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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SECOND SEMESTER – APRIL 2011
MT 2501/MT 2500 - ALGEBRA, ANAL.GEO & CALCULUS - II
Date : 08-04-2011
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions:
 tan
Evaluate  x e
1
1. Evaluate
2.
4
x
(10 x 2 = 20)
x dx .
dx .
dy
1
 y cos x  sin 2 x .
dx
2
2
4. Solve: D  5D  6 y  e x .
1 1 1
5. Prove that the series 1     .......... is convergent.
1! 2 ! 3 !
3. Solve:


n3  1
.

n
n 0 2  1

6. Test for convergency the series
7. Find the general term in the expansion of 1  x  3 .
2
 1 1  n  3n 2  .
1  2 x  3x 2
8. Prove that the coefficient of x in the expansion of
is
x
n!
e
9. Find the equation of the sphere which has its centre at the point 6,  1, 2 and touches the
plane 2 x  y  2 z  2  0 .
10. Find the distance between the parallel planes 2 x  2 y  z  3  0 and 4 x  4 y  2 z  5  0
n
n
PART – B
Answer any FIVE questions:

11. Prove that
(5 x 8 = 40)

4
 log 1  tan  d  8 log 2 .
0
12. If I n   sin xdx ( n being a positive integer), prove that nI n   sin n 1 x cos x  n  1I n  2 .
n

Also evaluate

2

2
6
 sin xdx and
 sin
0
0

7
xdx .
13. Solve: D 2  1 y  x cos x .
14. Solve
d2y
 y  sec x.
dx 2
15. Test for convergency and divergency the series 1 
16. Show that the sum of the series 1 
2 x 32 x 2 4 3 x 3 5 4 x 4



 ..........
2!
3!
4!
5!
1  3 1  3  3 2 1  3  3 2  33
1


 .........  ee 2  1 .
2!
3!
4!
2
17. Show that if x  0 , log x 
x 1 1 x2 1 1 x3 1
 .
 .
 ..........
x  1 2 x  12 3 x  13
18. Find the equation of the plane passing through the points
2, 5,  3,  2,  3, 5 and 5, 3,  3 .
PART – C
Answer any TWO questions:
19. a) Evaluate

(2 x 20 = 40)
3x  2
(10 marks)
dx
4x 2  4x  5
b) Find the area and the perimeter of the cardiod r  a1 cos  .
20. a) Solve: x 2
(10 marks)
d2y
dy
1
.
 3x  y 
2
dx
dx
1  x 2
b) Discuss the convergence of the series
positive values of x .
21.a) Show that the error in taking
(10 marks)
1
1
1
1



 .......... for
2
3
1  x 1  2x
1  3x
1  4x 4
(10 marks)
1 x 2  x

as an approximation to 1  x is
2 x
4
x4
when x is small.
27

1
1
b) show that 
   log 2.
2
n 1 2n  1 2n 2n  1
approximately equal to
(10 marks)
(10 marks)
22. a) A sphere of constant radius k passes through the origin and meets the axes in A, B, C.
Prove that the centroid of the triangle ABC lies on the sphere 9 x 2  y 2  z 2  4k 2 .
(10 marks)
b) Find the shortest distance between the lines
x 3 y 8 z 3 x 3 y  7 z 6


;


.
3
1
1
3
2
4
Also find the equation of the line of shortest distance.
(10 marks)
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