2000

1) Given that the terms of a sequence {an} are such that each ak,k≤3, is the arithmetic mean of its two immediately preceding terms. Show that the sequence converges. Also find the limit of the sequence.

[12M]

2) Determine the values of x for which the infinite product ∏∞n=0(1+1x2n) converges absolutely. Find its value whenever it converges.

[12M]

3) Suppose f is twice differentiable real valued function in (0,∞) and M0,M1 and M2 the least upper bounds of |f(x)|,|f′(x)| and |f′′(x)| respectively in (0,∞). Prove for each x>0,h>0 that f′(x)=12h[f(x+2h)−f(x)]−hf′(u) for some u∈(x,x+2h). Hence show that M21≤4M0M2

[15M]

4) Evaluate ∬S(x3dydz+x2ydzdx+x2zdxdy) by transforming into triple integral where S is the closed surface formed by the cylinder x2+y2=a2,0≤z≤b and the circular discs x2+y2≤a2, z=0 and x2+y2≤a2,z=b

[15M]

5) Suppose f(ζ) is continuous on a circle C. Show that ∫Cf(ζ)dζ(ζ−z) as z varies inside of C, is differentiable under the integral sign. Find the derivative. Hence or otherwise, derive an integral representation for f′(z) if f(z) is analytic on and inside of C.

[15M]

1999

1) A sequence {Sn} is defined by the recursion formula Sn+1=√3Sn;S1=1. Does this sequence converge? If so, find limit Sn.

2) Find the shortest distance from the origin to the hyperbola x2+8xy+7y2=225, z=0.

[20M]

3) Show that the double ∫∫Rx−y(x+y)3dxdy does not exist over R=[0,1;0,1].

[20M]

4) Show that the double integral ∬Rx−y(x+y)3dxdy does not exist over R=[0,1;0,1].

1998

1) Show that the function f(x,y)=2x4−3x2y+y2 has (0,0) as the only critical point but the function neither has a minima nor a maxima at (0,0).

2) Test the convergence of the integral ∫∞0e−axsinxxdx, a≥0.

3) Test the series ∑∞n=1x(n+x2)2 for uniform convergence.

4) Let f(x)=x and g(x)=x2. Does ∫10fdg exist? If it exist, then find its value.s

1997

1) Show that a non-empty set P in Rn each of whose points is a limit-point is uncountable.

[10M]

2) Show that ∭Dxydxdydz=a2b2c26 where domain D is given by x2a2+y2b2+z2c2≤1.

[10M]

3) If u=sin−1[(x2+y2)1/5], prove that x2∂2u∂x2+2xy∂2u∂y⋅∂x+y2∂2x∂y2=225tanu(2tan2u−3)

[10M]

1996

1) A function f is defined in the interval (a,b) as follows: f(x)=q−2, when x=pq−1f(x)=q−3, when x=(pq−1)1/2 where p,q are relatively prime integers; f(x)=0, for all other values of x. Is f Riemann integrable ? Justify your answer.

[15M]

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

[15M]

3) Evaluate ∫π/20∫π/20sinxsin−1(sinxsiny)dxdy

[15M]

1995

1) Let f be a continuous real function on R such that f maps open interval onto open intervals. Prove that f is monotonic.

[15M]

2) Suppose f maps an open ball U⊂Rn into Rm and f is differentiable on U. Suppose there exists a real number M>0 such that |f′(x)|≤M for all x∈U. Prove that |f(b)−f(a)|≤M|b−a| for all a,b∈U

[15M]

3) Find and classify the extreme values of the function f(x,y)=x2+y2+x+y+xy

[15M]

4) Suppose α is a real number not equal to nπ for any integer n. Prove that ∫∞0∫∞0e−(x2+2xycosα+y2)dxdy=α2sinα

[15M]