Mathematics

Paper-II

Time Allowed: Three hours Maximum Marks: 300
Thefigures in themarginindicatefhllmarks for thequestions:
Note: CandidatesshouldanswerQuestion nos. 1and5whichare compulsOlyandany three from the rest selecting at least one from each section.
SECTION-A

1. Answer any five from the following: 12x5=60

ShowthatifevelYelementofthegroup G is its own inverse, then G is abelian.


Prove that every finite Integral domain is a field.


Show that the sequence where


111
S
n 11+1 11+2 n+l1
is convergent.


Show that the funcation defined by xl is uniformly continuous on the interval 1].


Findwhether thefimction


Z3
is analytic.

Find the residues ofthe fimction



at its poles.

Findbasic feasiblesolutionoftheLinearProgramming
Max. z 3x1 5x2
such that

1 23
4x1 3X2 X4= 12


2. Answer all the following five parts 12x5=60

Prove that the set G to, is a finite abelian group of order 5 under additionmodulo 5as thecomposition inG


Show that the additive group ofintegers
...}
is isomorphic to the additive group
G ..., 1m, 2m, 3m, ...}
where m is any constant integer.



If M and N are normal subgroups of a group then show that MnN is also a normal subgroup ofG


Show that a field has no zero divisors.


Ifa, dare elements ofaringR, then evaluate


3. Auswerallthefiveparts: 12x5=60

Ifthe sequences {sJ converges to prove thatthe sequence sJ oscillates.


Show thatj{x) is not uniformly continuous on but it is uniformly


continuous on 00) where a >O.

Examine the convergence ofthe series
2 34
X xxx

2 34
x x1+x1+ x

Show that each constant fimctionj{x) c is Riemann integrable on any interval
b].

Find the maxima and minima ofthefimctionj{x,y)=x3 +y3-3x-12y+20
4. Solve all the four parts 15x4=60

Derive C-R equations for an analytic fimction.


Find the Laurent series for the fimction


with centre at z 1.
1-z


In conformal mappingj: G wherej(z) is analytic. Then prove thatjpreselves angle at each point ofG


Usingcontourintegrationevaluate


SECTION-B


5. Solve any five ofthe following: 12x5=60

From (ax y)2 fi-om the paltial differential equation.


Solve the partial differential equation






ax ax

Solvethefollowingequationby falseposition(Regula-falsi)method
x3

Thepopulation ofatmvn in the decennial census was given below:
Year x 1901 1911 1921 1931 1941
Population y in thousands 46 66 81 91 101

Find the population for the year 1905.

EvaluatethefollowingintegralbySimpson'sone-thirdrulefor 0.125
1
1 1
-dx



WriteaprograminBasictowritethefollowingnumbersinascendingorder
1




Dellve the equation ofcontinuity in Cmtesian co-ordinates.


6. Solve all the five pmts 12x5=60

Constructpartialdifferentialequationfrom
fCr z 0



Find the solution of



az
ax +.JYay ..fi

Solve by Charpit's method
az az)2


ax ay


Find the genel1u solution of


Construct the partial differential equation for a vibrating string fixed at the two ends.



3

7. Solve all the five parts 12x5=60

Findrealroots oftheequations
x2 -y2 x2 16
by Newton-Raphsonmethod.



Using Lagrange's Interpolation fOll11ula find the value of log 301 from the follow ingtable


Evaluate the integral x dx byGauss qurature fonnula.


Findthenumericalsolutionofthedifferentialequation


x 300 304 305 307
logx 2.4771 2.4829 2.4843 2.4871

dy
dr: =1 for yeO)
atx 0.2, 0.4, 0.6 using Runge -Kutta method by taking h 0.2

Explainthefollowingincomputer:
bits, bytes, words

Illustrates use of Algorithms and Flow charts in solving equations by bisec tion method.

8. Solve all the fourpalts: 15x4=60

A mass of fluid is in motion so that the lines of motion lie on the surface of coaxial cylinders. Findtheequationofcontinuity.


Find the velocity potential and stream function for a liquid flowing past a fixed elliptic cylinderwithveloicityvparallel to themajoraxis.


ec) Write Hamilton equations and explain.
Findmoment ofinertiaofa thin homogeneous rod oflength I about its end point.



