Whitby Community College Your account expires on: 8 Nov, 2013

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1 Page of 7 04/03/03 To print higher-resolution math symbols, click the Hi-Res Fonts for Printing button on the jsmath control panel Whitby Community College Your account expires on: 8 Nov, 03 DRV's Exam questions Syllabus: EdExcel Questions: 0 Time: hour 35 minutes Total Marks: 80 This paper contains a set of questions followed by the corresponding mark schemes. The time you should spend on each question together with its worth in marks is also given. The content of this paper is based on material from a wide selection of national and international examination boards and organisations. You are advised to have: a set of geometrical equipment, pen, HB pencil, eraser. Check if you are allowed a calculator. Some examinations, but not all, allow calculators, including graphical models. NOTES: The following browsers have been tested with this facility: Mozilla Firefox, Microsoft Internet Explorer and Google Chrome. Microsoft Internet Explorer (IE) was tested on versions 7 to 0 - IE6 is no longer officially supported by See the website for the small font problem with IE7 upwards, and IE8 was tested in IE7 compatibility mode. The latest versions of Google Chrome have a customised print preview which allows you to easily adjust the margins and whether or not headings are printed. Best results are when the background printing of images and colours is enabled (not available in Chrome). There are known printing format issues with the Opera web browser and we do not recommend using this browser. Many of the questions use the jsmath applet. This requires special fonts to be installed for successful printing. These fonts can be downloaded from: Use the Download the TeX fonts option. Full instructions for their use can be found at: is available on annual subscription, where you will find 807 pages of content, including 906 interative questions that can be printed and 957 on-line assessments. Visit to find out further details about subscribing.

2 Page of 7 04/03/03 03 MathsNet A-Level Plus. Licensed to Whitby Community College. Page: DRV's Exam questions Questions: 0 Time: hour 35 minutes Total Marks: 80 Q - ID: 68 The discrete random variable X has the probability function shown in the table below. x P (X x) Find [3 marks, 4 minutes] Q - ID: 337 The discrete random variable X can take only the values, or 3. For these values the cumulative distribution function is defined by (x + k) F (x) for x,,3 where k is a positive integer. Find k. Find the probability distribution of X. [5 marks, 6 minutes] Q3 - ID: 55 The discrete random variable X has E(X) 3 and E(X ) 6 Find the value of Var(X). Find E(X). A circle has radius X + 8. Find in terms of values for the mean and variance of the circumference, C, of the circle. The area S of the circle with radius X + 8 is to be expressed in the form (X + ax + b) where a and b are constants. Find values for a and b. (d) Find in terms of the value of E(S). [0 marks, minutes] Q4 - ID: 343 [6 marks, 7 minutes] A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.4 of finding oil, independently of all other sites. The random variable X represents the number of sites investigated. The probability distribution of X is shown below. r 3 4 P (X r) Find the expectation and variance of X. It costs 7000 to investigate each site. Find the expected total cost of the investigation.

3 Page 3 of 7 04/03/03 03 MathsNet A-Level Plus. Licensed to Whitby Community College. Page: 3 Q5 - ID: 758 The discrete random variable X has the probability function shown in the table below. x P (X x) Find E(8X + ) [4 marks, 5 minutes] Q6 - ID: 834 The discrete random variable X has the probability function shown in the table below. x P (X x) Find P (9 X ), F () E(8X + 7) Q7 - ID: 460 A discrete random variable X only takes positive integer values. It has a cumulative distribution function F (x) P (X x) defined in the table below. x F (x) Determine the probability function P (X x), of X. Calculate E(X) and show that Var(X) 6 9 Given that Y X + 8, find the mean and variance of Y. [ marks, 4 minutes] Q8 - ID: 597 The discrete random variable X has the following probability distribution. x P (X x) Given that E(X) 3, find the values of and. Write down F ( 8). Evaluate Var(X). Find the value of (d) E(4X 3), (e) Var(4X + 3). [5 marks, 8 minutes]

4 Page 4 of 7 04/03/03 03 MathsNet A-Level Plus. Licensed to Whitby Community College. Page: 4 Q9 - ID: 760 A discrete random variable X has the probability function shown in the table below. x P (X x) a a Given that E(X) 4 7, find a. Find the value of Var(X). Find the value of P (X 5 5). Q0 - ID: 563 The discrete random variable X has probability function k(x 9) x 4, 5, 6 P (X x) 0 otherwise where k is a positive constant Show that k. 50 Find E(X) and Var(X). Find Var(5X 3). [ marks, 3 minutes]

5 Page 5 of 7 04/03/03 03 MathsNet A-Level Plus. Licensed to Whitby Community College. Page: 5 DRV's Exam questions - Mark Scheme - ID: 68 P (X) [3 marks, 4 minutes] A - ID: 337 [5 marks, 6 minutes] (b F (3) P (X ) P (X ) P (X 3) (3 + k) (3 + k) 3 + k 8 k 5 36 F () F () F () 49 F (3) F () x P (X x) 3 5 A3 - ID: 55 (d) Var(X) C (X + 8) S E(S) E(X ) E (X) 7 E(C) E(X) + 6 Var(C) 4 Var(X) 8 8 (X + 8) (X + 6X + ) E( (X + 6X + )) (E(X ) + 6E(X) + ) 8 B [0 marks, minutes] A4 - ID: 343 E(X) E(X ) Var(X) 9 4 ( 777) 53 Expected cost B [6 marks, 7 minutes]

6 Page 6 of 7 04/03/03 03 MathsNet A-Level Plus. Licensed to Whitby Community College. Page: 6 A5 - ID: 758 E(8X + ) 8E(X) + 8 xp+ 8( ) + 47 A [4 marks, 5 minutes] A6 - ID: 834 P (9 X ) F () P (0) + P () + P () P (8) + P (9) + P (0) + P () E(X) E(8X + 7) E(8X + 7) 97 4 B A7 - ID: 460 function x P E(X) xp(x) E(X ) x P(X) Var(X) E(x ) E(x) 6 9 E(Y ) E(X)+897 Var(Y ) Var(X) A B [ marks, 4 minutes] A8 - ID: 597 [5 marks, 8 minutes] P (X) E(X) (04 ) (d) (e) F ( 8) Var(X) E(4X 3) Var(4X + 3) E(X ) E(X) 5 (3 ) 89 4E(X) Var(X) 46 4 MA

7 Page 7 of 7 04/03/03 03 MathsNet A-Level Plus. Licensed to Whitby Community College. Page: 7 A9 - ID: 760 E(X) 4 7 E(X ) P (X 5 5) a + 6 ( a) 4 7 a a + 6 ( a) 6 7 Var(X) 6 7 (4 7) Var(X) 0 6 P(4)+P(5) ID: 563 P E(X) E(X ) Var(X) Var(5X 3) k(4 9) + k(5 9) + k(6 9) k( ) k(50) 50 k XP(X) 4k(4 9) + 5k(5 9) + 6k(6 9) 70 7 k( ) X P(X) 6k(4 9) + 5k(5 9) + 36k( 6 9 ) k( ) Var(X) [ marks, 3 minutes]