1988

1) Show that the Cauchy product of ∑∞n=0(−1)n√n+1 with itself diverges.

2) Prove that the sequence {Sn(x)} where Sn(x)=nxe−nx2 is not uniformly convergent in any interval (0,k),k>0.

3) Evaluate
∬(1−x2a2−y2b2)1/2dxdy over the area of the ellipse x2a2+y2b2=1.

4) Discuss the convergence of the improper integral ∫x0tx−1e−tdt.

5) Show that a local extreme value of f given by f(ˉx)=xk1+….+xkn, ˉx=(x1,x2,…,xn)
subject to the condition x1+x2+……+xn=a, is akn1−k.

6) The function f:R2→R1 is given by
f(x,y)={x2yx2+y2 when (x,y)≠(0,0)0 when (x,y)=(0,0) Prove that at (0,0) f is continuous and possesses all directional derivatives but is not differentiable.

1987

1) Let

2) If f is continuous on a closed and finite interval [a,b] then show that f is uniformly continuous on [a,b].

3) Test for convergence ∫−x20dx.

4) If f is monotonic and g is bounded and real valued function Riemann integrable over [a,b], then Prove that there exists a c∈[a,b] such that

∫b0f(x)g(x)dx=f(a)∫c0g(x)dx+f(b)∫bcg(x)dx

5) Test for uniform convergence of the sequence {Sn(x)} where Sn(x)=nx(1−x)n when 0′′x′′′1.

6) Find the maximum and minimum value of f(x)=xy subject to the condition that x2+y2+xy=a2.

1986

1) If f(x)=x2 for all x∈R, then show that f is uniformly continuous on every closed and finite interval, but is not uniformly continuous on R.

2) Test the uniform convergence of the series ∑∞n=1xn(1+nx2).

3) If fand g are differentiable on [a,b] and f′,g′ are Riemann integrable over [a,b], then show that ∫b0f(x)dg(x)+∫bag(x)df(x)=f(b)g(b)−f(a)g(a).

4) If f is monotonic and g is Riemann integrable over [a,b], then show that there exists a C∈[a,b] such that

∫b0f(x)g(x)dx=f(a)∫g(x)dx+f(b)∫bcg(x)dx

5) Find the maximum and minimum values of f(x,y)=7x2+8xy+y2 where x, y are connected by the relation x2+y2=1.

1985

1) Examine the convergence of the integral ∫∞0sinx∞xndx.

2) State and prove the second mean value theorem for Riemann integrals.

3) Show that for the function f(x,y)={x2y2x2+y2;(x,y)≠(0,0)0;x=0,y=0
(i) fx is not differentiable at (0,0)
(ii) fyx is not continuous at (0,0)
(iii) fxy(0,0)=fyx(0,0)

1984

1) If f is monotonic on [a,b] and if α is continuous on [a,b], then prove that ∫bafdx exists.

2) If f and α′ are integrable in the sense of Riemann on [a,b], then prove that ∫fdα=∫b0f(x)α′(x)dx.

3) Show that the maximum and minimum values of the function u=x2+y2+xy, where ax2+by2=ab(a>b>0) are given by 4(u−a)(u−b)=ab.

4) Discuss the continuity and differentiability at (0,0) of the function f(x,y)=x2tan−1(y/x)−y2tan−1(x/y); f(0,0)=0.

Also examine if fxy and fyx are equal at (0,0).

1983

1) If f(x)≥0 and f decreases monotonically for x≥1, then prove that ∫∞1f(x)dx converges iff ∑∞n=1f(n) converges.

2) Examine the convergence of the integral ∫10(xp+x−p)log(1+x)dxx.

3) Obtain a set of sufficient conditions such that for a function f(x,y), ∂2f∂x∂y=∂2f∂y∂x.

4) Find the maximum and minimum values of x2+y2+z2 subject to the conditions x+y+z=1; xyz+1=0.