LOYOLA COLLEGE (AUTONOMOUS), CHENNAI  600 034
B.Sc. DEGREE EXAMINATION  PHYSICS
THIRD SEMESTER APRIL 2003
MT 3100/ MAT 100 MATHEMATICS FOR PHYSICS
28.04.2003
Max.: 100 Marks
9.00  12.00
PART  A
01.
02.
03.
04.
05.
06.
07.
08.
09.
10.
(10  2 = 20 Marks)
1
Define Laplace transform of f(t) and prove that L(eat) =
.
sa
1
Find L1  2  .
s 
e m m r
Prove that the mean of the Poisson distribution Pr =
, r = 0, 1, 2, 3 ….. is
r!
equal to m.
Mention any two significance of the normal distribution.
x  sin x
Find the Lt
.
x0
x3
Find L (1+ t)2 .
 1 
Find
L-1 
.
2 
 s  1 
Write down the real part of sin   i   .
Prove that in the R.H. xy = c2, the subnormal varies as the cube of the ordinate.
If y = log (ax +b), find yn.
PART  B
(5  8 = 40 Marks)
Answer any FIVE questions. Each question carries EIGHT marks


s
11. (a) Find L-1 

2
 ( s  2)  1
1  e t 
(b) Find L 
.
 t 
 Sin 2 t 
 .
12. Find L 
 t 
2 1
2
1
13. Define orthoganal matrix and prove that the matrix  2 1 2 is
3
 1  2 2
orthoganal.
1 3 7


14. Verify cayleyHamilton theorem and hence find the inverse of  4 2 3 
1 2 1


ax
1 a x
1 a x
 
15. (i) Prove that
  
  ....  log a  log x.
a
2 a 
3  a 
3 5 7
(ii)
Find the sum to infinity of series 1     .......
2! 3! 4!
2
3
1681
1682
 a  b cos    c sin 
Lt
1 .
5
16. (i) Find  approximately to the nearest minute if cos  =
(ii) Determine a, b, c such that
 0

17. If cos (x + iy) = cos  + i sin, Show that cos 2x + cosh 2y =2.
4
0
18. What is the rank of 
0

0
2 2 3
4 2 3
.
0 0 5

0 4 5
PART  C
(2  20 = 40 Marks)
Answer any TWO questions. Each question carries twenty marks.

19. (a) If y = x  1  x 2

m
show that (1  x 2 ) y n  2  (2n  1) xyn  1  (n 2  m 2 ) y n 0 .
(b) Find the angle of intersection of the cardioids r = a(1+cos) and r = b(1cos).
20. (a) Certain mass produced articles of which 0.5 percent are defective, are packed
in cartons each containing 130 article. What Proportion of cartons are free
from defective articles, and what proportion contain 2 or more defectives
(given e-2.2 = 0.6065)
(b) Of a large group of men 5 percent are under 60 inches in height and 40
percent are between 60 and 65 inches. Assuming a normal distribution find
the mean height and standard deviation.
21. (a) Find the sum to infinity of the series
5
5.7
5.7.9


 ..... 
3.6 3.6.9 3.6.9.12
(b) From a solid sphere, matter is scooped out so as to form a conical cup, with
vertex of the cup on the surface of the sphere, Find when the volume of the
cup is maximum.
1
sin 5  5 sin 3  10 sin  
16
1
b) Prove that sin4 cos2 = 5 cos 6  22 cos 4  cos 2  2 
2
22. a) Prove that sin5 =
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