Mathematics (Main Examination)





Paper-I

TimeAllowed: Three hours Maximum Marks: 300
The figures in the margin indicate fiill marks for the questions.
Questions nos. 1 and 5 are compulsory. Candidates should answer any three questions from the rest, selecting at least one each section.
SECTION-A

1. Answer any five ofthe following: 12x5=60
Let V be a vector space over a field F. Suppose that
S x2, ••. xJ
is a subset of non-zero elements of V. Then show that S is linearly dependent if and only ifthere is a k such that k nand


Show that the matrix 21 is unitary, while the matrix
5 2-4i
COS 8 -sin8
sin 8 cos8

o0
is Olthogonal.


Examinethefollowingfunctionformaximumandminimum: y 2 sin x cos 2x in the intelval 2n].
22

Computethearea ofaregionboundedbytheellipse Y:1
2
a b­

byusing integration.

Evaluatethelimit
lim tan x
7t
X 2" tan3x




State and prove Taylor's theorem.


Find equation of ellipse whose centre is at the origin and passes through the points and 1).


2. Answer all the five parts: 12x5=60
Prove that the four vectors <Xl and
23
in JR3 fonn a linearly dependent set.

Findthe rankofalllatrix
12

243 5
6


Findtheminimalpolynomial ofthe following matrix


Find all the eigen values and cOlTesponding eigen vectors ofthe matrix


State and prove Cayley-Hamilton theorem






3. Answer all the five parts: 12x5=60
Find the deIivative ofthe function


4)3 eX


State and prove mean-value theorem.


Find the asymptotes ofthe curve


x2 2x-1

x


Test the following function for maximumandminimum
x2-xy 3x -2y



DefineGammaandBetafunctionsandshowthat


B(x





4. Answer all the five parts 12x5=60

Find the equation to the circle which passes through the points and


Findequations ofthelinethroughthepoint (1,1,1)whichmeetboththelines


x-I y+l z-2
x=2y=3z



Find equations to the spheres which pass through the circle x2 y2 Z2 x 2y

3z 3 and touch the plane 4x+ 3y 15.



Find equation to the cone with vertex at the origin which pass through the curve givenby


r +y2 2ax b Ix my 11Z P
2 2
x v

Show that the plane 8x -6y 5 touches the paraboloid T-3=z. Find the coordinates ofthe point ofcontact.
SECTION-B

5. Answer anyfive ofthe following: 12x5=60
Solvethe differential equation
dy
dx x3y3


Findcompletesolutionofthedifferentialequation
d2y dv
2y e5x
dx2 dx .


Solvethe differential equation


Write the equation of motion of a pat1icle moving along a straight line with known initialvelocity andac.celeration.


One cubical box of one side equal to is placed on a fixed solid sphere such that the center of lower face of the box is in touch with the highest point of the sphere. Findtheminimumradiusofthesphere forwhichtheequilibriumis stable.


Findthe directional delivative of xy2




in the direction ofthe vector i +2j+2k

Show that
curl grad 0
3

6. Solve all the five parts: 12x5=60
Solvethe equation
dy
dx x+1


Find the general and singular solutions ofthe equation
1
dy dy (dy

dxdx dx


Solve the differential equation
xy) x dy dx 0



Solve the differential equation


d3y
y
3 ex 5ex

Solve the differential equation


7. Solve all the four parts: 15x4=60

In a Simple Harmonic Motion is amplitude is and period then find the relation between distance x fromthe center andvelocityvatthatpoint.


Find the velocity of projection and direction of a bullet which just crosses a wall at 50 metres away and 25 m high in horizontal direction.


If the maximum and minimum velocities of planet moving round the sun are respectively 30 and 29.2 km/sec, then find the eccentricity ofthe orbit.


State and explain Bernoulli's equation.


8. Solveallthefiveparts: 12x5=60

Provethederivative ofaconstantvectoriszero.


Find the angle between the tangents to the Cllive r fi 2tj -13k at the points 1 and


Show that curl curl 1=0 for 1=zi +xj +yk


Evaluate JJ[(x3 dydz +=dxdy -2x2ydzd't]


S
over the surface bOlmded by the planes x a.

Verify Stoke's theorem for yi zj xk where S is the upper half of r 1 and C is its boundary.