IAS Mains Mathematics 2003

8/20/2019 IAS Mains Mathematics 2003

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w w

w . e x a

m r a c e

. c o m

I .S.E-Mnins 200J

I

M A T H E M A T

=

I

=

8

= =

~ f

11me

;

11/

mvetl: 3 fi QIIrs ax imum

a

r ks: 3 )0

Candid

at

es

sho

ul

d auempl

Quesuon Nos

L

nnd

5

11

hi

cit

nre

comp

ulsory.

nttd

ally

tluee

of

tbe

re

m

nilli

tt

g.

quest

ions selecting at leas t uoe question from eucb $ettion

PAPER I

SECTION A

I .

Attempt IDt) fo.veoftlte foll

oM

ug

.

t

(lj Let S oo any non emply subset

of

a vecl<.>r space V

ove

r

the

tield F

Show Lhnt U e

se

t

tu

a

1

t

up

1

+a,a,.

:o

.al P.a,.a:· ··· · • .-1. •

e

,,r, S the subsp ce

generated

by S.

( 12)

(b)

tf 1=[;, :

:

1, h

en lind

the mat

ri> p r s e d ~

l I 21

2A

•t.

ION+ l-IN- M - •

3A" <

15A' ·2 1A' -+

<

N

+ A-

I

( 1

2)

(cf Let f real function de li ned

ns

follows :

fl,, }=x. -t :S >. < 1

1\x-l)=x.V £

R.

Show

tlt.nt f

is disconunuous at

every o

dd integer.

( 12)

(dJ For

all real numbers x.l'lx)

1s

g11e

n

as:

(e)

(I )

(at

(b)

f x

= • ·

{

e +osin .L .V< II

x r +.r-

2,

Find q lue a nnd b for

wh

ich r s diJrerentinblc ol

ax=

1

.

r

-

{

e• ..os

ln :r, · <

)

b(x- 1)-;-;r- 2 x:< tl

Filld 1alues of a

und

b for which f din erentmb leat x=t)

( 12)

.1\

l'ariable

plane rema

in

s at a

consllln

t distnnce unlt) from the

po i11l {I,

0. Ol nnd C\JIS the

co

ordi

nate axes

at

A. B

and

C. Fi

nd the

locus of

the

cen

tre

pf

the sph

ere

pass

in

g th

rough the

orig

in

and

the

poi

nt A,

B

3lld

C

( 1

2)

Fmd the equallon of the two straJgbt lines through the jl()tnl (I. J. l{ Jhat mlersect the li

ne

x-4

-lly -4,)= 2lz

-l

) al an angle or 60"

ll2)

Pr01'C that lhe eigen 1ectors corresponding to distincL e ~ g e values or a sq uare matnx are

linear ) independent,

( 2)

If H Is a He

rmtt

ian

matnx.

t

hen

show that A=

(H

+ttr

1

H-il) ts a

Ulli tar

y matrix. Also sho

thnt Cl'err

rrnit lly

ma trix

c.an be.expressed in th

is form

. provided I

is

n

ot

an eigenvalue

of

A

( L5)



w w

w . e x a

m r a c e

. c o m

.

[

6 l z]

(c)

If . 1

- z <

- 1 then

lind

a

diagonol matr

t D :rnd ;tmatri.x B

sucl1that

.A

=BDB' 'Vhent

' - 1 3

s·d '

1ote•

the

tran•poseofB.

liS)

(d) Reduce the quadmtic form give1t belo" to canonical fom1 nnd

fmd

llil

rnnkand s igtmtw

·e:

I ay· +- 9:<' I u· • 12yx I

6zx

•

* )

•

2x

u • 6zo.

(a)

l15)

A 1ectangul

ar

box. o

pen

at the

1011.

is lo hnve :1

volume

nf 4m

1

u.<inJ L.1granl e's methnd

uf

multipliors, .lind U1c dintcru.ions tJf ili<> bax so Uml the mattial of a given type n>quired tu

co m

1rucl Itmay be least,

( 15)

(b) Test

th

e convergence of

h

e inlegrall

(

) f'

•. " 1

.... ,

Evaluate

th

e 1

11

" \llll

f' f:

~

. : . . : : ; . ~ r ~ ~ ; ; ;

= ; c o

.•

~ r u

~ - /

(15)

(15)

I

d)

Find U1e

vo

lume goncroled by reoolvin&

H1

e ·arco,

bounded by t.ho curves

. , .

