LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION –PHYSICS
CV 08
THIRD SEMESTER – APRIL 2007
MT 3100 - ALLIED MATHEMATICS FOR PHYSICS
Date & Time: 28/04/2007 / 9:00 - 12:00
Dept. No.
Max. : 100 Marks
Section A
Answer ALL the questions (10 x 2 =20)
1) Evaluate lim
x0
sin x  x
.
x3
2) Find L t cos at  .
3) Find L
 1  2s  3 

.
 ( s  1)2  5 
4) If cos5 = Acos + Bcos3+ Ccos5 , prove that
sin5 = A sin -Bsin3 + Csin5.
5) If tan
x
y
 tanh , prove that cos x cosh y  1 .
2
2
6) If y = e 2x, prove that y4  64 y  0 .
7) Show that in the curve r = e  cot  , the polar subtangent is rtan.
8) Derive the expansion of log
1 x
.
1 x
a  bx (a  bx) 2
(a  bx) n

 ..... 
 ...
1!
2!
n!
10) Explain the concept of mutually exclusive events with an example.
9) Find the coefficient of x n in the series 1 
Section B
Answer any FIVE questions (5 x 8 = 40)
11) Determine a, b, c so that lim
 0
 [a  b cos ]  csin
5
 1.
12) Find L[t2 e -t cost].
13) Using Laplace transform solve
14) Prove that sin 5  
1
m
15) If y  y
1
m
dy
 y  2 , given y (0)  2 .
dx
1
(sin 5  5sin 3  10sin  ) .
16
 2 x , prove that ( x2  1) yn2  (2n  1) xyn1  (n2  m2 ) yn  0 .
1
1 0 3 
16) Verify Cayley Hamilton theorem for the matrix A   2 1 1 .
 1 1 1 


2
17) Sum the binomial series
3
5  1  5.7  1 
5.7.9  1 
 
  
   ...to  .
2!  3  3!  3 
4!  3 
18) Assuming that half the population own a two wheeler in a city so that the chance
of an individual having a two wheeler is ½ and assuming that 100 investigators
can take sample of 10 individuals to see whether they own a two wheeler, how
many investigators would you expect to report that three people or less were
having two wheelers.?
Section C
Answer any TWO questions only (2 x 20 = 40)
19) Solve
dy
dy
 2 x  3 y  0;
 3 x  2 y  0 , given x(0) = 0: y(0)=2 using Laplace
dx
dx
transform.
20) (a) Find the real and imaginary parts of sin (x+iy) and tan(u + iv).
If sin( x  iy )  tan(u  iv) , prove that tan x sinh 2v  tanh y sin 2u .
(b) If y  sin 1 x , prove that (1  x 2 ) y2  xy1  0 .
21) a) Find the angle at which the radius vector cuts the curve
(10+10)
l
 1  e cos .
r
b) A person is known to hit target in 3 out of 4 shots, whereas another person is
known to hit the target 2 out of 3 shots. Find the probability of the targets being
hit at all shots when they both try.
c) A bag contains 5 white and 3 black balls. Two balls are drawn at random one
after the other without replacement. Find the probability that both balls drawn
are black.
(10+5+5)
 1 1 3


22) a) Find the eigen values and eigen vectors of the matrix A   1 5 1 
3 1 1


29
 5 5
b) Find the first term with a negative coefficient in the expansion of 1  x 
 4 
(15+5)
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