Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download “I don`t believe in mathematics.” Albert Einstein A2 Maths with Statist
Transcript
“I don’t believe in mathematics.” Albert Einstein A2 Maths with Statistics Assignment (omicron) Due in w/b 19/1/15 14 questions including drill and challenge and exam style questions Drill Part A Evaluate, giving exact answers 4 8 (a) sin 2 xdx (b) 0 4 cos 2 xdx 8 (c) sec xdx 2 4 *note these two lower limits have a negative sign not very clear in print Part B Solve the following equations in the range 0 x 360o (a) cos 2 x 3sin x 2 (b) sec2 x 2tan x (c) cosec2 x 3 cot x 1 Part C Sketch the following functions: show clearly any asymptotes, vertical and horizontal. y 2 ln x 1 y 1 2e x (a) (b) (c) y 10e 2 x Part D Eliminate t from the following pairs of equations: 1 (a) (b) x 3 sin t , x tan t , y cos t y cos t Current work Trapezium rule 1 1. Find an approximate value to 3 decimal places for I = e x tan xdx using four strips. 0 3 2. For the integral I = sec xdx 0 a) Find the exact value of I. b) Use the trapezium rule to find an approximation of I using four strips. c) Find the percentage error of this approximation. Consolidation Statistics: 3. The random variable J has a Poisson distribution with mean 4. (a) Find P( J 10). The random variable K has a binomial distribution with parameters n = 25, p = 0.27. (b) Find P( K 1). D:\840976822.doc Updated: 05/05/2017 4. For a particular type of plant 45% have white flowers and the remainder have coloured flowers. Gardenmania sells plants in batches of 12. A batch is selected at random. Calculate the probability this batch contains (a) exactly 5 plants with white flowers, (b) more plants with white flowers than coloured ones. Gardenmania takes a random sample of 10 batched of plants. (c) Find the probability that exactly 3 of these batches contain more plants with white flowers than coloured ones. Due to an increasing demand for these plants by large companies, Gardenmania decides to sell them in batches of 50. (d) Use a suitable approximation to calculate the probability that a batch of 50 plants contains more than 25 plants with white flowers. Consolidation Pure 5. (a) Show that x 3 14 has a root lying between 2 and 3. (b) 3 Show that x 14 can be rearranged into the form x p x where p is a x2 2 constant to be found. (c) p xn starting with x0 2.5 find, to 3 xn2 2 3 significant figures, a root of x 14 . Justify the accuracy of your answer. Using the iteration formula xn1 Solve the following equations: 6. (a) 7. 5x 4 2x 3 (b) x2 x 6 (a) By writing sin 3 as sin (2 + ), show that sin 3 = 3 sin – 4 sin3 . (b) Given that sin = 3 , find the exact value of sin 3 . 4 D:\840976822.doc Updated: 05/05/2017 8. f(x) = 1 – 3 3 + , x –2. x2 ( x 2) 2 (a) Show that f(x) = x2 x 1 , x –2. ( x 2) 2 (b) Show that x2 + x + 1 > 0 for all values of x. (c) Show that f(x) > 0 for all values of x, x –2. 9. The curve C has equation x = 2 sin y. (a) Show that the point P 2, lies on C. 4 (b) Show that dy 1 = at P. dx 2 (c) Find an equation of the normal to C at P. Give your answer in the form y = mx + c, where m and c are exact constants. 10. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. (b) Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday. During the winter the mean number of yachts hired per week is 5. (c) Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter. During the summer the mean number of yachts hired per week increases to 25. The company has only 30 yachts for hire. (d) Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in summer. In the summer there are 16 Saturdays on which a yacht can be hired. (e) Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts. D:\840976822.doc Updated: 05/05/2017 Challenge Question Given that , find the range of x values if y is not a real number. Preparation: Read about Continuous Random Variables new S2 textbook pages 68-80 old S2 textbook pages 37-57. Drill Answers Part A (a) 1 8 4 (b) 8 1 2 2 (c) 2 Part B (a) 210, 330, 270 (b) 45, 225 (c) 90, 270, 30, 210 Part C Check using your graphic calculator or autograph Part D (a) y 2 x 2 1 (b) x2 y2 1 9 Answers (1) 1.329 (2a) ln(2+√3) (2b) 1.34 (3sf) (2c) 1.5% 3. a) 0.0081 b) 0.0039 4. a) 0.222 b) 0.261 c) 0.257 (5a) let f(x)= x3 -14 ,show change of sign 1 7 (6a) , (6b) –3, 2 7 3 (5b) 7 (5c) 2.41 9 3 16 7. a) proof 8. a), b) and c) proof 9. a) proof .10 a) if 10 b) Poisson is discrete, the Normal is Continuous c) 0.1247 b) d) 0.1977 b) proof d) 0.1357 c) y 2 x 2 4 c) 2.17 ( or 2 or 3) Challenge D:\840976822.doc Updated: 05/05/2017 D:\840976822.doc Updated: 05/05/2017 M THS ASSIGNMENT omicron Name COVER SHEET Current Maths Teacher Please tick honestly: Yes No - explain why. Have you ticked/crossed your answers using the answers given? Have you corrected all the questions which were wrong? How did you find this assignment? Use this space to outline any problems you’ve had and how you overcame them as well as the things which went well or which you enjoyed/learned from. D:\840976822.doc Updated: 05/05/2017
