LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – APRIL 2008
MT 3102 / 3100 - MATHEMATICS FOR PHYSICS
Date : 07/05/2008
Time : 1:00 - 4:00
Dept. No.
XZ 3
Max. : 100 Marks
SECTION A
Answer ALL questions.
(10 x 2 = 20)
1. Write the Leibnitz’s formula for the nth derivative of a product uv.
2. Prove that the subtangent to the curve y=ax is of constant length.
e 1
=
e 1
4. Find L[e-2tsin2t]
3. Prove that
1
2!
1
1!
 41!  61!  ..........
 31!  51!  ..........
5. If y = log ( 1+x ).then find D2y
6. Expand tan 7 in terms of tan
7. Prove that the matrix
cos 
sin 
 sin 
cos 
is orthogonal
8. If tan x
= tan h x
then show that cosx coshx = 1
2
2
9. Find the A.M. of the following frequency distribution.
x: 1 2 3
4
5
6
7
f : 5 9 12 17 14 10
6
10. Write the general formula in Poisson’s distribution.
SECTION B
Answer any FIVE questions.
(5 x 8 = 40)
11.If y=sin-1x, prove that ( 1-x2 )y2 –xy1 =o and (1-x2)yn+2-(2n+1)xyn+1-n2yn=o
12.Find the length of the subtangent, subnormal, tangent and normal at the point (a,a) on the
cissoid y2 =
x3
2a  x
13. Sum to infinity the series:1.3
1.3.5
1.3.5.7


 ...........
2.4.6.8 2.4.6.8.10 2.4.6.8.10.12
14. Verify Cayley Hamilton theorem for the matrix
1 1
2
A= 2
1
3
3
2 3
1
15. If sin (   i ) = tan ( x + iy) , Show that
tan 
sin 2 x

tanh  sinh 2 y
16. If sin (A + iB) = x + iy ,
x2
y2
x2
y2
Prove that (i)
(ii)


1

1
cosh 2 B sinh 2 B
sin 2 A cos 2 A
s3


17. Find L-1  s 2  4s  13 


18. Ten coins are tossed simultaneously. Find the probability of getting at least seven heads.
SECTION C
Answer Any TWO Questions.
(2 x 20 = 40)
2 4 34 4 4


 ..........  15e
2! 3! 4!
(b) Find the mean and standard deviation for the following table, giving the age
distribution of 542 members.
Age in years
20-30 30-40 40-50
50-60
60-70 70-80 80-90
19. (a) Prove that 1 +
No. of members
3
61
132
153
140
sin 7
 64cos6 - 80 cos4 + 24 cos2 - 1
sin 
(b)Expand sin3 cos4 in terms of sines of multiples of angles.
51
2
20.(a) Prove that
(10 + 10)
21.a)Find the maxima and minima of x5-5x4+5x3+10
b)Find the length of the subtangent and subnormal at the point `t’ of the curve
x = a(cost + t sint),
y = a(sint-tcost)
(10 + 10)
2
dy
d y
dy
 0 when t = 0.
 2  3 y  sin t , given that y =
22. a)Solve the equation
2
dt
dt
dt


1
b)Find L s ( s  1)( s  2) 


-1 
(15 + 5)
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