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'iest Code: PAPER-II: IAS (M)/25-9-16

MATHEMATIGSbyK. VENKANNA

FULL LENGTH TESTTest - 06

Time: Three Hours Maximum Marks: 250

CONSOLIDATED QUESTION PAPER.CUM.ANSWER BOOKLET

INSTRUCTIONS

1. This question paper-cum-answer booklet has 48 pages and has

3!_!4BIIEUBEIBI_questions. pteme ensure that the copyof the questionpaperrum-answer booklet you have received contains all the questions.

Write your Name, Roll Number, Name of the Test Centre and Mediumin the appropriate space provided on the right side.

A consolidated Question Paper-cum-Answer Booklet, having spacebelow each parUsub part of a question shall be provided to them forwriting the answers. Candidates shall be required to attempt answer tothe parVsub-part of a question stricfly within the pre-defined space. Anyattempt outside the pre-defined space shall not be evaluated. ,,

Answer must be written in the medium specified in the admissionCertificate issued to you, which must be stated clearly on the right side.No marks will be given for the answers written in a medium other thanthat specified in the Admission Certificate.

Candidates should attempt Question Nos. 1 and 5, which are mmpulsoryand any THREE of the remaining questions selecting at least ONEquestion from each Section.

The number of marks carried by each question is indicated at the endof the question. Assume suitable data if considered necessary andindicate the same clearly.

Symbols/notations carry their usual meanings, unless otherwise indicated.

All questions carry equal marks.

All answers must be written in blue/black ink only. Sketch pen, pencil orink of any other colour should not be used.

All rough uork should be done in the space provided and scored out finally.

The candidate should respect the instructions given by the invigilator.

The question paper-cum-answer booklet must be returned in its entiretyto the invigilator before leaving the examination hall. Do not removeany page from this booklet.

READ INSTRUCTIONS ON THE LEFTSIDE OF THIS PAGE CAREFULLY

Name.

UT54V KAUShALRoll No.

ORN

ENqLT5H

Do not write your Roll Number or Nameanywhere else in this Question Paper-cum-Answer Booklet.

Test Centre

Medium

o.

7.

8.

9.

'10.

11.

12.

I have read all the instructions and shallabide by them

Arr"r{@Signature of the Candidate

I have verified the information filled by thecandidate above

Signature of the invigilator

IMPORTANT NOTE:Whenever a question is being attempted, all its partsi sub-parts must be attempted contiguously. This means that before movlng on to the nextquestion to be attempted, candidates must finish attempting all parts/ sub-parts of the previous question aftempted. This is to be Jtrictly followed.Pages left blank in the answer-book are to be clearly struck out in ink. Any answers thai follow pages left blank may not be given credit.

P.T.O.

(}tr IIIATIEMATICAI,

)\

3of48

INDEX TABLE

QUESTION No. PAGE NO. MAX. MARKS MARKS OBTAINED

I (a)

(b)

(c) t(d)

(e)

f (a)

(b)

(")

(d)

-t (a) l?(b) t,(.) l((d)

J G)(b)

(.)

(d)

t (a)

(b) -7

(c)

(d)

(e)

5 (a) t

(b)

(c)

(d) I

(a)

(b) T(c)

(d) R

\ (^)

(b)

(c)

(d)

Total Marks

'.o. P,T,O.

q

IY

F

aT

t1

'o1uqo,aql?tursu! M '!uo..t!uE@o!02{Fsu! :rrrur

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m+ltoo 1e?en ,e6eN r6purleu pto I]EZ:@!J]O peeH

I

I

'o- r_xdx lpqlqons'gurxuorlelnr'.,edepulC.(9Sr):o pu€ (tZi:d tol .eSuI ([)

'3=,,D leqrqrns>l.ra8alur e.rrlrsodlsallpruseqlpuu.rr=[, I t o)=, rrl r]

\r t z t)v - NolrSts

(or)

6of48

1. (b) r, z[J-] = {u + u.I-6 I a,b,eZ} Euclidean domain ? Justiff your answer. (10)

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7of48

(c) (i) Give an example of an infinite set which is not bounded and having limit points.(ii) Give an example to show that the intersection of an infinite collection of open sets is not necessarily an openset.

,i> tuyyiotoL, 5 = { p * * I p, vt eZ+|.(10)

Walu\, n3Li.t(-t, r) ,\ 7,

\ / ,

--tt''

-9l^I/1L

si"eZt

ct oA

S= .nL-t I

!s; = "[o].' -.,| ,

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8of48

1" (O Iff(z):u+ivisananalyicfunctionofz:x* iv.and u - v : e* (cos y- sin y), find f (z) in terms of z.

t=u+i* + t*- iu-qr- +(t+t)7

{4n o,"^/U?;n =>Q+-i){ b ry"I

0x -/-

= 6,-,:) * i (u+tt) -.ci)

C

la*wfo'lub ' t

*, (2, o)

Zri) re

dx - L !Pr(+o)dz =J

ru;e' dz* L

Jez d<

; A;tu17+c

*ly ,= t cr I

0",= :ItL

t;n 4 ert'"*t-1

P.T,O.

