## Friday, January 04, 2019 2013
(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 is the positive integer. 1
(b) Find the value of 2 (c) Show that the subnormal at any point of a parabola is of constant length. 3
Or
Find the curvature of the ellipse and at (d) If then show that 4 Or
If is the angle between the radius vector and the tangent at any point of the curve then prove that 2. (a) State Darboux’s theorem in differential calculus. 1
(b) If a function is continuous in and differentiable in such that then prove that is strictly increasing on . 2
(c) State and prove Cauchy’s mean value theorem. 3
Of
If is continuous on and differentiable in and then show that (d) Prove that 4
Or
Expand in powers of for 3. (a) State and Euler’s theorem on homogeneous function of two variables. 1
(b) If then prove that 4
Or
If , then prove that 4. (a) State Schwarz’s theorem. 1
(b) If then show that 2 (c) If then prove that 3
(d) Prove that has neither maximum nor minimum at 4
Or
If , where , then prove that GROUP – B
(Integral Calculus)
(Marks: 20)

5. (a) If then write the relation between and 1
(b) Show that 2
(c) Evaluate: 3
(d) Deduce the reduction formula for 4
Or
Prove that 6. (a) If then write the length of arc where A and B have abscissa a and b respectively. 1
(b) Find the whole length of the curve 5
Or
Find the volume of the solid generated by revolving the ellipse about the major axis.
(c) Find the surface of the solid formed by revolving the cardioids about the initial line.    4
GROUP – C
(Riemann Integral)
(Marks: 25)

7. (a) What is the essential condition for a function to be Riemann integrable in [a, b]? 1
(b) Prove that for any two partitions and of  . 2
(c) Show that every monotonic function on is Riemann integrable. 5
Or
If and  are lower and upper bounds of on , then show that 8. (a) State first mean value theorem of integral calculus. 1
(b) A function having primitive may not be continuous. Justify it. 3
(c) If and g are integrable on and keeps the same sign over , then show that and where m, M are bounds of g on . 3
9. (a) Define Abel’s test for convergence of integral of a product. 1
(b) Show that Converges to 0 2
(c) Test for convergence (any one): 3
1. 2. 10. (a) Prove that 4 (b) Prove that ***