Annamalai University 2011-1st Year B.Sc Mathematics , ( ) ( ) ( Part - III ) ( Group - A - Main ) ( - I ) 530Analysis - I - Question Paper

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Name of the Candidate :

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B.Sc. DEGREE EXAMINATION, 2011

(MATHEMATICS)

(FIRST YEAR)

(PART-III)

(GROUP- A-MAIN)

(PAPER -1)

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 any non-empty set of real numbers which is bounded above has a supremum.

(b) Prove that is irrational.

2. (a) Prove that any Cauchy sequence of real numbers is convergent.

(b) Discuss the convergence of the series.

3.    (a) Find , if

(i) y = sin³ (x²) x + 4 ^y =

(b) Differentiate sec ¹

with respect to

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

6x

curve y = ~? ^at the point (2, 4) x -1

(b) Find the radius of curvature of the

2

curve y =    at the point (2, 0).

5. (a) Find y if y =

n

(b) If y = (x +    )^m, prove that

2    ^dy ?

(1 + X ) + X -- m y = 0

dx

6.    (a) State and prove Rolles theorem.

(b) If x is positive, show that

1 2

x X < log (1 + x) < X

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

x³ + 3x² - 24x + 20.

(b) Prove that the volume of the greatest right circular cone that can be inscribed in a 8

given sphere is of the volume of the sphere.

8.    (a) Evaluate :

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