4a')y

g,J,

2y

= x. :tnd x = U. obnut the y-axi•.

(•)

(15)

Find tho volume

of

tit" l<;trahcdronformtd by tltc

fouT

plllltes k

1

roy

IlL

" p,

lx

+ my

0.

my

I 112.

" 0,

Md 112. I

lX =

0

15)

h A .lphcre of oo nst:mt radius r p••scs through

Ute

origin 0 nntl cut<

llto

1:0-llrilinalc 4 l \ ~ $ at

A,

ll

nnd

C. Find the l o c l l ~ of he fool of

l1e

-porpondicular

from 0

tu the plane ABC.

( I

5

(c) l' ind tl1e e q u ~ L i l l n i 0

1

lhe

Of inl ll St<:Lion Oi' thc

plane

X

-

7y -

57

ond lht C()IIC ~ y 14-

Z. (-

30xy

=o.

(d) Find the equatio

n•

of the lineof\ho riesl hetween the

)

- 1,

= I, X- ) lind :< z= ,

y

= 0 the inleiSt:eliOn of

W ()

planes.

ai CTION

(15)

(IS)

S. Ahem

pi

•nY five

out

of the

fo ll

vwing;

(•) Shnil

lha

l lhc

c r g < m n l r a j ~ I C > r y

ul'n y ~ l c : m ot' lirl n l l l l p R < > ~ i

q

clforlhll onal,

(e

)

rA

·

vlve:

~ - ; : - y l o g y :

tu

·

(12)

IS)

"sphere

vtweiS)Il

\V and

d i u .

~ u ~ - . w]thln. lixcdsphc:rital shell

ot

..dius b. A partjele oJ'

weight 1\

.is

fixed

10

tbe l' l 't end of

Ute

vot

1i.:<.U dillmcter.

Ptove

IIIAL

I Je

<>quilibrium

l

. W b la

t 3 1 l e a f -

l f

"

(12)

(d) A particlc desoribco the curve

r ~ o(l <

Q )S

h 0) (cosll •

2)

unller a

fon:e F

lo

tl1

e

pole. Shc1w thnt

ll•e

low

cr

t' foroe

i•

F l1r

1

( 1

2)



w w

w . e x a

m r a c e

. c o m

6.

7.

c:

) Slww lh>t if.o' . h' and

e

ore

th

e r < . - e i

t o c ~ b l

to tbe non·ooplun

il

r

o n ~ l l

h un<l e, then any

vwto

r r

n•

•y be e t ~ s i t d • R r :

t

)n •

(r b )h + : j

l1

2)

(f) ?nn'f

lha

lth

e dlverg.:nce

of •

vector

fie

ld

is

invariant

w.r. 10

tmn sfo•mat

io

ns.

12)

(a)

(15)

(b) So ll·e the difl'erential equ"tion

p;<

1

-

y

1

)

(

JI

" -')' )

=

p

t

I)

. wh ere

p

=

dv

/

dx

.

by

it

Lu Clairuut s f

or

m using • ui tablo: gubstituliou• .

15)

(c ) Snl\'c : I - -c

1

)y

1

1 ( 1 < :t)y ' y

2[lllJ

I - x)l

d)

(o)

l11c di

ffc:rct>

i:t

l

equation

~

·

b

y

- 6) ~ x• scc

1

g

hy vMiu t

io

n nf par.une

lcmo

.

( I

S)

An

elastic string

of oan1ra l

leng

th

a+ b. where a b, 3nd

modulus

of y

).,

hus a

p:u

>tic

lc

uf moss on all4ched

to

it al a d i itanco a ftum

end wbk h

s

f't

"<ed to "

poinl A

of

a

sn1oo U1 horiz

onl >l

plan

a. T ho other e

nd

of

the

string 15 IL d to a point B

Ill

thoi lhe

string

s

JU$

t

un

rtre

tchad.

rr

he

~ r t i

b

<>

held ••

a

and

th

en

released.

ti

nd

the

period

ic

l

im

0

nnd

the

dls bnce in w

th

e

particle

wall oscillate to fro.

(15)

(b)

f

parti

cle

slides down n sm

oo

th cycloid. s wrting 1

-

om a p

oint

wh

O Ic

anm

nl

d

i.s

tanoe from

ihe

venex

is

b, prove that Its sp

eed at

any lime t

os

2-<b l s in(2l\T1'}.

where

I'

is

th

e

time

of

curuplete

oscillation uf

tl•

e po

rticle.

