in the form r · n = ρ. (8 marks) 9. Find in the form y = f(x) the general solution of the differential equation d2y dx2 − 3dy dx + 2y = 4x2 (10 marks) 10. (a) Show that the substitution z = 1 y3 transforms the first order differential equation dy dx + y x = −2y4, x > 0, y > 0 (?) into the linear first order differential equation dz dx − 3 x z = 6 (??) (3 marks) (b) Find the general solution of the differential equation (??). (5 marks) (c) Hence, find the general solution of the differential equation (?). Give your answer in the form y3 = f(x). (1 mark) Given that y = 1 at x = 1. (d) Find the particular solution of the differential equation (?). (1 mark) 7 (End)






























































































































































































































































































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