LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
THIRD SEMESTER – NOVEMBER 2012
MT 3102/3100 - MATHEMATICS FOR PHYSICS
Date : 07/11/2012
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
SECTION A
ANSWER ALL QUESTIONS.
(10 x 2 = 20)
01. Find the nth derivative of y  log(4 x  8) .
02. Find the slope of the curve r  a (1  cos  ) at  

2
.
p
03. Write the expansion for 1  x  q .

 3 -1 2 4 


04. Find the rank of the matrix  -6 2 -4 -8  .
 -3 1 -2 -4 


05. Find the Laplace transform of cos at .
 1 
06. Find L1 
.
2
  s  a  
07. Write down the expansion of cos5 in powers of cos .
08. Show that cosh 2 x  cosh 2 x  sinh 2 x .
09. Two dice are thrown. What is the probability that the sum of the numbers is greater than 8?
10. Define Normal distribution.
SECTION B
ANSWER ANY FIVE QUESTIONS.
11. Find the nth differential coefficient of cos x cos2x cos3x .
log x
12. Find the maximum value of
for positive values of x.
x
 2 2 0


13. Find the characteristic roots of the matrix  2 1 1  .
 7 2 3 


t
e ,0  t  4
14. Find the Laplace transform of f (t )  
.
0 , t  4
(5 x 8 = 40)
15. Express sin 5  in a series of sines of multiple of  .
16. Four cards are drawn at random from a pack of 52 cards. Find the probability that
(i) they are a king, a queen, a jack and an ace.
(ii) two are kings and two are queens.
(iii) two are black and two are red.
17. A car hire firm has two cars, which it hires out day by day. The number of demands for a car on
each day is distributed as a Poisson distribution with mean 1.5. Calculate the proportion of days on
which (i) neither car is used, (ii) the proportion of days on which some demand is refused.
18. X is a normal variable with mean 30 and standard deviation 5. Find the probabilities that
(i) 26  X  40, (ii) X  45.
SECTION C
ANSWER ANY TWO QUESTIONS.
19. (a) If y  sin 1 x , then prove that 1  x 2  yn2   2n  1 xyn1  n 2 yn  0 .
(2 x 20 = 40)
(b) Find the length of subtangent and subnormal at any point t on the curve x  a  cos t  t sin t  and
y  a  sin t  t cos t  .
(12 + 8)
 2 1 1 


20. (a) Verify Cayley-Hamilton theorem for the matrix A   1 2 1 and also find A1 .
 1 1 2 


2
2
1  3 1  3  3 1  3  3  33


 ... . (12 + 8)
(b) Find the sum to infinity of the series 1 
2!
3!
4!
21. (a) Express cos4 in terms of sin  .
(b) Find the mean and standard deviation for the following data:
x
10
20
30
40
50
60
f
15
32
51
78
97
109
(10 + 10)
2
d y
dy
dy
22. (a) Solve the equation 2  4  5 y  5 given that y  0,  2 when t  0 .
dt
dt
dt
(b) Ten coins are thrown simultaneously. Find the probability of getting at least seven
heads?
(12 + 8)