Indira Gandhi National Open University (IGNOU) 2005 B.Sc Physics Mathematics Methods in - 3 - Question Paper
PHE-14
BACHELOR OF SCIENCE (B.Sc.)
Term-End Examination
December, 2005
PHE-14 : MATHEMATICAL METHODS IN
Time: 2 hours Maximum Marks ; 50
Note : Attempt all questions. The marks for each question are indicated against ft. Symfeofs have their usual meanings.
(b) Write down the quadratic form corresponding to the following symmetric matrix :
(3
(c) Show that the dot product of two 3-D vectors is a scaJar.
(d) Show by operating on a rectangle that = ou.
(e) Obtain the analytic, function whose real part is
u(x, y) * x
(f) Determine the type of singularity and the region of convergence of
2 '
(g) Show that f(x} - 1, x > 0 cannot be represented by a Fourier integral.
(h) Plot c/q(jc) as a function of x.
2, Determine the eigenvalues and eigenvector of the hermitian matrix
(0 < O'
H
I 0 (
V2
(a) Show that electrical conductivity tensor o transforms
as a contravariant tensor of rank 2. 4
(b) Show that the set of all matrices of order m x n is a group under addition of matrices. Is this group abeli&n ? 6
3. Attempt any two parts : 2x$=*lQ
(a) If C is a circle ( n described in the positive sense and
calculate g(3).
(b) Evaluate the integral L s- where C is a circle i'_i o J 4 + z2
z - 3.
C
(c) Determine the Laurent series for
f{z) ~ s-~ valid for 1 < (z | < 3.
4. (a) Calculate the Laplace transform of the function
' 1, 0<t<it/2
At)
0, othewise
(b) Using Laplace transforms, solve the initial value problem
y" + 5y' + 4y = 0, y(0) = y'(0) = 0. 5
Consider an infinite metal plate placed in the xy plane. Its edge along y-axis is maintained at temperature 0 C and the temperature in the edge along x-axis is given by
1 0C, x > 2
Determine the steady-state temperature distribution of the plate using Fourier transform method. 10
5. Attempt any one part : 10
(a) The generating function lor Hermite polynomial is
) - e**-'2 = ]T
n = Q
Show that H* {x) = 2nH Jx) and n w n 1'
Hnfr) = 2xHfj _ t Of) H'-1 (x).
(b) Expand the function
\ 0 < x < 1
m -
0, l<x<0 in terms of Legendre polynomials.
PHE-14 4
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