Find the Harmonic conjugate of 3x^2 y-y^3 and construct the respective analytic function. SAMBALPUR UNIVERSITY INSTITUTE OF INFORMATION AND TECHNOLOGYМАТHЕMАТICS II (В.ТЕСН. III)(ASSIGNMENT)Q1.i. Define Analytic function. Show that |Z| is no-where analytic, where z=x+iy .Justify youranswer.23ii. Find the Harmonic conjugate of 3x y-y and construct the respective analytic function.1+iiii. Calculate the principal part of 1- 3i1/ziv. Find out the singular points of the function fz=in ]z|s2 and categorise them.e3(z-1) (sin2z)v. Evaluate the integral:dz, C:z-T|=4sinzQ2.i. Calculate all possible Laurent series expansion of fz=_z.3Z +zii, a. What is a singular point? Categorise them. Give examples of each category.b. Calculate the residue of the the function cosecz.cosechz.Q3. (a) State Baye's theorem.(b) Of the 15 girls in a class, 6 have blue eyes. If 4 of the girls are selected at random what is theprobability that (i) all have blue eyes, (ii) at-least 2 have blue eyes.(c) Two jet planes bomb a target one after other. The probability of each correctly scoring ahit 0.4 and 0.3 respectively. The score will bomb only if the first misses the target, then findout the probabiliy that the target is the target is hit.(d) Determine the probability of getting 8 exactly twice in the 3 tosses with a pair of fair dice.(e) Give a short note on Poisson distribution.(f) What is statistical estimation? Categorize it. And give an example of point estimation.(g) State and prove Law of total probability.(a) Calculate the standard deviation of Hypergeometric probability distribution.Q4.(b) If X be a Binomially distributed random variable with mean 2 and variance 4/3, then findthe distribution of X.(c) The mean number of defective parts in a sample of 20 is 2. Out of 2000 such samples howmanywouldyou expect to contain at least 3 defective parts?(d) Suppose on an average one person per 1000 makes a calculation error in income taxreturn. If 10000 forms are selected at random and verified, find the probability that 7 or 8 or 5of the forms will be in error.(e) The amount of pollutant X released by an industry should lie between 30 and 35. Assumethat X is normally distributed with mean 33 and standard deviation 3. The industry gets aprofit of Rs. 100 when 30 (f) Find P(X66.75) and P(X

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Find the Harmonic conjugate of 3x^2 y-y^3 and construct the respective analytic function. SAMBALPUR UNIVERSITY INSTITUTE OF INFORMATION AND TECHNOLOGYМАТHЕMАТICS II (В.ТЕСН. III)(ASSIGNMENT)Q1.i. Define Analytic function. Show that |Z| is no-where analytic, where z=x+iy .Justify youranswer.23ii. Find the Harmonic conjugate of 3x y-y and construct the respective analytic function.1+iiii. Calculate the principal part of 1- 3i1/ziv. Find out the singular points of the function fz=in ]z|s2 and categorise them.e3(z-1) (sin2z)v. Evaluate the integral:dz, C:z-T|=4sinzQ2.i. Calculate all possible Laurent series expansion of fz=_z.3Z +zii, a. What is a singular point? Categorise them. Give examples of each category.b. Calculate the residue of the the function cosecz.cosechz.Q3. (a) State Baye's theorem.(b) Of the 15 girls in a class, 6 have blue eyes. If 4 of the girls are selected at random what is theprobability that (i) all have blue eyes, (ii) at-least 2 have blue eyes.(c) Two jet planes bomb a target one after other. The probability of each correctly scoring ahit 0.4 and 0.3 respectively. The score will bomb only if the first misses the target, then findout the probabiliy that the target is the target is hit.(d) Determine the probability of getting 8 exactly twice in the 3 tosses with a pair of fair dice.(e) Give a short note on Poisson distribution.(f) What is statistical estimation? Categorize it. And give an example of point estimation.(g) State and prove Law of total probability.(a) Calculate the standard deviation of Hypergeometric probability distribution.Q4.(b) If X be a Binomially distributed random variable with mean 2 and variance 4/3, then findthe distribution of X.(c) The mean number of defective parts in a sample of 20 is 2. Out of 2000 such samples howmanywouldyou expect to contain at least 3 defective parts?(d) Suppose on an average one person per 1000 makes a calculation error in income taxreturn. If 10000 forms are selected at random and verified, find the probability that 7 or 8 or 5of the forms will be in error.(e) The amount of pollutant X released by an industry should lie between 30 and 35. Assumethat X is normally distributed with mean 33 and standard deviation 3. The industry gets aprofit of Rs. 100 when 30 (f) Find P(X>66.75) and P(X<62.83) if a random sample of size 36 is drawn from an infinitepopulation with mean 63 and standard deviation 9 assuming means are normallydistributed.(g) A company claims that a mean life time of tube lights is 500 hrs. Is the claim of thecompany tenable if a random sample of 25 tube lights produced by a company has mean 518hrs and standard deviation 40 hrs. Company is satisfied if t falls between tand t-0.010.01

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