Tamil Nadu Open University (TNOU) 2009-1st Year B.Sc Mathematics " ELEMENTS OF CALCULUS " (/ with Computer Applications ) UG 475 BMS/BMC 11 - Question Paper
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B.Sc. DEGREE EXAMINATION -JANUARY 2009.
First Year
Mathematics/Mathematics with Computer Applications
ELEMENTS OF CALCULUS
Time : 3 hours Maximum marks : 75
SECTION A (5 x 5 = 25 marks)
Answer any FIVE questions.
dy dx2
1. If y =acos5x + 6sin5x show that - + 25y = 0 . 17 7 2
2. Verify Eulers theorem for the function u (x, y) = x3 + y3 + 3x2y + 3xy2.
3. Find the envelope of the family of straight lines xcosa + j'sinar = p where a is the parameter.
1
5. Evaluate J J dr dO .
0 0
Show that the sequence {an} where an = for
6.
n
every n e N converges to 0.
7. Define
(a) Monotonic sequence
(b) Cauchy sequence.
8. Test the convergence of
SECTION B (5 x 10 = 50 marks) Answer any FIVE questions.
If y = acos(logx) + b sin(Iogx), prove
that
9.
x2yn + 2 +(2ti + 1 )xyn + l +(n2 + 1) yn =0
10. Find the maximum and minimum values of / (x, y) = 2 (x2 - /) - x1 + j4.
11. Show that the radius of curvature of the curve
2 2 (a - x) 1 . a
y = a - at (a, 0) is .
x 2
12. (a) Evaluate Jx (1 - x)10 dx .
0
(b) Prove that
2a a a
j" f (x) dx -U (x) dx + |f (2a - x) dx .
0 0 0
13. Establish a reduction formula for In = Jcosx dx
jt/2
where n e N and hence find J cos6 x dx .
0
14. If lim an = a and lim bn = b then prove that
n co ft co
lim (<an bn) = ab.
n > co
15. (a) Test the convergence of the series
3 5 7
+ H-- + co
12-22 22-32 32-42
(b) Show that a series un of positive terms
n
n - 1
either converges or diverges to go but never oscillates.
16. Let 'Luri and 'Zvn be the two given series of positive numbers such that un < kvn for every n e N where k is a positive number. Then prove that
(a) If I,vn is convergent Swn is convergent
(b) If Zu,h is divergent then 'Lvn is divergent.
4 UG-475
Find the surface area of a sphere of radius a.
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