2014

(November)

MATHEMATICS

(Major)

Course: 301

[Analysis – I (Real Analysis)]

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions

GROUP – A

(Differential Calculus)

(Marks: 35)

(b) Find the limit: 2

(c) Define sub tangent. Show that the sub tangent at any point of a parabola varies as the abscissa of the point of contact. 1+2=3

(d) If then prove that

Or

Show that the radius of curvature at the origin of the conic is

2. (a) Give an example of a continous function in a domain which has neither infimum nor supremum therein.

(b) If a function satisfies the conditions of Lagrange’s mean value theorem and also then show that is constant on.

(c) If then prove that

Or

Using Maclaurin’s theorem, expand in an infinite series.

(d) Discuss the applicability of the Rolle’s Theorem for in.

3. (a) Define homogeneous function of two variables. 1

(b) If then show that 4

Or

If u is a homogeneous function, of degree n, of x and y, then show that

4. (a) State Young’s theorem. 1

(b) If then show that 4

(c) If and are twice differentiable functions and then prove that 5

c is a constant.

Or

Find the extreme values of

GROUP – B

(Integral Calculus)

(Marks: 20)

5. (a) Write the reduction formula for 1

(b) Prove that 2

(c) Show that 4

(d) Evaluate:

Or

Prove that

6. (a) If and be the vectorial angles of A and B respectively, then write the formula for arc 1

(b) Find the perimeter of the cardioids 5

(c) Find the area of the surface of revolution formed by revolving the curve about the initial line. 4

Or

Find the volume of the solid generated by revolving the asteroid

about the x-axis.

GROUP – C

(Riemann Integral)

(Marks: 25)

7. (a) Define refinement of a partition. 1

(b) Show that the function defined by

is not integrable on any interval. 2

(c) Prove that every continuous function is Riemann integrable. 5

Or

If when)

, when

then show that although it has many points of discontinuity.

8. (a) State the fundamental theorem of integral calculus. 1

(b) If is a continuous function on, then show that there exists a number such that 3

(c) If a function is bounded and integrable on, then the function F defined as

Is continuous on. Prove it. 3

Or

Verify the mean value theorem for in the interval.

9. (a) State the Dirichlet test for convergence of integral of a product. 1

(b) Test for convergence of 2

(c) Prove that is convergent for 3

10. Answer any one of the following: 4

(a) Prove that

(b) Prove that

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