## Friday, January 04, 2019 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)

1. (a) If then write the value of 1
(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 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 ***