C.S.E. (MAIN)
MATHEMATICS- 2005
PAPER-I
Time allowed 3 hours Maximum MarkS 300
INSTRUCfiONS
SECfiON'A'
Q. 1. Attempt any five of the following
Find the values of k for which the vectors and are linearly independent in R4
(b)Let V be the vector space of polynomials in x of degree n over R. Prove that the set x2 ... is a basis for V. Extend this basis so that it becomes a basis for the set of all polynomials in x.

Show that the function given below is not continuous at the origin:
if xy
1 if xy not equal to 0

Let be defined as
<img src='./qimages/1143-1d.jpg'>

0
Prove that fx and fy exist at but f is not differentiable at
If nornals at the points of an ellipse whose eccentric angles are alpha,beta,gaama and delta meet in a point, then show that sin(beta+gaama) +sin (gaama+ alpha)+ sin (alpha+ beta) 0

A square ABCD having each diagonal AC and BD of length 2a, is folded along the diagonal AC so that the planes DAC and BAC are at right angle. Find the shortest distance between AB and DC.
Q.2.(a) Let T be a linear transformation on R3,whose matrix relative to the standard basis of R3 is <img src='./qimages/1143-2a.jpg'> Find the matrix of T relative to the basis. beta
Find the inverse of the matrix given below using elementary row operations only
<img src='./qimages/1143-2b.jpg'>

S is a skew-Hermitian matrix, then show that universe is a unitary matrix. Also show that every unitary matrix can be expressed in the above form provided is not an eigenvalue of A.
Reduce the quadratic form <img src='./qimages/1143-2d.jpg'> to the sum of squares. Also find the corresponding linear transformation, index and signature.

Q. If u x y uv y z and uvw then find
<img src='./qimages/1143-3a.jpg'>

Evaluate
<img src='./qimages/1143-3b.jpg'>
in terms of Beta function.

Evaluate <img src='./qimages/1143-3c.jpg'> where V is the volume bounded below by the cone x2 y2 z2 and above by the sphere x2 y2 z2 lying on the positive side of the y-axis.

Find the x-coordinate of the centre of gravity of the solid lying inside the cylinder x2 y2 2ax, between the plane z 0 and the paraboloid x2 y2 az.
Q.4. A plane is drawn through the line x y z 0 to make an angle sin-1 with the plane x y z 5. Show that two such planes can be drawn. Find their equations and the angle between them.
Show that the locus of the centres of spheres of a co-axial system is a straight line.
Obtain the equation of a right circular cyclinder on the circle through the points and as the guiding curve.
Reduce the following equation to canonical form and determine which surface is represented by it <img src='./qimages/1143-4d.jpg'>
SECTION'B'
Q. 5. Attempt any five of the following:
(a)Find the orthogonal trajectory of a system of co-axial circles x2 y2 2gx c where g is the parameter.
(b)Solve:
<img src='./qimages/1143-5b.jpg'>

A body of mass moving in a straight line is split into two parts of masses m1 and m2 by an internal explosion which generates kinetic energy E. If after the explosion, the two parts move in the same line as before, find their relative velocity.
If a number of concurrent forces be represented in magnitude and direction by the sides of a closed polygon, taken in order, then show that these forces are in equilibrium.

Show that the volume of the tetrahedron ABCD is 1/6 (AB x AC).AD. Hence, find the volume of the tetrahedron with vertices(2,2,2), and
<img src='./qimages/1143-5e.jpg'>

Prove that the curl of a vector field is independent of the choice of coordinates.
Q.6.(a) Solve the differential equation
<img src='./qimages/1143-6a.jpg'>

Solve the differential equation (x2 yp) yp)2 0 where p dy/dx,by reducing it to Clairaut's form by using suitable substitution.
Solve the differential equation
(sin x cos x sin y sin x 0 given that y =sin x
is a solution of this equation.

Solve the differential equation
x2 2xy' 2y xlogx,x 0 by variation of parameters.

Q.7.(a) A particle is projected along the inner side of a smooth vertical circle of radius a so that its velocity at the lowest point is u. Show that if 2ag u2 5ag, the particle will leave the circle before arriving at the highest point and will describe a parabola whose latus rectum is <img src='./qimages/1143-7a.jpg'>
Two particles connected by a fine string are constrained to move in a fine cycloidal tube in a vertical plane. The axis of the cycloid is vertical with vertex upwards. Prove that the tension in the string is constant throughout the motion.
Two equal uniform rods AB and AC, of length a each, are freely joined at and are placed symmetrically over two smooth pegs on the same horizontal level at a distance c apart (3c 2a). A weight equal to that of a rod, is suspended from the joint A. In the position of equilibrium, find the inclination of either rod with the horizontal by the principle of virtual work.

A rectangular lamina of length 2a and breadth 2b is completely immersed in a vertical plane, in a fluid, so that its centre is at a depth h and the side 2a makes an angle alpha with the horizontal. Find the position of the centre of pressure.

Q.8.(a) The parametric equation of a circular helix is r a cos ui a sin uj cu k. where c is a constant and u is a parameter. Find the unit tangent vector t at the point u and the arc length measured from u 0. Also find dt/ds, where s is the arc length.
(b)Show that
<img src='./qimages/1143-8b.jpg'>

where r is the distance from the origin and k is the unit vector in the direction OZ.
Find the curvature and the torsion of the space curve
x u3)
y 3au2
z·= a(3u u3)

Evaluate
<img src='./qimages/1143-8d.jpg'>

by Gauss divergence theorem, where S is the surface of the cylinder x2 y2 a2 bounded by z 0 and z b.