Calicut University 2005 B.Sc Mathematics II-DIFFERENTIAL AND INTEGRAL CALCULUS - Question Paper
FINAL YEAR B.Sc. DEGREE EXAMINATION, SEPTEMBER / OCTOBAR 2005
PART III - MATHEMATICS (MAIN)
PAPER II-DIFFERENTIAL AND INTEGRAL CALCULUS
( 2001 admissions)
D 10010 (Pages : 2) Name......................................
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FINAL YEAR B.Sc. DEGREE EXAMINATION, SEPTEMBER/OCTOBER 2005
Part IIIMathematics (Main) Paper IIDIFFERENTIAL AND INTEGRAL CALCULUS (2001 admissions)
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 IV is 20. JJ* J
1. Find the radius of curvature of 4ay2 = (2a x)3 at (a, a/2). (4 marks)
2. Find the radius of curvature of x = 3a cos 0 - a cos 30
y - 3a sin 0 - a sin 30. (4 marks)
nJ
3. Show that the evolute of the parabola x2 = Aay is 4 (y - 2a)3 = 27 ax2. (4 marks)
4. Find the asymptotes of the curve x3 + 2xy - xy2 - 2y3 + 4y2 + 2ry + y 1 =*0. (4 marks)
5. Examine the nature of the singular point on the curve (x + y)3 = 4% (y - x + 2)2. (4 marks)
6. Trace the curve (a2 + x2) y = a2x. (5 marks)
rt
\J 7. Obtain the reduction formula for lm%n = Jsinm x cos'1 x dx9 m, n being positive integers.
(4 marks)
2 2 X y
, / 8. Find the area of the ellipse + = L (4 marks)
az fr
/
9. Find the entire length of the curve x23 + y23 = a23. (4 marks)
10. The area between y2 = 4ax and x2 = 4ay revolves about the axis of x. Show the volume of the solid
- 96 3 formed is na .
5
I (4 marks)
711. Find the surface area of the solid generated by revolving the curve x - a cos30,y = a'sin3 about x-axis.
(4 marks)
12. Find the moment of inertia of a solid sphere about an axis passing through its centre.
t (5 marks)
Turn over
13
(4 marks
(4 marks
14
+
Unit III
Using Maclaurins scries, obtain series expression ofy = tan"1 x.
\3
g * r .u * , W 1 1 l*3 1 1-3-5 1 Sum to infinity the series 1-- +---------
2\2) 2-4 2 2-4*6 \2 )
15. Sum to infinity the scries* ---- -- r -.....-
(4 marks
(5 marks
;i: 4! 5!
16. Show that --- + ---- + ...=: 2 Io 2-1 1-2 2-3 3-4 4-5 be
17. Show that the function f (x, y) =
x + y~ is continuous at every point except a:
0 (x, y) = (0, 0)
origin.
22. Find th:; maximum or minimum values of 2 (.x2 - y2) - or4 + y4.
23. Evaluate jjx2 + y2 dxdy over the region S for which each ofx.y > 0 and x +y <> 1.
>6. If u log (tan x + tar. y -f tan j), show mat sin 2x + sin 2y sin 2z = 2.
- - ox ay oz
21. Fi:,d and as functions of/- un*J s if ;y - .v r 2v -r 2, x - r/s y = r2 + e* 2 = 2r.
or
-TT71 - * ly/v) that -r y ~ = 0.
ox dy
, prove mat a* ** y - = tan u.
oy
<):.u
(4 marks (5 marks
(5 marks
(5 marks
(4 mark:
(5 marks (5 marks
rj.
/ 2 2\
s + y
x + y
\ J )
If ii = sin"1
/
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Attachment: |
| Earning: Approval pending. |
