Annamalai University 2008-1st Year B.Sc Mathematics " 530 ANALYSIS - I " I 5229 - Question Paper

4

(b) Evaluate:

9. (a) If

. -i (y

u = sin ,

\ x ) ,

prove that

3²u    3²u

dx dy    dy dx

(b) If

prove that

3u 3u ^x - + y - = 3u log u.

dx    dy

10. (a) Find the sum to infinity of the series

1 1 1 T3 ⁺ 1 -2-3-5 ⁺ 1-2-3-4-5-7 ⁺ ""

(b) Find the sum to infinity of the series

1111 ~2~23⁺34~T5 ⁺

Name of the Candidate :

5 2 2 9 B.Sc. DEGREE EXAMINATION, 2008

(MATHEMATICS)

(FIRST YEAR)

(PART - III - A - MAIN)

(PAPER - I)

530. ANALYSIS -1

December ]    [ Time : 3 Hours

Maximum : 100 Marks

Answer any FIVE questions.

All questions carry equal marks.

(5 x 20 = 100)

1.    (a) State and prove Dedekinds theorem on

real numbers.

(b) Prove that (0, 1) is uncountable.

2.    (a) Discuss the convergence of the series

1

j _ j_ J___L ₊

2 ⁺ 3 4 ⁺

3. (a) Find

dy

dx

r\    o 2 3x

(l) y = 3x e cos x.

^X(X1}

(n) y =

X + 1

(b) Differentiate

-i ( ¹

sec I ₂

2x - 1 with respect to '{l

4. (a) Find the equation of the tangent to the

curve y = x³ at the point I _J_ _J_

2 8

(b) Find the centre of curvature of the curve

5. (a) If

x = cos t + t sin t; y = sin t - t cos t,

rr i d²y

find -

(b) If y^1/m + y ^1/m = 2x, prove that

(x²-l)y + (2n + 1) y , + (n²-m²) y = 0.

^v ''n + 2 ^y    ' 'n+l ^y    ' ^J n

6.    (a) State and prove Rolles theorem.

(b) if f(x) is a quadratic expression, show that the parameter 0 in the Lagranges mean value theorem is V2.

7.    (a) Determine the maxima and minima of

1 - x + x² 1 + X + x²

(b) Prove that the volume of the greatest right circular cone that can be inscribed in a given sphere is 8/27 of the volume of the sphere.

8.    (a) Evaluate:

SCC 7TX

_x 0 ^tan 3ftx

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Attachment: annamalai-university-2008-b-sc-mathematics-16702-1.pdf

Earning:   Approval pending.