Annamalai University 2010-1st Year B.Sc Mathematics , ( ) ( ) ( PART - III - A - MAIN ) ( - I ) 530ANALYSIS - I - Question Paper

B.Sc. DEGREE EXAMINATION, 2010
( MATHEMATICS )
( 1st YEAR )
( PART - III - A - MAIN )
( PAPER - I )
530. ANALYSIS - I

6

(b) Sum to infinity the series

i ₊ _i_ ₊ II + JL. ₊ 2L

3!    5! 7!    9!

Name of the Candidate :

12 0 2 B.Sc. DEGREE EXAMINATION, 2010

(MATHEMATICS)

(FIRST YEAR)

( PART-III-A-MAIN)

(PAPER - I)

530. ANALYSIS - I May ]    [ Time : 3 Hours

Maximum : 100 Marks

Answer any FIVE questions.

All questions carry equal marks.

(5 x 20 = 100)

1.    (a) Prove that-/! is irrational.

(b) State and prove Dedekinds theorem on real numbers.

2.    (a) Show that any Cauchy sequence of real

numbers is convergent.

,₊ _L ₊_L

1! 2!

is convergent.

(a) Find    if

dx

(i) y = 3x² e^3x cosx

x (x+ 1)

w ^y= vr

(b) Differentiate

sec ¹1rr

2x

with respect to

lim    logx

x 0 cosec x

(b) Evaluate

lim    , .tan 2x

(tan x)

x > ti/4

9. (a) Verify Eulers theorem for

u = x³ + y³ + z³ + 3xyz.

(b) If

-l (^{x2 +} y²

u = tan -

I x + y

Show that

9u    3u l

x - + y - = _

dx    dy 2

10. (a) Sum to infinity the series

2 2-5 2-5-8 + - + - +

6 6 12 6 12 18

Prove that

d²y    dy 2_t

^{(1 + x2)}17 ^{+ xmy = a}

6. (a) State and prove Rolles theorem, (b) If x is positive, prove that

1 + x<e^x< 1 + xe^x

7. (a) Find the maxima and minima of the function

x³ - 18x² + 96x + 4

(b) From a given circular sheet of metal, it is required to cut out a sector so that the remainder can be formed into a conical vessel of maximum capacity. Prove that the angle of the sector removed must be above 66.

(b) Show that the radius of curvature at any point of the cycloid

x = a (9 + sinG)

and y = a (1 - cosG)

9

is 4a cos

5. (a) If

x = cos t + t sin t

y = sin t - t cos t,

Turn over

Attachment: annamalai-university-2010-b-sc-mathematics-16640-1.pdf

Earning:   Approval pending.