IAS PYQs 2
1994
1) is defined as follows
Prove that and are continuous but is discontinuous.
[10M]
2) If and lie between the least and greatest values of prove that Where
[10M]
3) Prove that of all rectangular parallelopipeds of the same volume, the cube has the least surface.
[10M]
4) Show by means of beta function that
[10M]
5) Prove that the value of taken over the volume bounded by the co-ordinate planes and the plane is
[10M]
6) The sphere is pierced by the cylinder Prove by the cylinder Prove that the volume of the sphere that lies inside the cylinder is
[10M]
1993
1) Prove that and for is continuous and differentiable at but its derivative is not continuous there.
[10M]
2) If have derivatives when show that there is a value of lying between a and b such that
[10M]
3) Find the triangle of maximum area which can be inscribed in a circle.
[10M]
4) Prove that and deduce that ]
[10M]
5)
[10M]
6) Show that the volume common to the sphere and the cylinder is
[10M]
1992
1) If prove that and hence expand in powers of . Deduce the expansion of and .
[10M]
2) If , , , then prove that
[10M]
3) Find the dimensions of the rectangular parallelepiped inscribed in the ellipsoid that has greatest volume.
[10M]
4) Prove that the volume enclosed by the cylinders
is .
[10M]
5) Find the centre of gravity of the volume formed by revolving the area bounded by the parabolas and about the .
[10M]
6) Evaluate the following integral in terms of Gamma function: and prove that
[10M]
1991
1) Sketch the curve
2) Show that the function has as the only critical point and that has a minimum at this point.
3) Evaluate where is the interior of the triangle formed by the lines being all positive.
4) Find the equation of the cubic curve which has the same asymptotes as the curve and which passes through the points , and .
5) Prove, by considering the integral where is the square [0 or otherwise that .
1990
1.(a) If a function of the real variable has the first 5 derivatives 0 at a given value show tht it has a maximum or a minimum at according as the 6th derivative is negative or positive. What happens if only the first four derivatives are 0 but not the fifth?
1.(b) Show that has a critical point at if and only if, .
1.(c) Assuming that the condition in (2) holds, examine the nature of three critical points of detailing whether achieves maximum or minimum or else is stationary.
1.(d) Show that the function in (b) has atleast three (real) points of inflexion, irrespective of the condition in (2).
2) The functions in on [0,1] are given by .
For what values of p does the sequence converge uniformly to its limit f? Examine whether for and
1989
1) If is at least thrice continuously differentiable then show that , where lies between 0 and 1 and prove that
2) Prove that the volume of a right circular cylinder of greatest volume which can be inscribed in a sphere, is times that of the sphere.
3) Find the surface of the solid generated by the rotation of the Astroid ; about the axis of .
4) Evaluate over the interior of the tetrahedron with faces .