B.E./B.Tech. DEGREE EXAMINATION.
MA 132 — MATHEMATICS — II
(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
(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
(ii) Find the bi–linear transformation which maps onto . Hence find the fixed points. (8)
(b) (i) If is analytic such that , prove that
(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
(ii) Evaluate by Contour integration. (10)
(b) (i) Evaluate where C is . (6)
(ii) Evaluate by contour integration. (10)