M.Sc. DEGREE EXAMINATION, APRIL 2016
Physics
MATHEMATICAL PHYSICS I
(2011 onwards)
Time 3 Hours Maximum 75 Marks
Part A (10 2 20)
Answer all questions.
1. Obtain the value of a.b if a and b are perpendicular to
each other.
2. Evaluate divr if r ix jy kz k being the unit
vectors along Z axes, respectively.
3. When is a matrix said to be idempotent?
4. Define the rank of a matrix,
5. Show that
6. For Bessel functions, what is J1/ 2 equal to?
7. If is the Fourier transform of f then show that
the Fourier transform of the first derivative is
given by
8. If f is the Laplace transform of then show that
the Laplace transform of f
Sub. Code
521102
RW-10943
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9. What is the probability that a leap year selected at
random contains 53 Sundays?
10. Six cards are drawn at random from a pack of 52 cards
which contains equal number of red and black cards.
What is the probability that 3 will be red and three will
be black?
Part B 5 25)
Answer all questions, choosing either or
11. Prove that divculA 0 .
Or
Deduce the Laplacian in cylindrical coordinates.
12. Prove that every square matrix can be uniquely
expressed as the sum of a symmetric and a skewsymmetric
matrix.
Or
Show that the matrix




2 2
2 2
i
i is unitary.
13. Prove that
Or
For Legendre polynomials, show that
P Pn
n
n .
14. Find the finite cosine Fourier transform of
in the interval .
Or
Find the Laplace transform . Hence find
it for n 1/2 .
RW-10943
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15. Obtain the expression for the mean of binomial
distribution.
Or
Derive the second moment of Poisson distribution.
Part C 10 30)
Answer any three questions.
16. Evaluate F for the vector
F y 3xyj z2 over the surface of
hemisphere y2 z2 16 lying above x y plane, n
being the unit normal vector.
17. Diagonalize the matrix






0 0 1
sin cos 0
cos sin 0


.
18. State and prove the orthogonality relation for Hermite
polynomials.
19. State and prove the convolution theorem of Laplace
transforms.
20. Show that the mean deviation from the mean of the
normal distribution is about of its standard
deviation.