Tamil Nadu Open University (TNOU) 2008 B.Sc Mathematics 1st Year " Calculus & Classical Algebra " UG 313 - Question Paper

UG-313    BMS-01

B.Sc. DEGREE EXAMINATION -JUNE 2008.

(AY 2005-2006 CY 2006 batches only)

First Year

Mathematics

CALCULUS AND CLASSICAL ALGEBRA

Time : 3 hours    Maximum marks : 75

SECTION A (5 x 5 = 25 marks)

Answer any FIVE questions.

d² _y

1.    If y = acos5x + bsin5x show that 2 + 25y = 0 .

dx

2.    Find the angle between the radius vector and the

tangent at any point on the conic section = 1 + e cos 6 .

r

u    u

4. Prove that J f (x)dx = J f (t) dt.

5. Show that the sequence \¹ [> converges to 0.

6. If a_n is a convergent series then prove that

n=1

lim a_n = 0.

7.    Find the sum of first r coefficients in the expansion of (1 - x)^-3.

8.    Find the coefficient of xⁿ in the expansion of

a+bx

SECTION B (5 x 10 = 50 marks)

Answer any FIVE questions.

9.    If y = x²e^x show that

yn = 2^{n (n -} 1)y2 ^- n (n - 2)y1 + 2^{(n -}1) (n -2)y.

10.    Find the envelope of x cos³ a + y sin³ a = a where a is the parameter.

11.    Obtain a reduction formula for J sinⁿ xdx.

0

12.    Obtain a Fourier expansion for e^x in the interval - n < x < n.

13.    (a) State Leibnitz test for the convergence of an alternating series.

(b) Test the convergence of the series

(- 1)^n-1

n²

14. Test the convergence of the series 1 2 2 3 3 4

+-+-+......<x>.

3 4 5 4 5 6 5 6 7

15.    Find the sum to infinity the series

, 2 2 5 2 5 8 1 +---\---\---+..... 6 612 6 1218

16.    Find the sum to infinity the series

2³ 3³ 4³

1 + +-+-+.....<x> 2! 3 ! 4 !

3    UG-313

Attachment: tamil-nadu-open-university--tnou--2008-b-sc-mathematics-49084-1.pdf

Earning:   Approval pending.