LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
THIRD SEMESTER – NOV 2006
AA 03
MT 3100 - ALLIED MATHEMATICS FOR PHYSICS
(Also equivalent to MAT 100)
Date & Time : 28-10-2006/9.00-12.00
Dept. No.
Max. : 100 Marks
Section A
Answer ALL the questions (10 x 2 =20)
1) Evaluate lim
x0
1  cos x
.
2
x
2) Expand sin n and cos n
.
3) Prove that cosh 2 x  sinh 2 x  cosh 2 x .
4) Find yn when y  cosh 2 x .
5) Find L sin 3 2t 
6) Find L
 1  2s  3 


 ( s  1)5 
7) Show that, in the curve r  a , the polar subtangent varies as the square of the
radius vector.
8) Find the coefficient of x n in the expansion of
1
.
(1  2 x)(1  3x)
1 4 5


9) Find the rank of the matrix A   2 5 7 
3 6 9


10) Mention a relation between binomial and Poisson distribution.
Section B
Answer any FIVE questions (5 x 8 = 40)
1  et 

11) Find L 
 t 


1
12) Find L 1 

 s( s  1)( s  2) 
13) If x  iy  tan( A  iB) , prove that x 2  y 2  2 x cot 2 A  1 .
14) If y  sin(m sin 1 x) , prove that (1  x 2 ) y2  xy1  m 2 y  0 .
15) Find the slope of the tangent with the initial line for the cardioid r  a (1  cos  )
at  =

.
6
 1 1 2 
16) Find the inverse of the matrix A   2 1 3  using Cayley Hamilton
 3 2 3 


theorem.
2.3 3.5 4.7 5.9
7



...  2e  .
3! 4! 5!
6!
2
18) Suppose on an average one house in 1000 in Telebakkam city needs television
service during a year. If there are 2000 houses in that city, what is the
probability that exactly 5 houses will need television service during the year.
Section C
17) Show that
Answer any TWO questions only (2 x 20 = 40)
19) (a) Using Laplace transform solve ( D 2  5 D  6) x  e t when x (0) = 7.5,
x’(0) = -18.5.
(b) Prove that cosh5x = 16cosh5x – 20cosh3x + 5coshx.
(12+8)
20) (a) Expand cos4 sin3 in terms of sines of multiples of angle.
m


(b) If y   x  1  x 2  prove that


(1  x2 ) yn2  (2n  1) xyn1  (n2  m2 ) yn  0
(8+12)
 3 10 5 


21) (a) Find the eigen values and eigen vectors of the matrix A   2 3 4 
 3 5 7


2
(b) Find the greatest term in the expansion of (1  x)31 3 when x  .
(15+5)
7
22) (a) Find the angle of intersection of the cardioid r  a(1  cos ) and
r  b(1  cos  ) .
(b) Eight coins are tossed at a time, for 256 times. Number of heads observed at
each throw is recorded and results are given below.
No of heads at a throw 0 1 2 3 4 5 6 7 8
frequency
2 6 30 52 67 56 32 10 1
What are the theoretical values of mean and standard deviation? Calculate also
the mean and standard deviation of the observed frequencies.
(10+10)
______________