IA-JGPT-M-LBU-A I
MATHEMATICS Paper-I
ITime Allowed: Three Hours I IMaximum Marks: 200 I
QUESTION PAPER SPECIFIC INSTRUCTIONS
Please read each of the following instructions carefully before attempting questions.
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There are EIGHT questions in all, out of which FIVE are to be attempted.

Question no. 1 and 5 are compulsory. Out of the remaining SIX questions, THREE are to be attempted
selecting at least ONE question from each of the two Sections A and B.
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.
All questions carry equal marks. The number of marks carried by a question part is indicated against it.
Answers must be written in ENGLISH only.
Unless other-wise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary and indicate the same clearly.

SECTION-A
Q.1(a)Find the dimension and a basis of the solution space W of the system
x 2y 2z-s x 2y 3z S 3x 6y 8z s 5t 0 8
Q. 1(b)Find the characteristic equation of the matrix <img src='./qimages/1512-1b.jpg'>
and hence find the matrix
represented by A8-5A7 7A6 -3A5 A4 -5A3 8A2-2A I. 8

Q.l(c)Evaluate the integral <img src='./qimages/1512-1c.jpg'> by changing the order of integration. 8

Q.1(d) Find the surface generated by the straight line which intersects the lines y z a and
y+z and is parallel to the plane x+y =0. 8
Q.1(e) Find C of the Mean value theorem, if x(x and C has
usual meaning. 8

Q. 2(a)Let V be the vector space of 2 x 2 matrices over IR and let <img src='./qimages/1512-2a.jpg'>
Let be the linear map defined by Find a basis and the dimension of
(i)the kernel of W of F 10
(ii)the image U of F.
Q.2(b)Locate the stationary points of the function x4 y4 -2x2 4xy -2y2 and determine their nature. 10
Q.2(c)Find an orthogonal transformation of co-ordinates which diagonalizes the quadratic form
2x2 -4xy 5y2. 10
Q.2(d)Discuss the consistency and the solutions of the equations
x+ay +az ax-ay +4z for different values of a. 10

Q.3(a)Prove that if a0,al,......,an are the real numbers such that
<img src='./qimages/1512-3a.jpg'>
then there exists at least one real number x between 0 and 1 such that
<img src='./qimages/1512-3a1.jpg'>

Q.3(b)Reduce the following equation to its canonical form and determine the nature of the conic
4x2 4xy y2-12x-6y 5=O. 10
Q.3(c)Let P be a subfield of complex numbers and T a function from F3->p3 defined by
X2+3x3, 2x1-x2,-3x1+X2-x3). What are the conditions on
such that be in the null space of T Find the nullity of T. 10

Q.3(d)Find the equations to the tangent planes to the surface
7x2-3y2-z2+21=0, which pass through the line 10
Q.4(a)Evaluate
<img src='./qimages/1512-4a.jpg'>

Q.4(b)Let <img src='./qimages/1512-4b.jpg'>
be a Hermitian matrix. Find a non-singular matrix P such
that pt HP is diagonal and also find its signature. 10
Q.4(c)Find the magnitude and the equations of the line of shortest distance between the lines
<img src='./qimages/1512-4b.jpg'>
Q.4(d)Find all the asymptotes of the curve
x4-y4+3x2y +3xy2 xy =0.


SECTION-B
Q. 5(a)Solve: <img src='./qimages/1512-5a.jpg'>
Q.5(b) A particle is performing a simple harmonic motion of period T about centre O and it passes through
a point where OP=b with velocity v in the direction of OP. Find the time which elapses before it returns to P. 8
Q.5(c) F being a vector, prove that
<img src='./qimages/1512-5c.jpg'>
Q.(d)A triangular lamina ABC of density row floats in a liquid of density sigma with its plane vertical,
the angle B being in the surface of the liquid, and the angle A not immersed. Find
p/sigma in terms of the lengths of the sides of the triangle. 8
Q.(e)A heavy uniform rod rests with one end against a smooth vertical wall and with a point
in its length resting on a smooth peg Find the position of equilibrium and discuss the
nature of equilibrium. 8
Q.6(a)Solve the differential equation
<img src='./qimages/1512-6a.jpg'>
by changing the dependent variable
Q.6(b)Evaluate <img src='./qimages/1512-6b.jpg'> and s is the surface bounding the region and z=3.
6(c)Two bodies of weights w1 and W2 are placed on an inclined plane and are connected by a light string
which coincides with a line of greatest slope of the plane; if the coefficient of friction between
the bodies and the plane are respectively mew1 and mew2 find the inclination of the plane to the horizontal
when both bodies are on the point of motion, it being assumed that smoother body is below the other. 14

Q.7(a)Solve <img src='./qimages/1512-7a.jpg'>
Q.7(b)A body floating in water has volumes v1,v2 and v3 above the surface, when the densities
of the surrounding air are respectively P1 P2 P3 Find the value of
<img src='./qimages/1512-7b.jpg'>
Q.7(c)A particle is projected vertically upwards with a velocity in a resisting medium
which produces a retardation kv2 when the velocity is v. Find the height when the particle
comes to rest above the point of projection. 14
Q.8(a)Apply the method of variation of parameters to solve
<img src='./qimages/1512-8a.jpg'>
Q.8(b) Verify the Divergence theorem for the vector function
<img src='./qimages/1512-8b.jpg'>
Q.8(c)A particle is projected with a velocity v along a smooth horizontal plane in a medium whose
resistance per unit mass is double the cube of the velocity. Find the distance it will describe in time t. 13