## Friday, January 04, 2019

2015
(November)
MATHEMATICS
(Major)
Course: 301
[Analysis – I (Real Analysis)]
Full Marks: 80
Pass Marks: 32 (Backlog)/24 (2014 onwards)
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) Evaluate: 2
(c) Find the length of the subnormal to the curve at the point 3
(d) If, then show that 4
Or
Find the radius of curvature at the point on the cycloid
2. (a) Write the statement of Darboux’s theorem. 1
(b) Show that
is continuous at 2
(c) Show that 4
Or
Expand in an infinite series using Maclaurin’s series.
(d) Verify Rolle’s Theorem for 3
3. (a) State the Euler’s Theorem on homogeneous function of two variables. 1
(b) If
then show that
4
Or
If
then show that
4. (a) Write the statement of Schwartz’s theorem. 1
(b) if and , then show that
4
(c) If z is a function of x and y and, then prove that
5
Or
Prove that
has a minimum value at .

GROUP – B
(Integral Calculus)
(Marks: 20)

5. (a) Write the value of
1
(b) Prove that 2
(c) Prove that 3
(d) Show that 4
Or
Using reduction formula, evaluate
6. (a) Write the formula for length of an arc between two pointsandwhen the curve is given in parametric form. 1
(b) Find the length of the arc of the curveandfrom to     4
Or
Find the perimeter of the cardioids
(c) Find the volume and surface area of the solid of revolution formed by rotation of the parabola about x-axis and bounded by 5
Or
Find the volume and surface of the solid of revolution of the ellipse
GROUP – C
(Riemann Integral)
(Marks: 25)
7. (a) Every bounded function defined on an interval [a, b] is Riemann integrable. State True or False. 1
(b) Prove that a constant function is always Riemann integrable. 3
(c) State and prove the necessary and sufficient condition for a function to be Riemann integrable. 4
Or
Prove that is a function is monotonic on [a, b], then it is Riemann integrable on [a, b].
8. (a) Define primitive of a function. 1
(b) If is bounded and integrable in, and and are the bounds of in , then prove that
3
(c) If
Both exist and keeps the same sign throughout the interval, then prove that there exists a number between the bounds of such that
3
9. (a) Give example of an improper integral of second kind. 1
(b) Test for convergence of 2
(c) Prove that
converges. 3
10. Answer any one of the following: 4
1. Prove that
Hence deduce
1. Prove that
and.
***