15)

(c)

. \

ladder on a horizontal

fl

oor le:u

iJ og

oi

n.sto

vertical \vnU.

1 he

ooofficic

nt

s of Fiction of

lh

e

flc

wr ;

mel

tho

w

oll

wi

th

the la

chler il f

i>

JI

nn

e

J'•

p c : . : t 'If •

mon, whMe

c1

eight

is

o

tic

iie.

thot nf

th

e

ladd

er.

wo

nt

tc1

cli

l1l

h up U•e

ladde r. lind the min

i

mum so

fe aos le o

fth

e

ladder wi

th

u

u,

noot

·.

J5)

d)

r\n e

ll

iM•

~

•

: os

itl

:nri

erHcd v

c:rt

k olly in

11 fl uid wi th llli

semi-osi

s-

of length •

lumzonlaL If iL o<m tl't> he

>t

n

d"ffth

h.

li

nd

th

e depth

uf th

e centre

of pn:s

surc.

15)

(o)

L

c:1 llte position vector nf • p•rt

ic

le

m

ol

in

g em a plane curve be r

(t

), where t

s

the time. J'incl

thQconJponcnl. of tl. uccxle

ru

tlo11 olnt•g the ra cli•l and v r s e direction, ,

(b)

(o )

Prove

that

Jd""tity

VA· ;

2{A

.VJ,I

1A (v •

I) whereV

l • [-i ...

~ ~ ~ <l=

(1

5)

( l SI

l'

in

d

th

e rad

ii of

curv

ot

urc tmd to

rsiO

n at •

pOint of

inter$

edio

n

of the c ~ y

.: ,

y

=

x

tan

ht

Jc).

0 5)

•

( IS)



w w

w . e x a

m r a c e

. c o m

C.S.ll -M

nin,2003

M THEM TICS

- - - - ~ = = = = =

7me Allowed: 3

l m rs

Ma.\ illilllll t ~ r k s JOfl

Can d1

da1es

should a

lt

emp l Quesllon Nos. and

S

~ l ] c l J are

compu

lsory.

and

any lh r

ee

of the re

mmmn

&

questionsselectmg al

Jeas

l one quesuon

fmm

each secuon.

PAPER

II

SE TION

I, Answer

any

five

of

he fo llowing:

lf H

i s ~ subg

ro

up

of a group G sJ • H

fo

r

e 1C

I) xe G, then

pro1

•e lh$1 H is a n

ormal

subgroup of

G.

(J : )

{b) Show lhal the ring Z[ij = (a+b1) aeZ,be'Z,r=

._r-j)

of Gaussian integers IS a Euclidenn

dom nin.

( 12)

(c)

Lei

n be a posrlil'll

real

number

and

(x,

1

)

n sequence of

ratio

n

al numbe

rs such rhat

lim

>, =U

Show

hill

hm

n = I

( 12

I

d)

If a continuous function of:- sa

11sli

es the

funch on nl

eq

ua

lion I

Tx

+ y)

=

f(x) + I())

lh

en show

lli U

/'(x tu..- where

11

is

consln

nt

( 12)

(e) Determine all the bilinear r r ~ n s f o m 1 a t i o o s which transform the unll circle

1: 1-1

i

nt

o the uml

ctrcle

I'

S I

(1

2)

(1)

For the lbllo11ing system

of

eq

uations·

, c s ~ =

(a)

(b)

x

r x

•

3x = 4

O ~ t e n i n e

(i) all basic solutions

(ii) all

bas1c feasib

lesolmions

(i

ii) a feasible so

lu

tions wh

icl11s 01

a baste feasi ble sol utions

(i) Le

t R

lite

ring of all

rcal

-1•alued

con

tinuous

func

tions

on

lhe

cl

osed interval

10

. I I

Lei M=V x)eRf/(1/

3)=1)1

Sh

ow

t

hu

tM 1s a muxtmal1deal ofR_

I I

i l l

(ii) Let M and N be

two

1deals of a nng R. Sho" ~ 1 a 1 Mu N

IS

an 1deal of R 1f

:llld

only 1f

ei

Iher

s : ;

N orNcM

W)

Sho" l h ~ t Q( ,fi i) is asplimng field for 3x• +x'-J w

here

Q ts the ueld of

ra

ttonal

nu mbe rs.

