Calicut University 2007 M.Sc Physics Phy 101- MATHEMATICAL - Question Paper
FIRST SEMESTER M.Sc. DEGREE EXAMINATION, JANUARY 2007
Physics
PHY 101- MATHEMATICAL PHYSICS
(2003 admission onwards)
(Pages 2)
Name.
Reg. No..................................
FIRST SEMESTER M.Sc. DEGREE EXAMINATION, JANUARY 2007
Physics Phy 101MATHEMATICAL PHYSICS (2003 Admission onwards)
Time : Three Hours Maximum : 80 Marks
Section A
Answer any five questions. Each question carries 4 marks t
1. Prove that the necessary and sufficient condition for a square matrix to posses an inverse is that it is nonsingular.
cos 6 sin 6 0 sin 6 cos 6 0 0 0 1
2. Diagonalise the matrix
3. Obtain Laplace equation in Cartesian coordinates using variable separable method.
OQ
4. Show that C J /(*) where C stands for the Lapl&ce transform
oo
5. Evaluate / cos a:2 dx.
6. Using the generating function of the Laguerre polynomial Ln(x), establish that L'n(x) - nL'n_x{x) + nLn-iix) = 0.
7. Find the Fourier of the function shown in the figure. - *
T
2
3T
2
, where a is a constant.
(s2 + a2)2
[5 x 4 = 20 marks]
8. Use convolution theorem to evaluate
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D 27876
Section B /r
Answer any two questions. Each question carries 20 marks '
Obtain the expression for (a), grad rp and (b). div A in general orthogonal curvilinear coordinates. (5+7)
Hence deduce the expressions for (a), grad tp and (b). div A in spherical polar coordinates. (5+7)
(ii)
Or (fill (lit
Use Forbenius method to solve 2X2 r + x'+ (1 2?)y = 0. (10)
(b)(0
(ii)
10(a)(i)
(ii)
(b)(i)
(H)
dx2 ax
Evaluate where n is a positive integer. . (10)
By series method, solve the Legendre differential equation and obtain the Legendre polynomial. (8+4)
Obtain Rodrigues formula for Legendre polynomial and hence find the value of P0(x)yPi(x) and P2(x). (5+3)
Or
What is Fourier series? Evaluate and discuss the coefficients of Fourier series when f(x) is (1). an even function (2). an odd function (2+6+2+2)
In an LR circuit L = 30 H and R = 30 Cl. An emf E = 150 V is applied across
the combination. Assuming the current is zero at time 4 = 0, find the current at
t > 0. , (8)
[2x20 = 40 marks]
Section C
Answer anr- questi -as. Each question carries 10 marks
11. Find the eigen values and normalised eigen vectors of the matrix
10 0 0 1 1 0 1 1
12. Show that (2n + 1 )xP = (m + n)P%L, r-m + 1)h-i-
ef*
13. Prove that Jn(x) = (2)nxn-r. . Jq(t
14. Using Fourier sine and cosine transformevaluate:
OO
OO
f cos nx
w- y
dn
0
... f n sin nx
(t,)' / ~&TrP
0
(5+5)
[2x10 = 20 marks]
r'
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