1994

1) Examine the

(i) absolute convergence
(ii) uniform convergence

of the series (1−x)+x(1−x)+x2(1−x)+… in [−c,1], where 0<c<1.

[10M]

2) Prove that S(x)=∑1np+nqx2,p>1 is uniformly convergent for all values of x and can be differentiated term by term if q<3p−2

[10M]

3) Let the function f be defined on [0,1] by the condition f(x)=2rx when 1r+1<x<1r,r>0. Show that f is Riemann integrable in [0,1] and ∫10f(x)dx=π26.

[10M]

4) By means of the substitution x+y+z=u,y+z=uv,z=uvw, evaluate ∭(x+y+z)nxyzdxdydz taken over the volume bounded by x=0,y=0,z=0,x+y+z=1.

[10M]

1993

1) Show that x=0 is a point of non-uniform convergence of the series whose nth term is n2xen2x2−(n−1)2xe−(n−1)2x2.

[10M]

2) Find all the maxima and minima of f(x,y)=x3+y3−63(x+y)+12xy

[10M]

3) Examine for Riemann integrability over [0,2] of the function defined in [0,2] by f(x)={x+x2, for rational values of xx2+x3, for irrational values of x

[10M]

4) Prove that ∫∞0sinxxdx converges and conditionally converges.

[10M]

5) Evaluate ∭dxdydzx+y+z+1 over the volume bounded by the coordinate planes and the plane x+y+z=1.

[10M]

1992

1) Examine f(x,y,z)=2xyz−4zx−2yz+x2+y2+z2−2x−4y−4z for extreme values.

[10M]

2) If Un=1+nxnenx−1+(n+1)x(n+1)e(n+1)x,0<x<1 Prove that ddxΣUn=Σddxun Is the series uniformly convergent in [0,1] ? Justify your claim.

[10M]

3) Find the upper and lower Riemann integral for the function defined in the interval [0,1] as follows: f(x)={√1−x2when x rational 1−x when x is irrational  and show that f(x) is not Riemann integrable in [0,1]

[10M]

4) Discuss the convergence or divergence of ∫∞0xβdx1+xαsin2x,α>β>0

[10M]

5) Evaluate ∬√(a2b2−b2x2−a2y2)√(a2b2+b2x2+a2y2)dxdy over the positive quadrant of the ellipse x2a2+y2b2=1

[10M]

1991

1) Examine whether the function y=tanx is uniformly continuous in the open interval (0,π2).

2) Evaluate ∫∞0log(1+a2x2)1+b2x2dx.

3) If the rectangular axes (x,y) are rotated through an angle α about the origin and the new co-ordinates are (ˉx,ˉy) show that for any function u, ∂2u∂x2+∂2u∂y2=∂2u∂ˉx2+∂2u∂ˉy2.

4) A rectangle is inscribed in the ellipse x2a2+y2b2=1. What is the maximum possible area of the rectangle?

1990

1) Discuss the convergence of ∫π/20logsinxdx and evaluate it, if it is convergent.

2) Find the point on the parabola y2=2x, z=0, which is nearest the plane z=x+2y+8. Show that this minimum distance is √6.

3) Show that ∭(a2b2c2−b2c2x2−c2a2y2−a2b2z2)1/2dxdydz over the volume bounded by x2a2+y2b2+z2c2=1 is equal to ≠2a2b2c24.

TBC

1989

1) The function f:[a,∞)→R is continuous and f(x)→l as x→∞, prove that f is uniformly continuous on [a,∞).

2) Find the values of x for which the series ∑∞n=1xa1+n2x2,x≥0,α≥0, converges uniformly on (i) [0,1] and
(ii) [0,∞)

3) Discuss the existence of the improper integral ∫∞−∞e−x2/2dx.

4) Show that the value of ∬(1−x−y)2x1/2y1/2dxdy taken over the interior of the triangle whose vertices are the origin and the points (0,1) and (1,0) is ≠410.

TBC