CS (Main) Exam:20l5 MATHEMATICS Paper—I

SECTION—A

Q. The vectors V1= V2 I V3 and V4 1 are linearly independent. Is it true Justify your answer. 10

Q. Reduce the following matrix to row echelon form and hence find its rank
src='./qimages/128-1b.jpg'>

Q. Evaluate the following limit:
src='./qimages/128-1c.jpg'>

Q. Evaluate the following integral:
src='./qimages/128-1d.jpg'>

Q. For what positive value of the plane ax 2y z 12 0 touches the sphere x2 y2 z2 2x 4y 2z 3 0 and hence find the point of contact. 10

Q. If matrix A <img src='./qimages/128-2a.jpg'> then find A30. 12

Q. A conical tent is of given capacity. For the least amount of Canvas required, for it, find the ratio of its height to the radius of its base. 13
Q. Find the eigen values and eigen vectors of the matrix
src='./qimages/128-2c.jpg'>

Q. If 6x 3y 2z represents one of the three mutually perpendicular generators of the cone 5yt 8zx 3xy 0 then obtain the equations of the other two generators. 13

Q. Let V R3 and T € for all a1 € be defined by
src='./qimages/128-3a.jpg'>

What is the matrix T relative to the basis
V2 v 3 12

Q. Which point of the sphere x2 y2 z2 1 is at the maximum distance from the point 13

Q. Obtain the equation of the plane passing through the points 3 and parallel to x-axis. 6

Verify if the lines

src='./qimages/128-3c-ii.jpg'>

are coplanar. If yes, then find the equation of the plane in which they lie. 7

Q. Evaluate the integral
src='./qimages/128-3d.jpg'>
where R is the rhombus with successive vertices as <img src='./qimages/128-3d-i.jpg'>. 12

Q. Evaluate <img src='./qimages/128-4a.jpg'>
where R 1 2]. 13

Q. Find the dimension of the subspace of R4, spanned by the set



Hence find its basis. 12

Q. Two perpendicular tangent planes to the paraboloid x2 y2 2z intersect in a straight line in the plane x 0. Obtain the curve to which this straight line touches. 13

Q. For the function

<img src='./qimages/128-4d.jpg'>

Examine the continuity and differentiability. 12

SECTION—B

Q. Solve the differential equation
src='./qimages/128-5a.jpg'>

Q. Solve the differential equation
(2xy4ey 2xy3 y)dx (x2y4ey x2y2 3x)dy 0. 10

Q. A body moving under SHM has an amplitude ‘a’ and time period T If the velocity is trebled, when the distance from mean position is 2/3 a the period being unaltered, find the new amplitude. 10

Q. A rod of 8 kg is movable in a vertical plane about a hinge at one end, another end is fastened a weight equal to half of the rod, this end is fastened by a string of length to a point at a height b above the hinge vertically. Obtain the tension in the string. 10


Q. Find the angle between the surfaces x2 y2 z2 9 0 and z x2 y2 3 at 2). 10

Q. Find the constant a so that y)a is the Integrating factor of
(4x2 2xy 6y)dx (2x2 9y 3x)dy 0 and hence solve the differential equation. 12

Q. Two equal ladders of weight 4 kg each are placed so as to lean at A against each other with their ends resting on a rough floor, given the coefficient of friction is m(Mue) . The ladders at A make an angle 60° with each other. Find what weight on the top would cause them to slip. 13

Q. Find the value of X and ji so that the surfaces <img src='./qimages/128-6c.jpg'> may intersect orthogonally at 2). 12

Q. A mass starts from rest at a distance ‘a’ from the centre of force which attracts inversely as the distance. Find the time of arriving at the centre. 13

Q. Obtain Laplace Inverse transform of

<img src='./qimages/128-7a-i.jpg'>

Using Laplace transform, solve

<img src='./qimages/128-7a-ii.jpg'>

Q. A particle is projected from the base of a hill whose slope is that of a right circular cone, whose axis is vertical. The projectile grazes the vertex and strikes the hill again at a point on the base. If the semivertical angle of the cone is h is height, determine the initial velocity u of the projection and its angle of projection. 13

Q. A vector field is given by

F (x2 (y2

Verify that the field F is irrotational or not. Find the scalar potential. 12

Q. 7(d)Solve the differential equation

src='./qimages/128-7d.jpg'>

Q. Find the length of an endless chain which will hang over a circular pulley of radius so as to be in contact with the two-thirds of the circumference of the pulley. 12

Q. A particle moves in a plane under a force, towards a fixed centre, proportional to the distance. If the path of the particle has two apsidal distances b then find the equation of the path. 13

Q. Evaluate<img src='./qimages/128-8c-i.jpg'> where C is the rectangle with vertices 0),(Where n is Pie)

src='./qimages/128-8c-ii.jpg'>

Q. Solve
src='./qimages/128-8d.jpg'> 13