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# MA6251 Mathematics-IIÂ Model Question Paper

### Department : Common for All Branches

MA6251 Mathematics-II Model Question Paper

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

Time: 3 HrsÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Â Â Â Â Â Â Â Â Â Â Â  Â Â Â  Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Maximum: 100 Marks

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Part A (10×2=20 Marks)

• 1) Find the Directional Derivative of at Â 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 â B (5 x 16 = 80)

• 11) (i) Â Â Â  a) Â Â Â Â Â Â Â  Show thatÂ  Â Â is an irrational vector for any value ofÂ  Â butÂ  solenoidal only if
1. b)Â Â Â Â Â Â Â Â  Verify Greens theorem in the XY- plane for Â where C is Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  the boundary of the region given by .

(Or)

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

• 12) (i) a) Â Â Â Â Â Â Â  Solve
1. b) Â Â Â Â Â Â Â  Solve

(Or)

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

1. b) Â Â Â Â Â Â Â  Solve .
• 13) (i) a) Â Â Â Â Â Â Â  Find the Laplace transform ofÂ  .
1. Â b) Â Â Â Â Â Â  Solve by using Laplace transform Â given

(Or)

(ii) Â Â Â  a)Â Â Â Â Â Â Â Â  Find the Laplace transform of

With

1. b) Using convolution theorem find
• 14) (i) a)Â Â Â Â Â Â Â Â  ProveÂ  that an analytic function with constant modulus is constant.
1. b) Find the Mobius transformation that maps the points Â into Â  What Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  are the invariant points of this transformation?

(Or)

(ii) Â Â Â  a) Â Â Â Â Â Â Â  Prove that

1. b) Show that under the transformation Â the image of the hyperbola Â is the Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  lemniscate
• 15) (i) a)Â Â Â Â Â Â Â Â  Evaluate Â where C is Â using Cauchyâs integral formula.
1. b) Â Â Â Â Â Â Â  Find the Laurentâs series expansion for Â valid in the annular region

(Or)

(ii) Â Â Â  a) Â Â Â Â Â Â Â  Using contour integration evaluate

1. Â Â Â Â Â Â  Â Â Â Â  b) Â Â Â Â Â Â Â  Prove that

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