Calicut University 2006 B.Sc Mathematics II-DIFFERENTIAL AND INTEGRAL CALCULUS - Question Paper
FINAL YEAR B.Sc. DEGREE EXAMINATION, SEPTEMBER / OCTOBAR 2006
Part III - Mathematics (Main)
PAPER II-DIFFERENTIAL AND INTEGRAL CALCULUS
(Improvement - 2000 and earlier admissions)
5002
Name,
Reg. No...............................................
AL YEAR B.Sc. DEGREE EXAMINATION, SEPTEMBERyOCTOBER 2006
Part IIIMathematics (Main) Paper IIDIFFERENTIAL AND INTEGRAL CALCULUS (Improvement2000 and earlier admissions)
: Three Hours Maximum : 75 Marks
There are four units.
imum marks that can be earned from Unit I is 20, Unit II is 20, Unit III is 10 Cind Unit IV is 25.
Find the radius of curvature0 on the curve given by the equation : x - c sin 0(1 + cos 2 0) and y = c cos 2 0(1- cos 20)
* * (4 marks)
1 1 4 4 J
2. Find the radius curvature phe curve 4x + \/y-"= 1 at
ark-*)-
v** **/
3)Prove that the asymptotes of xy2 = c2Jx2r\La.\ vertipdij of a .---(5 marks)
4. Prove that the evolute of the hyperbola 2tggj/a2 is (x + y)% - {x - y)% = 2a . (6 marks).
"RFinarks/-
(6 marks) (4 marks)
5. Tr?.ce the curve y2 = (jc - 1) (x - 2)2.
<> *
G. Show that the evolute of - :L- = 1 is (ax)'3 + {by)7$ = j a* + 6* I
a b* '
Examine for double points of the curve x4 - 2 ay3 - 3 a*y2.....2 a2x2 + a4 = 0.
Unit n
n
n
r*2 (1 - xZ ] dx prove that In =
(5 marks)
. evaluate I,. n + 1 n 1 7
I
8. If I =
n
0
9. Find the area cut off from the parabola y2 = 4 ax by the line y = mx. (5 marks)
10. Find the reduction formula for j;tm (log x)ndx hence evaluate Jx4(log x)3cx. (5 marks)
Url. Find the length of an arch of the cycloid whose equations are x = a (t - sin t)t y = a (1 - cos t).
m
(5 marks)
12. * Find the volume generated by the revolution of the loopy2 = x4 (x + 2) about the X-axis. (5 marks)
13. Find the area of the surface formed by revolving the ellipse x2 + 4 y2 = 16 about the X-axis.
(5 marks)
14. Find the moment of inertia of a circular disc about any diameter. - (5 marks)
Turn over
2
</
Unit HI
15.* Using Maciaiirins theorem expand log (1 + x).
c t ~ 2. 3 3.5 4.7 5.9
16. Sum to muruty - +--+ - + - + . . . .
J 3! 4! 5! 6!
.
1 1 1 ' * 1 v
17. Find the sum of the series- - - + - - - + .
1.2 2.3 3.4 4.5
V2 f)2 A2 o
18. If r2 = (x - a)2 + (y - 6)2 + (z - c)z prove that+ + ~ = .
dx dy dz r
3 3
19. If log u = --- show that x + y = 2 ii log u.
$ dx J dy h
20. Given that a: + y + z = a find the maximum value of xfn y zp.
* - y
+ yJ
21. Verify Eulers theorem on homogeneous functiou for u - sin
22. Expand sin1 x in powers of* by Taylors thtforem.
23. If w = - - ; x = cosh i ; v - oiiih i iind .
2 2 ' * * -}* x + y
fa 2 a V a - x ---
! ?.
24. Evaluate J j a - - y dy dx.
0 o
25. Find the maxima and minima of fix, y) = x3 * y1 - 3 x - 12 y + 20.
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| Earning: Approval pending. |
