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```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SECOND SEMESTER – APRIL 2008
MT 2500 - ALGEBRA, ANAL.GEO & CALCULUS - II
Date : 23/04/2008
Time : 1:00 - 4:00
Dept. No.
XZ 54
Max. : 100 Marks
PART – A
(10 x 2 = 20)
1. Evaluate  Sin 2 3x dx
2. Write the value of
3.
4.
5.
6.
7.

a 2  x 2 dx
Is ( x 2 y  2 xy2 )dx  ( x 2  3x 2` y)dy  0 exact?
Solve ( D 2  3D  5) y  0
State Raabe’s test.
Define uniform convergence of a sequence.
Find the Coefficient of x 32 in the expansion of ( x 4 
1 15
x3
)
1 12
)
x
9. Write the intercept and normal forms of the equation of a plane.
10. Find the Centre and radius of the sphere x 2  y 2  z 2  2 x  6 y  4 z  35  0
8. Write down the last term in the expansion of ( x 
PART – B
11. Evaluate
(5 x 8 = 40)
x  Sinx
 1  Cosx dx
12. Solve ( D 2  16) y  Cos4 x
13. Test the Convergence of

1
n 2 1
14. Find the sum to infinity of the series 1 
3 3.5 3.5.7


.....
4 4.8 4.8.12
1  3 1  3  32

 ....
15. Sum the series 1 
2!
3!
16. If a, b, c denote three Consecutive integers, show that
1
log b  1 log a  1 log c  1  1 .
 ...
e
e
e
2
2
2ac  1 3 (2ac  1)3
17. The foot of the perpendicular drawn form the origin to the plane is (12,-4,-3); find the equation
of the plane.
18. Find the equation to the sphere through the four points (0,0,0), (a,0,0), (0,b,0), (0,0,c) and
1
PART – C
(2 x 20 = 40)
ax
19. a) Evaluate  Cosbxdx
e
b) Find the area of the cardioid r  a(1  Cos ).
(12+8)
20. a) Prove that the series
1 a 1.3 a(a  1) 1.3.5 a(a  1)( a  2)
1 . 
.

 ....
2 b 2.4 b(b  1) 2.4.6 b(b  1)(b  2)
1
is convergent if a  0, b  0 and b  a 
2
1 1.3 1.3.5

 ...
b) Sum the series 1  
4 4.8 4.8.12
21. Find the image of the point (1,3,4) in the plane 2 x  y  z  3  0 . Hence prove that the image of
x 1 y  3 3  4 x  3 y  5 3  2




the line
is
.
1
2
3
1
5
 10
22. Through the circle of intersection of the sphere x 2  y 2  z 2  25 and the plane
x  2 y _ 2 z  9 two spheres S1 and S 2 are drawn to touch the place 4 x  3 y  30 . Find the
equations of the spheres.


2
```
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