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2007 Anna University Chennai General B.E MATHEMATICS — II Question paper

Course: B.E   University/board: Anna University Chennai

Second Semester
(Common to all branches except Information Technology)
Time : Three hours Maximum : 100 marks
Statistical Tables permitted.
Answer ALL questions.
PART A — (10 ´ 2 = 20 marks)
Express in polar co–ordinates.
Is the vector , Irrotational?
Find where .
Prove that real and imaginary parts of an analytic function are harmonic functions.
Find the image of under the transformation ?
State Couchy’s integral theorem.
What is a removable singularity? Give an example.
For the set of numbers 5, 10, 8, 2, 7 find second moment.
The two regression equations of the variables x and y are : and find the mean and .
PART B — (5 ´ 16 = 80 marks)
(i) A survey of 200 families having 3 children selected at random solve the
following results :

Test the hypothesis male and female births are equally likely at 5% level of significance using test. (8)
(ii) A group of 10 rats fed on diet A and another group of 8 rats fed on diet B, recorded the following increase in weight in gms.

In diet A superior to diet B at 5% level of significance? (8)
(a) (i) Find the area of the region bounded by using double
integrals. (6)
(ii) Evaluate . (4)
(iii) Evaluate using Beta and Gamma function. (6)
(b) (i) Change the order of integration and evaluate . (6)
(ii) Evaluate . (6)
(iii) Find using Beta and Gamma functions. (4)
(a) (i) If find , if .
(ii) Find the circulation of about the closed curve C in the xy plane where . (8)
(b) (i) Evaluate where and S is the
surface of the cube using
divergence theorem. (6)
(ii) Verify Stokes theorem for over the surface . (10)
(a) (i) If is analytic find given that
. (8)
(ii) Find the bi–linear transformation which maps onto . Hence find the fixed points. (8)
(b) (i) If is analytic such that , prove that
. (8)
(ii) Prove that the transformation maps the upper half of the z–plane onto the upper half of the w–plane. What is the image of under this transformation? (8)
(a) (i) Expand in a Laurent series valid in and
. (6)
(ii) Evaluate by Contour integration. (10)
(b) (i) Evaluate where C is . (6)
(ii) Evaluate by contour integration. (10)

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