ASSIGNMENT BOOKLET MTE-03 Mathematical Methods (MTE-03) (Valid from 1 st July, 011 to 31 st March, 01) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira Gandhi National Open University Maidan Garhi New Delhi-110068 (For July, 011 cycle)
Dear Student, Please read the section on assignments in the Programme Guide for Elective courses that we sent you after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous evaluation, which would consist of one tutor-marked assignment for this course. The assignment is in this booklet. Instructions for Formatting Your Assignments Before attempting the assignment please read the following instructions carefully. 1) On top of the first page of your answer sheet, please write the details exactly in the following format: ROLL NO: NAME: ADDRESS: COURSE CODE:. COURSE TITLE:. ASSIGNMENT NO.. STUDY CENTRE:.... DATE:.... PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. ) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is valid only upto March, 01. If you have failed in this assignment or fail to submit it by March, 01, then you need to get the assignment for the January, 01 cycle and submit it as per the instructions given in that assignment. 7) It is compulsory to submit the assignment before filling in the exam form. We strongly suggest that you retain a copy of your answer sheets. We wish you good luck.
Assignment (MTE-03) (July 011 March 01) 1. Which of the following statements are true? Give reasons for your answers. i) The function f (x) = sin x is monotone in the interval [ 0, π ]. x If f :R R is defined by f (x) =, then f is increasing on R. 1 + x 1 x i sin is a homogeneous function of x and y y iv) The function f (x) = x has a minima at x = 0. Course Code: MTE-03 Assignment Code: MTE-03/TMA/011-1 Maximum Marks: 100 v) Set { p,q, r} and { p, r,q, p} are equal. (10). a) In an exponential decay process given by M = M 0 e the original amount M 0 has been reduced by a factor 16 in 31 days. How many days did it take to be reduced by a factor of? What is the value of k? (4) b) Evaluate the following limits x + 3x + i) lim x 1 1 x lim x 0 kt 1 + x 1 x c) Find the values of x for which the function f (x) = x 3x, is increasing. () 3. a) A leading brokerage firm charges a 6 % commission on gold purchases in amounts from Rs.500 to Rs.3000. For purchases exceeding Rs.3000, the firm charges % of the amount purchased plus Rs.10. Let x denote the amount of gold purchased (in rupees) and let f (x) be the commission charges as a function of x. i) Describe f (x). What is the domain of f? Find f (1000) and f (5000). (5) b) Draw the graph of the function f : R R :f (x) = x 6x + 5. Obtain the coordinates of the points at which the graph meets the x and y axes. (5) 4. a) A, B, C, D are the points i k, i + j, i 3k, 3i j k respectively. Show that the projection of AB on CD is equal to that of CD on AB. Also find the cosine of their inclination. (5) b) Check the function x 1, 0 x 1 f (x) = 1, x > 1 for continuity and differentiability at x = 1 c) Find the domain of f (x) = (x ) (x 3). () 5 5. a) Expand the binomial ( 1 y). () b) Five animals are assigned five different treatments. In how many ways can this be done? () c) The sum of three numbers in A.P. is 7 and their product is 88, find the numbers. 3 (4) 3
d) Suppose that a ball is thrown straight up into the air and its height after t second is 4 + 48t 16t meters. Determine how long it will take for the ball to reach its maximum height and determined the maximum height. 6. a) Evaluate the following integrals 1 i) 1 (1 + x ) tan x 1/ 1/ 4 x x () () b) Solve the following differential equations i) dy = cot (y + x) 1 dy ( x + y + 3) = x + y + 1 7. a) From the frequency distribution table given below find i) the mean the median i mode iv) variance v) standard deviation. 1 L L 50 5 53 55 56 58 59 61 6 64 f 5 10 1 8 6 (5) b) A fair coin is tossed five times. Find the possibilities that a head appears i) exactly three times at least two times i at the most four times. c) In a certain Poisson frequency distribution the frequency corresponding to 3 successes is one third the frequency corresponding to 4 successes. Find its mean and standard deviation. () 8. a) There are five children in a family of parents AB BB. The children of such parents must have genotype AB or genotype BB. Find the probability that two of the children have genotype AB and three others have genotype BB. b) Suppose the diameter x of a rod has normal distribution N (, 0.16). If the diameter x satisfies 1.8 x. 1, then it is non-defective. Find the probability that the rod is nondefective. (4) c) A bag contains 6 white and 4 red balls. One ball is drawn at random and put aside without noticing its colour. Now another ball is drawn from the bag at random. What is the probability that the second ball is white? 9. a) The following data were obtained in a study of the relationship between the resistance (ohms) and the failure time (minutes) of certain overloaded resistors Resistance 48 8 33 40 36 39 46 40 30 4 44 48 39 34 47 Failure time 45 35 39 45 36 35 36 45 34 39 51 41 38 3 45 Find the coefficient of correlation. (5) b) Suppose that the temperature is normally distributed with expectation 50 C and variance 4 C. What is the probability that the temperature T will be between 48 C and 53 C? What is the probability that T 5 C? (5) 10. a) A continuous random variable X has the p.d.f. 4
Ax(6 x), 0 < x < 6 F(x) = 0, otherwise Calculate the mean and standard deviation of X. Construct the distribution function F (x) 1 and hence evaluate P X >. (6) 4 b) The probability that a certain plant will die within x hours in a certain environment is 1 estimated to be [1 (1 + x ) ]. Determine the probabilities that the plant will die within hours and that it will survive more than 3 hours. Find the corresponding density function. (4) 5