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Punjab Technical University 2005 M.Tech Electronics and Communication Engineering Advanced Mathematics for Engineers - Question Paper

Sunday, 14 April 2013 05:55Web

ADVANCED MATHEMATICS FOR ENGINEERS EC 501 (M. Tech. first Sem.) May 2k5

Max Marks 100

Note: Attempt any 5 ques.. All ques. carry equal marks.

1. (a) Prove that as the solid figure of revolution for a provided surface area the sphere has maximum quantity.
(b) Using Rayleigh-Ritz method, obtain the potential at a point due to a charged sphere of radius a.
(c) A mass suspended at the end of a light spring having a spring constant K is set into vertical motion. Use Lagranges’s formula to obtain the formula of motion of the mass. 7,6,6

2. (a) obtain the inverse of the matrix

Matrix 1
(b) Solve the subsequent system of equations by Gauss-Siedel Method

X + y + 2z = 4
2X - y + 3z = 9
3X - y - z = 2

3 (a) Draw the flow chart for solution of simultaneous formula using Gauss Elimination Method.
(b) Write down an algorithm for determination of eigenvalues by iteration.

4 elaborate the properties of z-f transform? discuss them with examples.

5 List different properties of Fourier transform and discuss them with the help of examples.

6. (a) define discrete Fourier series and Fourier transform. obtain the Fourier series of f(x) = ekx in (0,2?) where k >0.
(b) obtain circular convolution of the sequence:

Fig. 1

7. What is the relationship ranging from Fourier transform and z-transform? obtain z-transform of provided equation:
x(n) = [3(2n) – 4(3n)] u(n)
Find ROC also

8. Write short notes on the following:
(a) Conformal mapping
(b) Schwarz’s Christofel transformations.
(c) Laplace transform.

ADVANCED ENGINEERING MAHEMATICS MCE 501 MAY 2K5

ADVANCED ENGINEERING MAHEMATICS MCE 501 MAY 2K5

(M. Tech. 1^ST SEMESTER)

Time 3 hours Maximum Marks: 100

Note:. Attempt any Five questions. All questions carry equal marks.

(a) Find the Fourier Sine Transform of e^-|x|

Hence evaluate

(b) Find the solutoion of the integral equation

where a is constant.

2. (a) Find the complex form of Fourier Integral representation for the function:

Verify that it is same as that obtained by using Fourier Cosine Integral representation.

(b) Use Fourier Sin Transform to solve the equation

Under the conditions u(0,t) = 0, u(x,0)=e^-x and u(x,t) is bounded.

3. (a) State and prove Parsevals identity for Fourier Transform.

(b) Using Grouts reduction method, solve the system of equations:

x + y +z = 3

2x y +3 z = 16

3 x + y x = -3

4 (a) Explain how Jocobis method is used to obtain numerical solution of a system of linear equations. What is the condition of convergence for any choice for the first approximation?

(b) find the dominant eigenvalue and the corresponding eigenvector of the matrix:

by power method with unit vector as the initial vector.

5. (a) Find the transformation which will map the interior of the infinite strip bounded by the lines g = 0, n = b on to the upper half of the z-plane.

(b) Find the bilinear transformation which maps the points z=1, I,-1 on to the points w=i,o, -i. Hence find the image of |z|<1

6. (a) Show that a_ij A^kj= Dd_i^k where D is a determinant of order three and A^ij are cofactors of a^ij.

(b) A contravariant tensor has components a, b, c in rectangular co-ordinates; find the components in spherical co-ordinates.

7. (a) If a tensor is skew symmetric w.r.t. the indices p and q in one co-ordinate system, show that it remains skew symmetric w.r.t. p and q in any co-ordinate system.