R13 SET - 1 PART-A. is analytic. c) Write the test statistic for the differences of means of two large samples. about z =1.

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1 R3 SET - II B. Teh I Semester Regular Examinations, Jan - 5 COMPLEX VARIABLES AND STATISTICAL METHODS (Eletrial and Eletronis Engineering) Time: 3 hours Max. Marks: 7 Note:. Question Paper onsists of two parts (Part-A and Part-B). Answer ALL the question in Part-A 3. Answer any THREE Questions from Part-B ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ PART-A. a) Write Cauhy Riemann equations in polar form. b) Find a and b if f(z) ( x xy + ay ) + i( bx y + xy) is analyti. ) Write the test statisti for the differenes of means of two large samples. d) Expand ( ) e z f z 3 ( z ) about z. e) Determine the poles of tanz and find the residue at the simple poles f) Find the bilinear transformation whose fixed points are and g) Three masses are measured as 6.34,.84,35.97 kgs with standard deviation.54,.,.46 kgs. Find the mean and standard deviation of the sum of the masses. h) A sample size was taken from population. Standard deviation of sample is.3. Find the maximum error with 99% onfidene (M+3M+M+3M+3M+3M+3M+3M) PART-B. a) Find the Analyti funtion whose real part is u( x, y) sin x. osh y + os x b) Show that the funtion f(z) z z is differentiable but not analyti at origin. 3. a) Evaluate ze z ( z i) dz 3 π integral formula b) Obtain Laurent s expansion for, where is a irle of radius 4 with entre at origin, by Cauhy f z) in z > ( ( z + ) ( z + ) of

2 R3 SET - π dθ 4. a) Evaluate 5 + 4osθ b) Evaluate osaxdx ( x + a ) 5. a) Disuss the transformation w osz. b) Find the Bilinear transformation whih maps z -,, onto w, i,3i. 6. a) A random sample of size 64 is a taken from normal population with mean 5.4 and S.D 6.8. What is the probability that the mean of samples will (i) exeed 5.9 (ii) less than 5.6 (iii) between 5.5 and 5.3. b) Find the 95% onfidene limits for mean of the population from whih sample was taken from 5,7,,8,6,9,7,,3,4. 7. a) A ollege management laims that 75% of all single women appointed for teahing job get married and quit the job in two years. Test this hypothesis at 5% level of signifiane if among 3 suh teahers, got married within years and quit then jobs b) In a test given two groups of students, the marks obtained are as follows First Group Seond group Examine the signifiant differene between the means of the marks of the two group at 5% level. of

3 R3 SET - II B. Teh I Semester Regular Examinations, Jan - 5 COMPLEX VARIABLES AND STATISTICAL METHODS (Eletrial and Eletronis Engineering) Time: 3 hours Max. Marks: 7 Note:. Question Paper onsists of two parts (Part-A and Part-B). Answer ALL the question in Part-A 3. Answer any THREE Questions from Part-B ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ PART-A. a) Define harmoni funtion and give an example 4 b) If is a simple losed urve then evaluate (sin3z + z + e z ) dz ) Write test statisti for the differenes of means of two small samples z e d) Find the residue of f ( z) 3 ( z ) at z e) Determine the poles of tanz and find the residue at simple pole f) Find the bilinear transformation whose fixed points are i and -i g) Define two types of Errors in sampling. h) A sample size was taken from population with S.D of sample is.3. Find the maximum error with 99% onfidene (M+3M+M+3M+3M+3M+3M+3M) PART-B. a) Find the Analyti funtion whose imaginary part is v( x, y) sin x sin y osh y + os x b) Show that the untion f(z) xy is not analyti at origin although CR equations are satisfied at the point 3. a) Evaluate z ze 3 ( z ) dz b) Obtain Laurent s expansion for where is the irle with radius 3 by Cauhy integral formula f ( z) in < z < ( z + )( z + ) of

4 R3 SET - π dθ 5 4sinθ 4. a) Evaluate b) Evaluate dx 6 (x + ) 5. a) Disuss the transformation w sinz b) Find the Bilinear transformation whih maps z, i, onto w -,-i, 6. a) Show that Sample mean is the unbiased estimator of population mean b) A random sample of size taken from normal population with mean 76 and S.D 6. What is the probability that the mean of samples will (i) exeed 78 (ii) less than 6 (iii) between 75 and a) The mean prodution of rie in a sample of fields is lb per are with S.D of lb. Another sample of 5 fields gives the mean lb and S.D lb. Find if the two results are onsistent at % level. b) The nine items of the sample had the following values: 45,47,5,5,48,47,49,53, and 5. Does the mean of nine items differ signifiantly from the population mean of at % level. of

