Anna University Model Question Paper BE II sem MATHEMATICS — II

MODEL PAPER

B.E./B.Tech. DEGREE EXAMINATION.

Second Semester

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)

  1. Express  in polar co–ordinates.
  2. Simplify
  3. Is the vector , Irrotational?
  4. Find  where .
  5. Prove that real and imaginary parts of an analytic function are harmonic functions.
  6. Find the image of  under the transformation ?
  7. State Couchy’s integral theorem.
  8. What is a removable singularity? Give an example.
  9. For the set of numbers 5, 10, 8, 2, 7 find second moment.
  10. The two regression equations of the variables x and y are :  and  find the mean  and .

PART B — (5 ´ 16 = 80 marks)

  1. (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)

  1. (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)

Or

             (b)     (i)      Change the order of integration and evaluate .       (6)

                       (ii)    Evaluate .                                                     (6)

                       (iii)   Find  using Beta and Gamma functions.                    (4)

  1. (a)     (i)      If  find , if .            
                                                                                                                             (8)

                       (ii)    Find the circulation of  about the closed curve C in the xy plane where .                              (8)

Or

             (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)

  1. (a)              (i)      If  is analytic find  given that

                                         .                                                              (8)

                       (ii)    Find the bi–linear transformation which maps onto . Hence find the fixed points.              (8)

Or

             (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)

  1. (a)     (i)      Expand  in a Laurent series valid in  and
             .                                                                                             (6)

                       (ii)    Evaluate  by Contour integration.                            (10)

Or

             (b)     (i)      Evaluate  where C is .                                           (6)

                       (ii)    Evaluate  by contour integration. (10)

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