Calicut University 2005 B.Sc Mathematics-DIFFERENTIAL&INTEGRAL CALCULUS - Question Paper
SECOND YEAR B.Sc. DEGREE EXAMINATION, MARCH / APRIL 2005
Part III - Mathematics (Main)
PAPER II-DIFFERENTIAL&INTEGRAL CALCULUS
C 4928 (2 Pages) Name...................................................
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SECOND YEAR B.Sc. DEGREE EXAMINATION, MARCH/APRIL 2005
Part IIIMathematics (Main) Paper IIDIFFERENTIAL AND INTEGRAL CALCULUS Time : Three Hours Maximum : 60 Marks
Maximum marks that can be earned from Unit I is 15, Unit II is 15, Unit III is 10 and Unit TV is 20.
Unit I
I. Find the radius of curvature of y2x = a2 (a -x) at a). (4 marks)
I. Find the radius of curvature at the point (x, y) on the curve x = a (0 - sin 0), y = a (1 - cos 0).
(4 marks)
3. Find the evolute of x = a cos3 0, y = a sin3 0. (5 marks)
4F*ind the asymptotes of S*2 + = y2 + x. . (3 marks)
5. Show that for the cissoid y2( 2a-x) = x3, the origin is a single cusp of the first species. (4 marks)
6. Trace the curve r = a sin3 0. (5 marks)
Unit ii
l
7. Obtain the reduction formula for Jxneax dx, n is a positive integer. Use it to find Jx2exdx .
o
. (4 marks)
8. Find the area of loop of the curve x = a (1 -12), y = at (1 -12). f (4 marks)
9. Prove that 8a is the length of one arch of cycloid whose equation is x = a (t sin t), y = a (1 - cos t).
(4 marks)
10. Find the volume of the solid obtained by the revolution of the loops ofy2 = x (2 x l)2. (4 marks)
II. Show the surface area of the solid obtained by revolving x = a (0 - sin 0), y = a (1 - cos 0) about
64 2 y = 0 is 7ia .
3
(4 marks)
12. Find the moment of inertia of a thin uniform rod about an axis passing through its centre and ' '/.orpcndicular to the rod.
(5 marks) Turn over
2
Unit III
13. Expand log (1 + ex) upto term containing x* using Maclaurins series.
(4 marks (4 marks)
(4 marks)
1 14 14 7
14. Sum to infinity the series + 1- + -1:-
10 10 . 20 10 . 20 . 30
1 1 +
15. Sum to infinity the series
+ .
2 3.1! 4.2! 5.3!
16. Show that---
(5 marks)
1.2.3 3.4.5
5.6.7
Unit IV
17. Find how close to origin one should take the point (ac, y) to make \f{x, y) - f(0, 0)| < e if
and E = 0.01.
* + y x2 + 1
fix>y) =
(4 marks) (5 marks)
( 5 marks)
si
2 2 -.2-2
o Tf.. i__X + y d u d u
8. If u = log-, prove that
dx dy dy dx
xy
\19 Ifz = er (x cos v - y oin y) show that + --4r = 0
-2
ox
J*!
. du du 1 prove that x + v = tan u
dx J dy 2
x + y
4x + Jy
-l
20. If u = sin
(5 marks)
v 21. Find where u = xy, x = cos t and y = sin t. (4 marks)
22. Find the extreme values of the function f(x,y) = xy-x2-y2-2x-2y + 4. (5 marks)
3. Evaluate JJ* yjxy - y2 dx dy where S is the triangle with vertices (0, 0), (10, 1) and (1, 1).
(5 marks)
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Attachment: |
| Earning: Approval pending. |
