IAS PYQs 2
1994
1) Examine the
(i) absolute convergence
(ii) uniform convergence
of the series in , where .
[10M]
2) Prove that is uniformly convergent for all values of and can be differentiated term by term if
[10M]
3) Let the function be defined on by the condition when . Show that is Riemann integrable in and .
[10M]
4) By means of the substitution , evaluate taken over the volume bounded by .
[10M]
1993
1) Show that is a point of non-uniform convergence of the series whose term is .
[10M]
2) Find all the maxima and minima of
[10M]
3) Examine for Riemann integrability over [0,2] of the function defined in [0,2] by
[10M]
4) Prove that converges and conditionally converges.
[10M]
5) Evaluate over the volume bounded by the coordinate planes and the plane .
[10M]
1992
1) Examine for extreme values.
[10M]
2) If Prove that Is the series uniformly convergent in ? Justify your claim.
[10M]
3) Find the upper and lower Riemann integral for the function defined in the interval [0,1] as follows: and show that is not Riemann integrable in [0,1]
[10M]
4) Discuss the convergence or divergence of
[10M]
5) Evaluate over the positive quadrant of the ellipse
[10M]
1991
1) Examine whether the function is uniformly continuous in the open interval .
2) Evaluate .
3) If the rectangular axes are rotated through an angle about the origin and the new co-ordinates are show that for any function , .
4) A rectangle is inscribed in the ellipse . What is the maximum possible area of the rectangle?
1990
1) Discuss the convergence of and evaluate it, if it is convergent.
2) Find the point on the parabola , , which is nearest the plane . Show that this minimum distance is .
3) Show that over the volume bounded by is equal to .
TBC
1989
1) The function is continuous and as prove that is uniformly continuous on .
2) Find the values of for which the series converges uniformly on
(i) and
(ii)
3) Discuss the existence of the improper integral .
4) Show that the value of taken over the interior of the triangle whose vertices are the origin and the points and is .
TBC