IAS PYQs 3
1988
1) Show that the Cauchy product of with itself diverges.
2) Prove that the sequence where is not uniformly convergent in any interval .
3) Evaluate
over the area of the ellipse .
4) Discuss the convergence of the improper integral .
5) Show that a local extreme value of f given by ,
subject to the condition is .
6) The function is given by
Prove that at (0,0) is continuous and possesses all directional derivatives but is not differentiable.
1987
1) Let
2) If is continuous on a closed and finite interval then show that is uniformly continuous on .
3) Test for convergence .
4) If is monotonic and is bounded and real valued function Riemann integrable over , then Prove that there exists a such that
=
5) Test for uniform convergence of the sequence where when .
6) Find the maximum and minimum value of subject to the condition that .
1986
1) If for all then show that is uniformly continuous on every closed and finite interval, but is not uniformly continuous on .
2) Test the uniform convergence of the series .
3) If fand are differentiable on and are Riemann integrable over , then show that .
4) If is monotonic and is Riemann integrable over then show that there exists a such that
=
5) Find the maximum and minimum values of where , are connected by the relation .
1985
1) Examine the convergence of the integral .
2) State and prove the second mean value theorem for Riemann integrals.
3) Show that for the function
(i) is not differentiable at
(ii) is not continuous at
(iii)
1984
1) If is monotonic on and if is continuous on then prove that exists.
2) If and are integrable in the sense of Riemann on then prove that .
3) Show that the maximum and minimum values of the function , where are given by .
4) Discuss the continuity and differentiability at (0,0) of the function ; .
Also examine if and are equal at .
1983
1) If and decreases monotonically for then prove that converges iff converges.
2) Examine the convergence of the integral .
3) Obtain a set of sufficient conditions such that for a function , =.
4) Find the maximum and minimum values of subject to the conditions ; .