5 R3 SET - 3 II B. Teh I Semester Regular Examinations, Jan - 5 COMPLEX VARIABLES AND STATISTICAL METHODS (Eletrial and Eletronis Engineering) Time: 3 hours Max. Marks: 7 Note:. Question Paper onsists of two parts (Part-A and Part-B). Answer ALL the question in Part-A 3. Answer any THREE Questions from Part-B ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ PART-A. a) Find the invariant points of + z w. z b) Find the Harmoni onjugate of log x + y. dz ) Evaluate, where : z 5. z 3 z e d) Find the residue of f ( z) ( z ) at z. e) Determine and lassify the singular point of f(z) z sin. z f) Write any three harateristis of Normal Distribution. g) Define Hypothesis, Critial region and Standard error. h) If we an assert 95% that maximum error is.5 and p. find the sample size. (M+3M+M+3M+3M+3M+3M+3M) PART-B. a) Find the Analyti funtion given that sin x v + u. osh y os x 3 x y( y ix) b) Show that the untion f(z) 6 x + y is not analyti at origin although CR equations are satisfied at the point. of

6 R3 SET a) Evaluate z e where is the unit irle by Cauhy integral formula + ( z ) dz b) Obtain Laurent s expansion for f ( z) in z < ( z + )( z + ) π dθ 4. a) Evaluate. 3 sinθ b) Evaluate dx. 4 (x + ) 5. a) Disuss the transformation w z. b) Find the Bilinear transformation whih maps z, i, on to w,i,. 6. a) Show that Sample variane is not the unbiased estimator of population variane b) A random sample of size 36 is taken from normal population with mean 55 and S.D 5. What is the probability that the mean of samples will (i) exeed 57 (ii) less than 6 (iii) between 55 and a) A sample of 45 items is taken from a population with mean 3 and S.D. Test whether the sample has ome from the population with mean 9. Also alulate 95% onfidene limits of the population mean. b) Two samples are drawn from two normal populations from the following data, test whether the two samples have the same variane at 5% level. Sample I Sample II of

7 R3 SET - 4 II B. Teh I Semester Regular Examinations, Jan - 5 COMPLEX VARIABLES AND STATISTICAL METHODS (Eletrial and Eletronis Engineering) Time: 3 hours Max. Marks: 7 Note:. Question Paper onsists of two parts (Part-A and Part-B). Answer ALL the question in Part-A 3. Answer any THREE Questions from Part-B 4. Probability tables Normal, t, F and hi square tables are required ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ PART-A. a) Find the invariant points of w. z i b) Find the Harmoni onjugate of x y + xy 3dz ) Evaluate, where : z. z +. z d) Evaluate ze dz where is the unit irle by residue theorem. e) Determine and lassify the singular point of f(z) sin. z f) Write any three harateristis of hi square Distribution. g) Write the test statisti for testing the equality of two population means for small samples and large samples. h) What is the maximum error one an expet to make with the probability.9, when using the mean of random sample 64 to estimate population mean with σ.6 (M+3M+M+3M+3M+3M+3M+3M) PART-B sin x. a) Find the Analyti funtion given that v + u. y y e + e os x b) Prove that an analyti funtion with onstant real part is onstant. of

8 R3 SET - 4 z ze 3. a) Evaluate ( z a) dz where the point a lies within the losed urve by Cauhy integral 3 formula. b) Obtain Laurent s expansion for f ( z) in z + > ( z + )( z + ) π dθ 4. a) Evaluate. 3+ osθ x dx b) Evaluate. ( x + ) 5. a) Disuss the transformation w e z. b) Find the Bilinear transformation whih maps z, i, - onto w i,,-i. 6. a) Write a short note on properties of Estimators. b) A random sample of size 5 is taken from normal population with mean 55 and S.D 5. What is the probability that the mean of samples will i) exeed 57 ii) less than 6 between 53 and 58 (iii) 7. a) A ollege management laims that 8% of all single women appointed for teahing job get married and quit he job within two years of time. Test this hypothesis at 5% level of signifiane if among suh teahers, got married within two years and quit their jobs. b) Two investigations study the inome of group of persons by the method of sampling. Following results were obtained Investigator Poor Middle Well A 6 3 B 4 4 Show that the sampling tehnique of at least one of the investigators is suspeted at 5% level. of