( 15)

(l

i)

.Pro1e

Ol U x +x+41s irreducible over F he

li

eld of integers module l l and pro"e

funher that ( , H ) s a field hnl'ing 121elementS

. r

~

( 15)

C

c)

Let R

be

a uruque

fac

tonzation

domam

(UFD).Ihen

pro1

·e

thai

Rlxl

ts

also UFO.

(10)



w w

w . e x a

m r a c e

. c o m

3. (o) Shnw that the ml Ximum \','J]ue nf x'y'z'

S\lbje<:t tn

the

co

ndition

x>

y'

+

1

<=

c•

L.

c'/27,

Interpret the

re>ull

(20)

(b) The axe.• 11f twt>

equal

eylioders intctllect at rig

ht

a n g l e ~ If • he

th

eir i 1 1 ~ . th

en

liuu tile

\ O iumeen

mmnn

10

th

e

cy

linden<

b)

the

qf l t i p l

e integr:tls.

e)

Ca)

fh)

(20)

Show

Ulllt s• : . is

t11•

ergcrtL

J-.X

Sin

r

(i)

( l

20)

Disom•

lh

c tr•mfi>rmotion rr lc . (e real)

sh••

wing

tltat

the vpper

h

nlf

oftbe

:

•lc

W-pla

ne

Cllm>O"ptmds

to the intcrioc

of

Ue

.semicircJc lymg t·o

tbe right

of

imaginary

axil

in

the

7.1'ln

n

a.

(IS)

J;•ing lhe ru.d.hod ofconto

l1t

integro ti

on

II) t>rove thai

f .

d J, b 1.,

:.

O)

j

1

c r

•

SlD t} •

(15)

An animal feed. .:omp.'lny

mi1St(lroduce

200

..£

of r

.mix

tu

re

comlsbns of n g t e d i u t s

X,

and

X.

d3ily.

X cooll

Rs.

3 per

k.g

and

X:

co

su

S per

kj

. No more

tltan

80 kg

tlf

X

1

<;an he

tt•ed, nnd at l

e<ll

t 60

kg

of X, mu.•t he:

a . ~ e d

Formu

lnt

c • linear

programming model tif prohl

om

and Simple ( • t h o d to d o t e r m i n ~

Ute

ingredients x, und x_o b.: ll$.:11

to

minimize

(15)

(iil

F'in

d the

()Jltimal $Olution fo

r the assignment pr

ob

lem with Ute following cos t matrix:

1

11 Ill nr v

.. { j

1 9 n

JJ

g

17

.:

(

I I

t 3 3 3

1

-1

I I

6

I I

E 8

)II

II

'

13

Indi

cate olenrly the rule you

nppl

y lo arrive nl the complete

assignm

en

t,

IIS)

SI CTION.

5. Answer an y tiVe of

h

e

fo

iiDl\"ing:

(a) Findtbegeuerol solut

fo

u

of

i} :+J ll .: 2°·.:

= ~ ~ c o s 2 . - . ~

. 3 y )

m· d<<lJ

· ilr

l2)

(b) Show U>al

tlt

e differenhal e<Juolion ofaU cones whk h h•ve

tlt

etrvertex l l h e ongin arepx -

q)- z.

Ve

rif)•that Y zx ><y 0 is • surf.lcc satisfying the ahove cquatiotL

12)

(c)

Ev

aluate

r

d<:

em ploying lht·eepoiols

(

dj

G

o:tu.••ian qundrnl11rq follnul>.

fwding

the

rcq

i1U·

cd woights and rcsidu.,;.

ll •

e

live

d

ec.lmol

plaollli for computation,

(i)

Convert the fo Ubwlng binary numbor

int

o octal

nnd

bexa dc-<:imals

ys

tc:m

:

101 110010.10010

li) Find tb . < : J ~ l i o n of he following

bin:lfy u u m b

ll .OOL

1

and 101.1

021

(b)

(6)



w w

w . e x a

m r a c e

. c o m

7

8.

(c.) A $Uiid body

of

dcn•ity p i

1

in

I Jo

shape

of h

e solid lllm1cd by the rcwlution o.f thu c:mlinitl

t

=

n( Itws tl}

al>oul

the in iliaI Un a. Show thai its moment of inertin about the s l:nlighl line

~ r o u g h U1e pole

and

J1"1'1l<fldkular to tbe initinllb1e

is

352

11

pn

,

lOS

(\2)

Cf)

FOJ

•n i n ~ > ( ) J n l > r e s s i b

h

omogcfiOU.< Uuid at. the

point (s. y,

z} tl••

\

'l>)

ocity

dis tdbut.to11 u

(a)

' '· d

I ' tl.t

n l d

II

tl' - ·

g1Venvyu ; - - ·

.v=-,

w : ,,

w t

crcr

tnotcs

'

I . I I J t n c e u u m J . · O . U ~ .

r r

Sbo\\

tl1<11

it

is :\

possible motion and dcll,'t'n>ine ll1e •

u1

fncc wltich is odhogonol

to

st

=UI

line.

