LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SECOND SEMESTER – April 2009
MT 2501 / 2500 - ALGEBRA, ANAL.GEO & CALCULUS - II
Date & Time: 23/04/2009 / 1:00 - 4:00 Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions:
(10 x 2 = 20)
p
p
1. Show that
2. Evaluate
3. Solve
2
ò f (sin x)dx = 2ò f (sin x)dx .
0
0
ò xe dx .
x
dy y + 2
=
.
dx
x- 1
d2y
dy
+ 5 + 4y = 0 .
2
dx
dx
5. If u1 + u2 + ... is convergent, show that lim ui = 0 .
4. Solve
i® ¥
6. State comparison test.
7. Find the coefficient of x n in
x- 1
.
ex
1+ x
.
1- x
9. Find the angle between the planes 2 x - y + z = 6
8. Write the series for log
2
2
x + y + 2z = 3 .
2
10. Find the radius of the sphere 2 x + 2 y + 2 z - 2 x + 4 y + 2 z - 15 = 0 .
PART – B
Answer any FIVE questions:
11. Evaluate
ò (3x -
(5 x 8 = 40)
2) x 2 + x + 1 dx .
12. Find the area of loop of y 2 = x 2
13. Solve ( x + 1)
a+ x
.
a- x
dy
+ 1 = 2e- y .
dx
14. Solve ( D2 - 4D + 3) y = e- x sin x .
15. Test the convergence of the series
1
3
5
+
+
+ ...
1.2.3 2.3.4 3.4.5
n
16. Show that the series
å
{(n + 1)r }
n n+ 1
ZA 05
is convergent if r < i and divergent if r ³ 1 .
17. Find the coefficient of x n in the expansion of
1
x+ 1
.
( x - 1)2 ( x - 2)
x- 3 y- 4 z + 2 x- 1 y + 7 z + 2
.
=
=
;
=
=
- 1
2
1
1
3
2
18. Find the shortest distance between the lines
PART – C
Answer any TWO questions:
(2 x 20 = 40)
p
19. (a) Derive the reduction formula for Im,n =
ò Sin
m
n
x Cos x dx and obtain
2
ò Sin
m
x Cos n x dx .
0
(b) Find the area of the surface of the solid generated by rotating the cardioid
v = a(1 + Cosq) about its line of Symmetry.
2
2
(12+8)
2
20. (a) Solve (a - 2 xy - y )dx - ( x + y) dy = 0 .
(b) Solve x 2
(c) Solve
d2y
dy
1
.
+ 3x + y =
2
dx
dx
(1- x)2
d2y
+ y = Sec x .
dx 2
(5+7+8)
¥
21. (a) Test the convergence of
å
n= 0
(b) Show that 1 +
n3 + 1
.
2n + 1
1 1
+ + ... is divergent.
2 3
(c) Discuss the convergence of the series
1+
(1!) 2
(2!) 2 2
x+
x + ...
(6+5+9)
2!
4!
2 12 3 13 4 14 5 15
1+ e
+
+ ... =
.
1! 2! 3! 4!
e
(b) Find the equation of the plane through the line of intersection of the planes
x + y + x = 1; 2 x + 3 y + 4 z = 7 and perpendicular to the plane x - 5 y + 3 z = 5 .
22. (a) Show that
(c) Find the equation of the sphere having the circle
x 2 + y 2 + z 2 - 2 x + 4 y - 6 z + 7 = 0; 2 x - y + 2 z = 5 for a great circle.
*****************
2
(8+7+7)