Tamil Nadu Open University (TNOU) 2006 B.Sc Mathematics 1st Year " Calculus And Classical Algebra " UG 743 - Question Paper
UG-743 BMS-01
B.Sc. DEGREE EXAMINATION - JUNE, 2006. First Year Mathematics CALCULUS AND CLASSICAL ALGEBRA
Time : 3 hours Maximum marks : 75
SECTION A (5 x 5 = 25 marks)
Answer any FIVE questions.
1. Differentiate y = (sin xY .
2. Find the equation of the tangent to the curve
y ~t~ at the point (2,4). x -1
3. Find the envelope of the family of straight lines
y = mx + for different values of m. m
5. If ux +u2 + .... +ua +..... is convergent then show
that lim u=0.
//co
6. Test the convergence of the series -.
n=0% +1
7. Define absolute convergence and conditional convergence.
8. Show that 2+16 +9 = nearly for
2x
sufficiently large values of x.
SECTION B (5 x 10 = 50 marks)
Answer any FIVE questions.
9. (a) If y = sin {m sin-1 x), prove that (l-x2)j'2-xj'1 +m2y = 0.
(b) Find lim 1~CS~*'- . (7 + 3)
j-->0 x
10. (a) What is the radius of curvature of the curve x1 +jy4 =2 at the print (1,1)?
(b) Find the pedal equation of the curve r = ae0cota. (6 + 4)
11. Obtain a reduction formula for Jsin" x , where n is a positive integer. Deduce a formula to evaluate
/r/2
Jsin" x dx . (6 + 4)
o
12. Express J(x)-x2 as a Fourier series with period 2k , to be valid in the interval -n to n . (10)
13. Show that the series -4- + i- + -4- +...... is
1* 2 3
convergent when >1 and divergent when k <1. (10)
14. (a) State Leibuitz test for checking the convergence of an alternating series.
(b) Discuss the convergence of the series
15. Sum the series
l2 l2 +22 l2 +22 +32 l2 + 22 +... + /Z2
--1---h.............. +.... H---h...
1! 2! 3! n\
16. Show that if x > 0,
. x-1 1 x2-l 1 x3-l
logX=-+--T- +--5- +..........(10)
x + 1 2 (x +1) 3 (ar + 1)
3 UG-743
Using Bernoullis formula, find j(2x2 +l)cos;r<2'x.
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| Earning: Approval pending. |