( 12)

( i)

n

_

.... ....

S I

• ,. • • .. '

<'• .

Q

vc:: -

,

- ~

=

..

t ~

ar· i y· ax

0•

•

(

15)

(ij)

SlliVe

the

tquntion

p' - q' ·2pq - 241) • 2.xy 0

l l ~ i Clmrpti

'• e t b t ~

l Also find lhc &ing

ulnr

<olutillfl

of

it

o

equnticm.

i f t e.x.ists.

(IS)

{II)

Find Ihe

deOc:etion

u(

t.

.

t)

(J[

i l

vibmting $Iring,

:Ill'

hod he

V

con

fi.x"d

.(lllinL•

(0.

II)

and (.'/,

0 , cvrr<:sp<luiliug to zero in iLi• l velocity atld toll

owi11g

n i l i ~ l deflection.

C•)

:

.

"" II

I

I

/i 3J-1

.-

1\X -

l w l w J . t

/l{x-3/) 1•/wn 1ls

.< ' >

3/

I

1\hero

h

ts com

l

tnnL

l"md the rositive rooLOtthe equo tmn

e

.

_

.t+ 2

(30)

Os lng Newton-R.1pbson mdtbod corre.zL to four decimal p l ~ ~ e s

Also

;h0\1

tbllt tllil

lollowlfli

>Chv'lllc for C'n:or

of so:cond

ordert

.. = ~ X (I+ 1

(301

(II) D r a w ~

flow

chan

and

write a

pro

,gumme BASlt'

fot

• .S impsan ·s 1

13"

1

nllefot lJII\}grntion

• I

J co

rrect

al Ill"',

.•. [

(a)

(:10)

A line c1reu lar lube,

rndnJS

c..

ties

on o f mootlJ horiZOntal p.lone.

an

d

CtJnlaim

lwo equal

p:u-lk:lcs

connected by an

elasuc string

in

the lube.

Ute>

nnlurnllenl Ut

ofwhtcb is

equ.1llo half

U1c ci'n;,umfercncc.. 'D1o particles ure

in

cunlllct and Jiutcnod

I0£<1llor

.

I Ju

string being

l ~ c h e d

round the

tu

be

.

Jf

tho

l)•n.de bect1me

d i . u n 1 1 e c ~ provo lh>t the ve

loci ty

<>f the mbe wht:n

t h ~

SU111J

h(l.•

rcg•iucd i t ~ .natura( length

L ~ ~ ~ A m

~ ~ l

Whon

M, m

urc tho

mois"' ol'Ute

lttb

c anJ ead1 p3rti

c.le

MpcCtiVc(y.

anil

J is

tho

m o i l u l u uf

cltisticit).



w w

w . e x a

m r a c e

. c o m

(b)

(

i)

30)

f\\ o SQUTC<lil. eaeh ofs

u-en

glh mare

l ~ t - e d

at the

p<Jint

s (-a, 0) and (a. 0)

1111d

n•ink ot'

su-en

{IUt

2m

i;;

placed •t

Ute

ori

{li

n. Sho1\ that

Ute

s

lre

am lin<:s are Ute curvell (xl +y

1

i

'" (

x

-

y -

>'ty) wbcre . j$ 4 nt-inhle

parAm

eter.

1hw

Also o w ll111t tltc Ouid

speed

al

-nn

y

point

~ . wltcre r,. r,

and

tJ

arc

lr '1

respe.;tively

the

dis_

tances of U

e point rom the sour

ces artd

sink.

1

15

(ii) An Infia.ito moss

of

fluid is

acted

upon

by •

force ur· l l per

unit

mnss

directed

to

the

ortgm.

lfinitia lly

d1

e fluid

is at

rest and th

er

e

~ cavity

in the

fot'l\1

or

a

sphore r =

o

in

show that

th

e

cavity wil.l

l i l l e d

up

aner an

mteo

v

•l

of

the

5 f t

15)

Documents

IAS Mains Mathematics 2003