12N / 238
CLASS: B.Sc. PHYSICS
St. JOSEPH’S COLLEGE (AUTONOMOUS) TIRUCHIRAPPALLI – 620 002
SEMESTER EXAMINATIONS – NOVEMBER 2012
TIME: 3 Hrs.
MAXIMUM MARKS: 100
SEM
SET
PAPER CODE
TITLE OF THE PAPER
I
2012
12UPH130401
MATHEMATICS – I
SECTION – A
Answer all the questions:
20 x 1 = 20
Choose the correct answer:
1. The nth derivative of sin(ax+b) is
a) n 
b) a n sin( ax  b )
n 
a sin  ax  b  
2 

c) n
d) a n cos( ax  b )
n 

a cos  ax  b  
2 

2.
a
If f(x) is an odd function of x, then  f ( x )dx 
a
a) 0
b)
a
 f ( x )dx
0
c)
a
2  f ( x )dx
d) 1
0
3. Differential equation is one which involves
a) constant
b) derivative
c) variable
d) zero
4. A diagonal matrix in which all the diagonal elements are equal is
called
a) scalar matrix
b) unit matrix
c) identity matrix
d) none
5. The series a + ar + ar2 + …  is convergence if
a) |r| < 1
b) r > 1
c) r  1
d) r  1
Fill in the blanks:
6.
 sin nx dx = _______.
7.
The nth derivative of (ax + b)m is _______.
8.
dy
 Py  Q is _______.
dx
The determinant of singular matrix is _______.
The integrating factor of
9.
10. If an tends to a finite limit ‘a’ as n, the series  an is said to be
_______.
State True or False:
11. The points at which f(x) is either maximum or minimum are called
extreme points.
b
12. b
 f ( x )dx   f ( a  x )dx
a
13.
a
1
d (tan x ) 
x dy  y dx
2
x y
2
14. A symmetric matrix remains unchanged if rows and columns are
interchanged.
15. The convergence or divergence of an infinite series is not affected
when each of its term is multiplied by a finite quantity.
Answer in one or two sentences:
16. State Leibnitz theorem.
2
4
2
17. Find  sin x cos x dx
0
18. Find the complementary function of (D2 + 5D + 6) y = ex.
1 2
 1  1
 ; B  
 Find AB.
19. If A  
3
4
2
1




20. Define series.
SECTION – B
Answer all the questions:
5 x 4 = 20
21. a. If y = x2 eax then find yn.
OR
b. Evaluate  x e x dx
22. a. Find reduction formula for I   tann x dx .
n
OR
32
2
b.
(sin x )
Evaluate 
dx
32
32
 (cos x )
0 (sin x )
12
23. a.
dy  1  y 2 

0
Solve
dx  1  x 2 
OR
2
x
b. Solve (D + 5D + 6) y =e
24. a.
1 1 1 1


Find the rank of the matrix A   4 1 0 2  .
 0 3 4 2


OR
b.
 1 1 3


Find the eigen value of the matrix A   1 5 1  .
 3 1 1


25. a. Determine the nature of the series
1 1
1
 un  1   2  ...  n 1  ...
3 3
3
OR
b.

n! 2n
n 1
nn
Discuss the convergence of the series 
SECTION – C
Answer any FOUR questions:
4 x 15 = 60
26. Find the maximum and minimum value of the function 2x3 – 3x2 –
36x + 10.
27. Establish reduction formula for I   sin n x dx and find
n
2
n
 sin x dx
0
28. Show that the solution of the differential equation
dy
d2y
 0 when t=0
which
is
such
that
y
=
0
and

4
y

A
sin
pt
2
dx
dt
1


A sin pt  p sin 2t 
2
 if p2. If p=2, show that
is y  
2
4 p
A(sin 2t  2t cos 2t )
y
.
8
29. Verify Cayley-Hamilton theorem for the matrix
 2 1 1 


A    1 2  1 and find A-1.
 1 1 2 


30.
Prove that the series
1
k
1

1
2
k
 ... 
1
n
and divergent if k1.
**************
k
 ... is convergent if k>1