Section-A
1. Define a function f of two real variab les in the xy-plane by
src='./qimages/209-1a.jpg'>
Check the continuity and differentiability of f at 0). 12
Let p and q be positive real numbers such that 1. Show that for real
numbers b 0
ab

Prove or disprove the following statement
If B b2, b3 b4 b5 is a basis for R5 and V is a two-dimensional
subspace of IR5, then V has a basis made of just two members of B.
Let R3 R3 be the linear transformation defined by
src='./qimages/209-1d.jpg'>

Find a basis and the dimension of the image of T and the kernel of T. 12
Prove that two of the straight lines represented by the equation
src='./qimages/209-1e.jpg'>
will be at right angles, if b+c -2.

2. Let V be the vector space of all
2 x 2 matrices over the field of
real numbers. Let W be the set
consisting of all matrices with zero
determinant. Is W a subspace ofV?
Justify your answer.
Find the dimension and a basis for
the space W of all solutions of
the following homogeneous system
using matrix notation

x1 +2x 2 +3x3 -2x4 +4x5
2x 1 +4x2 +8x3 +x4 +9x 5
3x1 6x2 13x3 +4x4 14x5 0

Consider the linear mapping
R2 R2 by
(3x+4y 2x-5y
Find the matrix A relative to the
basis and the matrix B
relative to the basis
If is a characteristic root of
a non-singular matrix then
prove that is a characteristic root of Adj A.

Let
<img src='./qimages/209-2c.jpg'>
be a Hermitian matrix. Find a nonsingular
matrix P such that D pT H P(bar) is diagonal.

3. Find the points of local extrema and
saddle points of the function f of two
variables defined by

y x3 y 3 63(x y 12xy
Define a sequence sn of real numbers
by
src='./qimages/209-3b.jpg'>
compute the value of this limit and justify your answer.
Find all the real values of p and q
so that the integral <img src='./qimages/209-3b.jpg'> converges.
4. Compute the volume of the solid enclosed between the surfaces

x2 y2 9 and x2 z2 9.

A variable plane is parallel to the plane

and meets the axes in C respectively. Prove that the circle ABC lies on the cone
src='./qimages/209-4b.jpg'>
Show that the locus of a point from which the three mutually perpendicular tangent lines can be drawn to the paraboloid x2 y2 2z O is x2 +y2

Section-B
5. Solve
src='./qimages/209-5a.jpg'>

Find the orthogonal trajectories of the family of curves x2 y2 ax. 12
Using Laplace transforms, solve the initial value problem
y" y 1 12

A particle moves with an acceleration <img src='./qimages/209-5d.jpg'> towards the origin. If it starts from rest at a distance a from the origin, find its velocity when its distance from the origin is

If
<img src='./qimages/209-5e.jpg'>

Show that the differential equation <img src='./qimages/209-6a.jpg'> is not exact. Find an integrating factor and hence, the solution of the equation.
Find the general solution of the equation y" l2x2 6x.
Solve the ordinary differential equation x(x l)y" (2x 2y x2 (2x

7. A heavy ring of mass slides on a smooth vertical rod and is attached to a light string which passes over a small pulley distant a from the rod and has a mass M fastened to its other end. Show that if the ring be dropped from a point in the rod in the same horizontal plane as the pulley, it will descend a distance before coming to rest.

A heavy hemispherical shell of radius a has a particle attached to a point on the rim, and rests with the curved surface m contact with a rough sphere of radius b at the highest point. Prove that if square root the equilibrium is stable, whatever be the weight of the particle.
The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is
src='./qimages/209-7c.jpg'>
8. Derive the Frenet-Serret formulae. Define the curvature and torsion for a space curve. Compute them for the space curve
X y t2, z t3

Show that the curvature and torsion are
equal for this curve.

Verify Green's theorem in the plane for
src='./qimages/209-8b.jpg'>
where C is the closed curve of the region
bounded by y x and y x2.
c)<img src='./qimages/209-8c.jpg'>
where S is the surface of the sphere
x2 y2 z2 a2 above the xy-plane.