civils mains 2013 MATHEMATICS (Paper

Time Allowed: Three Hours

Maximum Marks: 250

QUESTION PAPER SPECIFIC INSTRUCTIONS

Please read each of the following instructions carefully before attempting questions.

There are EIGHT questions divided into two SECTIONS and printed both in IDNDI and in ENGLISH.

Candidate has to attempt FIVE questions in alL

Question No. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at least ONE from each section.

The number of marks carried by a question!part is against it.

Answers must be written in the medium authorized in the Admission certificate which must be stated clearly on the cover of this Question-cum-Answer booklet in the space provided. No marks will be given for answers written in medium other than the authorized one.

Assume suitable data, if considered necessary, and indicate the same clearly.

Unless and otherwise indicated, symbols and notations carry their usual standard meaning.

Attempts of questions shall be counted in chronological order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the answer book must be clearly struck off.

SECTION

1. Answer all the questions:

Find the inverse of the matrix:
<img src='./qimages/186-1a.jpg'>
by using elementary row operations. Hence solve the system of linear equations
x 3y 10
2x-y 7z =21
3x 2y 4 10

Let A be a square matrix and be its adjoint, show that the eigenvalues of matrices AA* and A*A are real. Further show that trace trace

Evaluate <img src='./qimages/186-1b.jpg'> 10

Find the equation of the plane which passes through the points and and is parallel to the line joining the points 4). Find also the distance between the line and the plane. 10

A sphere S has points at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere S with the plane 5x 2y 4z 7 0 as a great circle. 10

Let Pn denote the vector space of all real polynomials of degree atmost n and P2->P3 be a linear transformation given by t p o Pz. Find the matrix of Twith respect to the bases x and x x 3 of Pz and P3 respectively. Also, find the null space of T. 10

Let V be an n-dimensional vector space and V V be an invertible linear operator. Xz, ..., Xn is a basis ofV, showthatp' IXz, ... is also a basis of V. 8

<img src='./qimages/186-2b.jpg'>
Let A be a Hermetian matrix having all distinct eigenvalues AI> ..., If Xz ... Xn are corresponding eigenvectors then show that the n x n matrix C whose kill column consists of the vector Xk is non singular. 8

Show that the Vectors XI -i. and =CO. 1-2i. in c3 are linearly independent over the field of real numbers but are linearly dependent over the field of complex numbers. 8

Using Lagrange's multiplier method, fmd the shortest distance between the liney 10 2x and the ellipse 20


Compute fxy and fyx for the function
<img src='./qimages/186-3b.jpg'>
Also, discuss the continuity of fxy and /yx at 0). 15

Evaluate If xy dA, where D is the region bounded by the line y and the parabola y2 2x 6. 15

Show that three mutually perpendicular tangent lines can be drawn to the sphere x2+ y2 Z2 r2 from any point on the sphere 2(x2 y2 Z2) =3r2. 15

A cone has for its guiding curve the circle x2+ y2 2ax 2by Z 0 and passes through a fIxed point c). If the section of the cone by the plane y 0 is a rectangular hyperbola, prove that the vertex lies on the fIxed circle x2+ y2 Z2 2ax 2by 2ax+2by+cz=0. 15
A variab,le generator meets two generators of the system through the extremities B and B of the minor axis of the principal elliptic section of the hyperboloid x2/a2+y2/b2-z2c2=1 in P and BP.B'P'=a2+c2.


SECTION

5.Answer all the questions:

y is a function of such that the differential coefficient is equal to cos(x sin(x y). Find out a relation between x and which is free from any derivative/differential. 10
Obtain the equation of the orthogonal trajectory of the family of curves represented by r n a sin nO, being the plane polar coordinates. 10

A body is performing S.H.M. in a straight line OPQ. Its velocity is zero at points P and Q whose distances from 0 are x and y respectively and its velocity is v at the mid-point between P and Q. Find the time of one complete oscillation. 10

5(d)The base of an inclined plane is 4 metres in length and the height is 3 metres. A force of 8 kg acting parallel to the plane will just prevent a weight of 20 kg from sliding down. Find the coefficient of friction' between the plane and the weight. 10

Show that the curve <img src='./qimages/186-5c.jpg'>

•Solve the differential equation (5.0 12x2+ 6y2)dx 6xydy =O. 10

Using the method of variation of parameters, solve the differential equation d2y/dx2+a2y=sec ax

Find the general solution of the equation x2d2y/dx2+xdy/dx+y=lnx sin(lnx)
By using Laplace transform method, solve the differential equation (D2 +n2 D2 subject to the initial conditions x 0 and dx at t in which n and a are constants. 15

A particle of mass 2·5 kg hangs at the end of a string, 0·9 m long, the other end of which is attached to a fixed point. The particle is projected horizontally with a velocity 8 m/sec. Find the velocity of the particle and tension in the string when the string is horizontal vertically upward. 20

A uniform ladder rests at an angle of 450 with the horizontal with its upper extremity against a rough vertical wall and its lower extremity on the ground. If }J. and are the coefficients of limiting friction between the ladder and the ground and wall respectively, then [md the minimum horizontal force required to move the lower end of the ladder towards the wall. 15

Six equal rods AB, BC, CD, DE, EF and FA are each of weight Wand are freely jointed at their extremities so as to form a hexagon; the rod AB is fixed in a horizontal position and the middle points of AB and DE are joined by a string. Find the tension in the string. 15

Calculate V2(rn) and find its expression in terms of rand r being the distance of any point from the origin, n being a constant and V2 being the Laplace operator. 10

A curve in space is defined by the vector equation t 2i+2tj t3Ie Determine the angle between the tangents to this curve at the points t and t 10
By using Divergence Theorem of Gauss, evaluate the surface integral 1 JJ(a2 x 2 b2y2 c2z2fr dS, where S is the surface of the ellipsoid a.x2 band c being all positive constants. 15
Use Stokes' theorem to evaluate the line integral Jc (-y3dx+x3dy-z3dz), where C is the intersection of the cylinder x2 and the plane x y z=1. 15