2004

1) Show that the sequence {fn}, where fn(x)=nxe−nx2 is pointwise, but not uniformly convergent in [0,∘).

3) Show that f(x,y)=x4+x2y+y2 has a minimum at (0,0).

2003

1) Evaluate I=∬(a2−x2−y2)1/2dxdy over the positive quadrant of the circle x2+y2=a2.

4) Let (a,b) be any open interval, f a function defined and differentiable on (a,b) such that its derivative is bounded on (a,b). Show that f is uniformly continuous on (a,b)

5) If f is a continuous function on [a,b] and if ∫b0f2(x)dx=0 then show that f(x)=0 for all x in [a,b]. Is this true if f is not continuous?

6) Find the maximum and minimum distances of the point (3,4,12) from the sphere x2+y2+z2=1.

2002

1) Evaluate ∭√1−zdxdydz through the volume bounded by the surfaces x=0,y=0,z=0 and x+y+z=1

2) Define a compact set. Prove that the range of a continuous function defined on a compact set is compact.

3) A function f is defined in [0,1] as f(x)=(−1)r−1 ; 1r+1<x<1r, where r is a positive integer show that f(x) is Riemann integrable in [0,1]& find its Riemann integral.

4) A function f is defined as f(x,y)=x2y2x2+y2,(x,y)↑(0,0)=0, otherwise. Prove that fxy(0,0)=fyx(0,0) but neither fyx nor fxy is continuous at (0,0).

2001

1) Change the order of integration in the integral ∫4a0∫2√axs24adydx and evaluate it.

2) If an=log(1+1n2)+log(1+2n2)+….+(1+nn2) find lim limx→∞an.

3) State the weierstrass M-test for uniform convergence of an infinite series of functions. Prove that the series ∑∞n=1xn(1+nx2) with α<0 is uniformly convergent on (−∞,∞).

2000

1) Change the order of integration in ∫2a0∫3axs24aϕ(x,y)dxdy

2) Test the convergence of the integral ∫10sin1x√xdx

3) Test for uniform convergence the series ∑∞n=12nx(2n−1)1+x2n

4) Prove that the function f(x,y)=x2−2xy+y2−x2−y3+x5 has neither a maximum nor a minimum at the origin.

5) Evaluate ∫∞0loge(x2+1)x2+1dx by using method of residues.