LOYOLA COLLEGE (AUTONOMOUS), CHENNAI  600 034.
B.Sc. DEGREE EXAMINATION  MATHEMATICS
SECOND SEMESTER  APRIL 2003
MT 2500 / MAT 501  ALGEBRA ANALYTICAL GEOMETRY,
CALCULUS  II
23.04.2003
9.00  12.00
Max : 100 Marks
PART  A
Answer ALL questions. Each question carries TWO marks.


2
2
0
0
(10 2=20 marks)
n
n
 sin xdx   cos xdx.
01. Prove that

2
 log tan xdx.
02. Evaluate
0
03. State cauchy’s root test for convergence of a given series.
2
04.
05.
06.
07.
2
1 1
1 1




Show that 1    ...  1  1    ...
2! 4!
3! 5!




2
3
4
y2 y3
x
x
x
If Y= x 


 ..... show that x  y 

 ....
2
3
4
2!
3!
Solve D 2  5D  6 y  e x
dy
Solve p 2  3 p  2  0 where p 
dx



2
08. Evaluate
 cos
6
xdx.
0
09. Find the equation to the plane through the point (3,4,5) and parallel to the plane
2x  3y  z  0
10. Find the equation of the sphere with centre (1, 2, 3) and radius 3 units.
PART  B
Answer any FIVE questions. Each question carries EIGHT marks.
a
11. If U n =
x
n
(5 8=40 marks)
e  ax dx prove that U n  n  a   u n 1  a(n  1) u n  2  0 .
0
log 1  x 
dx .
2
1

x
0
15 15.21 15.21.27


 ..... .
13. Sum the series
16 16.24 16.24.27
1
12. Evaluate

14. Sum the series


n  13
n 1
n!

xn .

15. Solve D 2  D  1 y  x 2 .
1


16. Solve D 2  4D  3 y  e  x sin x .


17. Test for convergence of the series

n 4  1  n 4 1 .
1
18. Find the perpendicular distance from P (3, 9, 1) to the line
x  8 y  31 z  13


.
8
1
5
PART  C
Answer any TWO questions. Each question carries TWENTY marks.
(220=40 marks)

2
19. a) Evaluate I =
 log sin xdx.
(10)
0
b) Find the reduction formulae for In =
x
n
cos axdx.
d2y
 y  sec x by variation of parameter method.
dx 2
d2y
dy
b) Solve x 2
 x  y  log x
2
dx
dx
2
1 1 2  2 2 1 2  2 2  3 2 12  2 2  3 2  4 2



 .....
21. a) Sum the series
1!
2!
3!
4!
20. a) Solve

b) Sum the series
1
 (2n 1)2n(2n  1)
(10)
(10)
(10)
(10)
(10)
n 1
1 1
1
 k  k  ... is convergent when k is greater than unity and
k
1
2
3
divergent when k is equal to or less than unity.
(10)
b) Find the equation of the sphere which passes through the circle
(10)
x 2  y 2  z 2  2 x  4 y  0, x  2 y  3z  8 and touch the plane 4 x  3 y  25
22. a) Show that the series
+ + + + +
2