MA6251 Mathematics-II Model Question Paper

Department : Common for All Branches

MA6251 Mathematics-II Model Question Paper

SUBJECT CODE/NAME: MA6251/Mathematics-II                                                 Date:

Time: 3 Hrs                                                                            Maximum: 100 Marks

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Part A (10×2=20 Marks)

• 1) Find the Directional Derivative of at  in the direction of
• 2) Prove that is irrotational.
• 3) Solve.
• 4) Transform the equation into a differential equation with constant coefficients.
• 5) Find
• 6) State convolution theorem.
• 7) Show that is harmonic.
• 8) Find the invariant points of the bilinear transformation
• 9) Calculate the residue of at its poles.
• 10) Evaluate where C is the unit circle

PART – B (5 x 16 = 80)

• 11) (i)     a)         Show that    is an irrational vector for any value of   but  solenoidal only if

1. b)         Verify Greens theorem in the XY- plane for  where C is                               the boundary of the region given by .

(Or)

(ii)     Verify Gauss Divergence Theorem for  taken over a       rectangular parallelepiped

• 12) (i) a)         Solve

1. b)         Solve

(Or)

(ii)     a)         Solve   using the method of variation of   parameters.

1. b)         Solve .

• 13) (i) a)         Find the Laplace transform of  .

1.  b)        Solve by using Laplace transform  given

(Or)

(ii)     a)         Find the Laplace transform of

With

1. b) Using convolution theorem find

• 14) (i) a)         Prove  that an analytic function with constant modulus is constant.

1. b) Find the Mobius transformation that maps the points  into   What                    are the invariant points of this transformation?

(Or)

(ii)     a)         Prove that

1. b) Show that under the transformation  the image of the hyperbola  is the                        lemniscate

• 15) (i) a)         Evaluate  where C is  using Cauchy’s integral formula.

1. b)         Find the Laurent’s series expansion for  valid in the annular region

(Or)

(ii)     a)         Using contour integration evaluate

1.             b)         Prove that

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