MATHEMATICS
Paper-I
(Time Allowed Three Hours) Maximum Marks " 300)
:INSTRUCTIONS
Each question is printed both in Hindi and in·English Answers must be written in thejmedium specified in the Admission Certificate issued to you, which must be stated clearly on the cover ofthe answer-book :in lhe space providedfor the purpose. NOimarks will .be givenfor the answers written in a medium other than that specified in the Admission Certificate. Candidates should attempt Question Nos. 1 and 5 which are compulsory, and any three ofthe remaining
.
..
questions' selecting at least one question from each
Section.
.The number,.ofmarks carried each question is
indicatedat.the end of the question.


Assume suitable data if considered necessary and
indicate the same clearly.
Symbols/notations carry 'their meanings, unless
otherwise indicated·


SECTION:A
1.(a)Let A be a non-singular, n x n square matrix. Show that A.(adj |A|.In. Hence show that|adj (adj 10
(b)Let <img src='./qimages/1073-1b.jpg'> Solve the system of equations given by AX=B Using the above, also solve the system of equations AT B where AT denotes the transpose of matrix A. 10
(c)find <img src='./qimages/1073-1b.jpg'> if it exists. 10
(d)Let f be a function defined on R such that and for all values of x in IR. How large can possibly be 10

(e)Find the equations of the straight line through the point to intersect the straight line and parallel to the plane 4x y 5z 0. 10
(f)Show that the equation of the sphere which touches
the sphere 4(x2 y2 z2) 10x-25y-2z 0 at the point and passes through the point is x2 y2 z2 2x -6y 1 0 10

lamda1,lamda2...lamda n be the eigen values of a n x n square matrix A with corresponding eigen vectors Xl,X2,...X n.If B is a matrix similar to A show that the eigen values of B are same as that of A. Also find the relation between the eigen vectors of B and eigen vectors of A. 10
(ii)Verify the Cayley-Hamilton theorem for the matrix <img src='./qimages/1073-1b.jpg'> Using this,show that A is non-singular and find A universe. 10
that the subspaces of IR 3 spanned by two sets of vectors and are identical. Also find the dimension of this subspace. 10
(ii)Find the nullity and a basis of the null space of the linear transformation given by the matrix
<img src='./qimages/1073-2b2.jpg'>

that the vectors and are linearly independent in R(3).Let be a linear transformation defined by 2y 3z, x 2y 5z, 2x 4y 6z). Show that the images of above vectors under T are linearly dependent. Give the reason for the same. 10
Let <img src='./qimages/1073-2c2.jpg'>and c be a non singular matrix of order 3 x3. Find the eigen values of the matrix B3 where B universe AC. 10

3.(a)Evaluate:
<img src='./qimages/1073-3a1.jpg'>
<img src='./qimages/1073-3a2.jpg'> 12)

(b)Find the points on the sphere x2 y2 z2 4 that are closest to and farthest from the point 20
Find the volume of the solid that lies under the paraboloid z x2+y2 above the xy-plane and inside the cylinder x2 y2 2x. 20
4.(a)Three points are taken on the ellipsoid x2/a2+y2/b2+z2/c2 1 so that the lines joining R to the origin are mutually perpendicular.Prove that the plane PQR touches a fixed sphere. 20
(b)Show that the cone yz+zx xy 0 cuts the sphere x2 y2 +z2 a2 in two equal circles, and find their area. 20
(c)Show that the generators through any one of the ends of an equiconjugate diameter of the principal elliptic section of the hyperboloid x2/a2+y2/b2/z2/c2 are inclined to each other at an angle of 60° if a2+b2 6c2. Find also the condition for the generators to be perpendicular to each other. 20
SECTION-B
5.(a)Obtain the solution of the ordinary differential equation if 1. 10

(b)Determine the orthogonal trajectory of a family of curves represented by the polar equation r a(l-cos theta), theta) being the plane polar coordinates of any point. 10
(c)The velocity of a train increases from 0 to v at a constant acceleration f1, then remains constant for an interval and again decreases to 0 at a constant retardation f2.If the total distance described is find the total time taken. 10

A projectile aimed at a mark which is in the horizontal plane through the point of projection, falls x meter short of it when/the angle of projection is alpha and goes y meter beyond when the angle of projection is beta. If the velocity of projection is assumed same in all cases, find the correct angle of projection. 10
For two vectors a bar and b bar given respectively by <img src='./qimages/1073-5e.jpg'>
u and v are two scalar fields and f is a vector field, such that <img src='./qimages/1073-5f.jpg'> 10

6.(a)Obtain Clairaut's form of the differential equation
dy/dx-y)(ydy/dx a2 dy/dx. Also find its general solution. 15
(b)Obtain the general solution of the second order ordinary differential equation 2y= x+ex cos where dashes denote derivatives w.r.to x. 15
(c)Using the method of variation of parameters, solve the second order differential equation d2y/dx2+ 4y= tan2x. 15
Use Laplace transform method
to solve the following initial value problem
<img src='./qimages/1073-6d.jpg'>

A mass of 560 kg. moving with a velocity of 240 m/sec strikes a fixed target and is brought to rest in 1/100sec. Find the impulse of the blow on the target and assuming the resistance to be uniform throughout the time taken by the body in coming to rest, find the distance through which it penetrates. 20
ladder of weight W rests with one end against a smooth vertical wall and the other end rests on a smooth floor.If the inclination of the ladder to the horizon is 60 0,find the horizontal force that must be applied to the lower end to prevent the ladder from slipping down. 20
After a ball has been falling under gravity for 5 seconds it passes through a pane of glass and loses half its velocity. If it now reaches the ground in 1 second, find the height of glass above the ground. 10 A particle of mass m moves on straight line under an attractive force mn2x towards a point o on the line, where x is the distance from O.If x a and dx/dt u when t find for any time t>0 10

8.(a)Examine whether the vectors Vu, Vv and Vw are coplanar,where u,v and w are the scalar functions defined by:
u
v=x2+y2+z2
and w yz +zx+xy. 15
4yi xj 2zk, calculate the double integral <img src='./qimages/1073-8b.jpg'> oyer the hemisphere given by x2 y2 z2 a2 15
r be the position vector of a point, find the value(s) of n for which the vector
rn r
irrotational, solenoidal 15
(d)Verify Gauss' Divergence Theorem for the vector
x2i y2j z2k
taken over the cube
15