Unit I: Asymptotic Notation

Step by Step

Question Papers

Step by Step

Question Papers

(Winter 2014)

Q. a) Solve

6

b) Solve

7

ORQ. a) Solve

6

b) Apply the method of variation of parameters to solve

7

(Summer 2015)

Q. a) Solve

6

b) Solve by method of variation of parameters

7

OR

Q. a) Solve

6

b) Solve

7

(Winter 2014)

Q. a) Find the Laplace transform of:

4

b) Find the inverse Laplace transform of

4

c) Use convolution theorem to find:

6

ORQ. a) Find the inverse Laplace transform of:

7

b) Use Laplace transform method to solve the equation , when x(0) = 0 and x'(0) = 1

7

(Summer 2015)

Q. a) Find the Laplace Transform of

4

b) Using Convolution Theorem find inverse Laplace Transform of

5

c) Find Laplace Transform of:

5

, 0 < t < a , a < t < 2aOR

Q. a) Evaluate:

4

b) Solve the differential equation using Laplace Transform:

, y(0) = y'(0) = 1

5

c) Show that:

5

(Winter 2014)

Q. a) Solve

i)

4

ii)

4

b) Find the z-transform of sin (5k + 3).

5

OR

Q. a) Solve:

4

b) Find the inverse z-transform of: if | z | > | 2 |.

4

c) Solve the difference equation

, y(0) = 0, y(1) = 1 by z-transform.

5

(Summer 2015)

Q. a) Solve the difference equation:

i) , y0 = 0, y1 = 1

4

ii)

4

b) Solve yn+2 4 yn = 0 using z-transform, given y0 = 0, y1 = 2.

5

OR

Q. a) Solve the difference equation:

i)

3

ii)

5

b) Find inverse z-transform of Using direct division method.

5

(Winter 2014)

Q. a) Solve the following equations:

4+4i)

ii)

b) Find the Fourier sine transform of

5

ORQ. a) Solve the following equations:

4+4i) z = px + qy + sin (p + q)

ii) z2 (p2 + q2) = x2 + y2.

b) Find the Fourier transform of:

5

(Summer 2015)

Q. a) Solve the following partial differential equation:

4+4i) xp + yq = nz

ii) x(y2 z2)p + y (z2 x2)q + z(y2 x2) = 0 b) Find Fourier sine transform of:

f(x) = x , 0 < x < 1

= 2 x, 1 < x < 2

= 0 , x > 2

6

OR

Q. a) Solve the following partial differential equation:

4+4i)

ii)

b) Find Fourier Transform of

f(x) = 1 x2, | x | < 1

= 0 , | x | > 1 hence evaluate

6

(Winter 2014)

Q. a) Show that the function u(x, y) = 4xy 3x + 2 is harmonic. Construct the corresponding analytic function f(z) = u(x, y) + i v(x, y). Express f(z) in terms of complex variable z.

6

b) Expand the following function in Laurent's series:

, for 1 < | z | < 3.

7

ORQ. a) Find the bilinear transformation which maps the points z = 1, i 1 into the points w = i, 0, i. Hence find the image of | z | < 1.

6

b) Determine the analytic function

w = u + iv, if v = log(x2 + y2) + x 2y.

7

(Summer 2015)

Q. a) Show that the function u = e2xysin (x2 y2) is harmonic and determine the analytic function f(z) = u + iv as an analytic function of z.

7

b) Find the bilinear transformation which maps the point z = 1, 0, 1 from z-plane into w = 0, i, 3i in w-plane.

6

OR

Q. a) If f(z) is an analytic function of z, prove that:

7

b) Expand for 1 < | z | < 2.

6

(Winter 2014)

Q. a) Find the divergence and curl of

4

b) Show that the vector is solenoidal.

3

c) Evaluate , where s is the surface of the plane 2x + y + 2z = 6 in the first octant and .

7

OR

Q. a) Evaluate the line integral , where C is the square formed by the lines y = 1 and x = 1.

7

b) Find the directional derivative of ,( = 4e2x y + z at the point (1, 1, -1) in the direction towards the point (- 3, 5, 6).

7

(Summer 2015)

Q. a) Find the directional derivative at (1, 2, 3) of V = xy + yz + zx in the direction of the vector 3i + 4j + 5k.

6

b) Find the divergence and curl of the vector = xyzi + 3x2yj + (xz2 y2z)k. 7

OR

Q. a) Determine the constant a so that the vector = (x + 3y)i + (y 2z) j + (x + az)k is solenoidal.

3

b) If u = (v where u, v are scalar fields show that .curl= 0.

4

c) If = (2x2 3z)i 2xyj 4xk, then evaluate: , where V is bounded by the planesx = 0, y = 0, z = 0 and 2x + 2y + z = 4.

6

UNIT I

UNIT VI

UNIT V

UNIT IV

UNIT III

UNIT II

Previous University Examination Question Papers

550

Engineering Mathematics III Engineering Mathematics III

545

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