CS MAINS MATHEMATICS
I I
Question P.aper Specific Instructions
Plf!ase read each ofthe following instructions. carefully before attempting questionS
There are EIGHT questions divided in TWO SECTIONS and printed .both in HINDI· and in
ENGLISH.
Candidate has to FIVE questions in'all.
'Qu!!stions 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 ofmarks carried by a question /part is indicated 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 a medium other than the authorized one.
Assume suitable da.ia, ifconsidered 1J.ecessary, and indicate the same clec:zrly.
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
Booklet must be clearly struck off.




SECTION A
10x5=50
Find one vector in R3 which generates the intersection of V and where V is the xy. plane and W is the space generated by the vectors and 1).
Using elementary row or column operations, find the rank of the matrix
src='./qimages/280-1b.jpg'>
Prove that between two real roots of eX cos x 1 a real root of eX sin x 1 0 lies.


Evaluate: src='./qimages/280-1d.jpg'>
Examine whether the plane x y Z 0 cuts the cone yz zx xy in perpendicular lines.

Q2. Let V and Wbe the following subspaces of R4
V b 2c d and
W a b
Find a basis and the dimension of V n w.

Investigate the values of lamda and mue so that the equations.x y Z x 2y 3z 10, x 2y lamdaZ mue have no solution, a unique solution, an infinite number of solutions.

ii) Verify Cayley Hamilton theorem for the matrix A 4 2 and hence find its inverse. Also, find the matrix represented by A5 4A4 -7A3 11A2 I.

By using the transformation x y uv, evaluate the integral ff {xy x dx dy taken over the area enclosed by the straight' lines y and x 1. 15


Q3. Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a. 15
Find the maximum or minimum, values of x2 y2 z2 subject to the conditions ax2 by2 cz2 and lx my nz =o. Interpret the result geometrically. 20

Let A <img src='./qimages/280-3c1.jpg'> Find the eigen values of A and the corresponding eigen vectors. 8
Prove that the eigen values of a unitary matrix have absolute value 1. 7

Q4. Find the co-ordinates of the points on the sphere
x2 y2 z2 4x 2y the tangent planes at which are parallel
to the plane 2x y 2z =1.

Prove that the equation ax2 br cz2 2ux 2vy 2wz d represents a cone if d. 10

Show that the lines drawn from the origin parallel to the normals to the central conicoid ax2 by2+ cz2 at its points of intersection with the plane Lx my nz p generate the cone

Find the equations of the two generating lines through any point cos thita, b sin thita, of the principal elliptic section of the hyperboloid by the plane z o. 15


SECTIONB.
Q5. Answer all the questions 10x5=50
Justify that a differential equation of the form: x f(x2 dx f{x2 dy where f(x2 y2) is an arbitrary function of (x2 is not an exact differential equation and is an integrating factor for it. Hence solve this differential equation for f(x2 y2) (x2 y2)2. 10

Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of tangency. 10
A particle is performing a simple harmonic motion (S.H.M.) of period T
about a center O with amplitude a and it passes through a point
where OP b in the direction OP. Prove that the time which elapses
before it returns to P is T/pie 10

Two equal uniform rods AB and AC, each of length are freely jointed at A and rest on a smooth fixed vertical circle of radius r. If 2thita is the angle between the rods, then find the relation between r and thita by using the principle of virtual work. 10
Find the curvature vector at any point of the curve src='./qimages/280-5e.jpg'> Give its magnitude also. 10

Q6. 10
Solve by the method of variation of parameters
5y sin x
Solve the differential equation:
src='./qimages/280-6b.jpg'><br><br>
Evaluate by Stokes' theorem
src='./qimages/280-6c.jpg'><br><br>
where r is the curve given by x2 y2 z2 2ax 2ay x y 2a starting from and then going below the z-plane. 20

Q7. Solve the following differential equation:
x(d2y/dx2)
when ex is a solution to its corresponding homogeneous differential
equation. 15
A particle of mass hanging vertically from a fixed point by a light inextensible cord of length is struck by a horizontal blow which imparts to it a velocity 2 root gl. Find the velocity and height of the particle from the level of its initial position when the cord becomes slack. 15
A regular pentagon ABCDE, formed of equal heavy uniform bars jointed together, is suspended from the joint and is maintained in form by a light rod joining the middle points of BC and DE. Find the stress in this rod. 20

Q8.
Find the sufficient condition for the differential equation
dx dy 0 to have an integrating factor as a function of y). What will be the integrating factor in that case? Hence find the integrating factor for the differential equation
(x2 xy) dx xy) dy
and solve it. 15


A particle is acted on by a force parallel to the axis of y whose
acceleration (always towards the axis of is muey-2 and when y it is projected parallel to the axis of x with velocity root 2mue/a Find the parametric equation of the path of the particle. Here mue is a constant. 15
Solve the initial value problem
(d2y/dt2) 8 e-2t sin 0
by using Laplace-transform. 20