# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 ```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
FIRST SEMESTER – NOVEMBER 2012
MT 1100 - MATHEMATICS FOR PHYSICS
Date : 03/11/2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
SECTION A
(10x2 =20)
1) Find the nth derivative of y  sin(ax  b) .
2) Write down the formula for subtangent and subnormal.
1 1 1
   ...
e  1 2! 4! 6!
3) Prove that
.

e  1 1  1  1  ...
1! 3! 5!
3 −1 2
4) Find the rank of the matrix (−6 2 −4).
−3 1 −2
1
−𝑎𝑡
5) Show that [𝑒 ] = 𝑠+𝑎 .
6) State the formula for Laplace transformation of a periodic function.
7) Write down the expansion for sin 𝑛𝜃.
8) If 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1, Show that cos ℎ2 𝑥 − 𝑠𝑖𝑛ℎ2 𝑥 = 1.
9) What is the chance that a leap year selected at random will contain 53 Sundays?
10) Define Binomial distribution.
SECTION B
11) Find the angle of intersection of cardioids r  a(1  cos  ) and r  b(1  cos  ) .
12) Find the minimum and maximum value of the function 2 x 3  3x 2  36 x  10 .
24 34 44
   ...   .
2! 3! 4!
14) Show that the system of equations
xyz  6
x  2y  3z  14
x  4y  7z  30
13) Find the sum to infinity series 1 
are consistent and solve them.
e  t , 0 &lt; t &lt; 4
15) Find the L(f(t)) if f (t )  
0 , t &gt; 4
(5x8 =40)
 s 
16) Find a) L1 
b) L(cos4 t ) .
2 
 ( s  2) 
17) Prove that cos8θ = 1- 32sin2 θ + 160sin4 θ-256sin6 θ+128 sin8 θ.
18) Find the moment generating function for the Poisson distribution and hence find its
mean and variance.
SECTION C

2
19) a) If y  x  1  x
 then Prove that (1  x ) y
(2x20 = 40)
m
2
n2
 (2n  1) xyn 1  (n 2  m 2 ) yn  0 . b) Find
3
.
(10+10)
( x  1)(2 x  1)
3
1
1
20) If 𝐴 = [−1 5 −1] then
1 −1 3
a) Find the characteristic value and characteristic vector of the matrix.
b) Verify Cayley Hamilton Theorem and find A-1.
(10+10)
5
3
21) a) Express cos θ sin θ in terms of sines of multiples of θ.
b) Separate into real and imaginary parts of tan-1(α+iβ).
(10+10)
dx
dy
22) a) Solve
 2 x  3 y  0,
 y  2 x  0 with x(0)  8, y(0)=3 using Laplace transform.
dt
dt
b) An urn contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the
probability that: (i) two of the ball drawn is white; (ii) one is of each colour,
(iii) none is red.
(14+6)
the nth derivative of
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