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l. (e) ObtainthedualoftheLpproblem:Min. z: Xr * x, + x:. subject to the constraints :

x, - 3x, + 4x3: 5,xr_2x, <3,2xr_x, > 4;xr,x, > 0 and x, is unrestricted.

p*;,"ol ; : (undry, as, ai - q-i, , @O; 4l >, ) )

,,,w h)

qY

* %t- 3r2 * Y*i

- tt * 9x, -Y4-U * 2cr^+ Dr/

a\tztr-2d

+

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>5

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9of48

10 of 48

D,ral: mo*>izr, Z' = 5U,-5Ur'TVr+Yhh: Vt'Vt- Vt 4s I

'31,+?UL*rUr+? s I

Yg1 -Vy" -Yrt ( I

h,Vz,t). W>o-v;HtV{z +'4 <-l

Thro ,cam bc t.rrn W al:

rrqllL+fu; Z'= Sf i - SXs+ YW

ryrr h: Vi -yslt'- 3yi lz4r+ zyrl <)

\1,ti - U4- '\Us,

>.I

Y

Tht oilrw d4;J+t/v- Lff

2" (a) IfRis aringrrithidentitysuchthat(xy)2:x2f forallx, y eR,thenshowthatRiscommutative' Setanexample

to show that the above result may be false iiR does not have an identity' (14)

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11 ot 48

P,T.O.

12 of 48

Give an example of a homomorphism f ' R + ft' such that I is the unity of R, but f(1) is not the unity of R' .

(06)

Letthe function fbe defined on [0, I] as follows : -1l

f(x):2o*n.n . <x<-.r:1.2.3.......' r+I r

Prove that f is R-integrable in [0, 1] and eyaluat" f f (*)a* .

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(12)

E

'tr c.

13 of 48

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14 ot 48

2. (O Use the method of contour integration to prove that

1 dx - ;r(U + zc)r - - wnereb >u.e >u"

' '(*'* b')(x' + c')' 2bcr(b+ c;: """" (18)

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15 of 48

H€ad Otfic€:25/8. Otd Rajinder Nagar Market, Dethi-60Branch Officoi 10t106, Top Ftoor, Mukherj€e Tomr, Mukherro€ Nagar, D€thi-g

,a) LetGbetheg llt 'l -

'* 1L; i.ll*r"'.u,u e R. b . o) *a

"={[; i]l*n*"*."] ShowthatHisasubgroupofG.rsHanormalsubgroupofG?Justifliyour

answer.

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16 of 48

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17 of 48

(b) (i) Discuss the convergence of the inhnite product

f|[,. ul" )nr\ ^ *i,l'

(n) If x > 0, show ttrut .L < loe(1 + x) < xq l+x ,

) T(r+ a,) ry ^4n>ln

---2C(AY1 -

eL&+l

A-d Anil

^l R^t-

), le-l 7l , l^,,*,n4fr

=l CA4/vil'ytA'

lr"la l , )""-"' $-F

n42n

"( +l(ett '

2.t1+2a +,nlt

2L

An.or+t

6ryr_ =anrl

l"n"t/""|'

;k---

*J,/

L'

l-

n=tt o"--).*,

x-,-1 , An: t yZ +

ilr"t"(r

P,Mf

"rrrPt

Prt {t*), @A+";

U lan<ru^V a

{(x,t-{(o)1- 0

nU'^^W"14'l

-- )t(c)0

I--- |

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18 of 48

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3. (c)

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Solve the following linear prograrnming problem by simplex metlod.Max. z: - 2x, - x,, subject to 3x, * x, : l, 4x, * 3x, ) 6, x, + 2x, < 4,and x,, x, > 0.

lrulA a.v\d, ffi/f*/ wni,e*bo:

l= *2q-o(2 t 0a1*oaz- frAt-Mhz(o: Txt t x2 t O0Z t 2bz + frt + 0AZ

y ry + bt<z *oat ' a2 +oA) * Az

oq+2xz *at loa, *041 *0Az

Zlra(zt AtrAt, hr*r7g

Iryn&at b,/2A f1ru;lb /20/4"h"h' " At=3s At-- 6, a t = 4, zL =D

3

6

V

4iTF

- fvl

-M

o

(z)

Y

I

2i6.\,

-24

-l42

oA,

odt

0

-lD

- fvt

A1

Az

4l

I

3

L

0

o

I

b;9

6

IDDIoa

l->elz

\,

19 of 48

cie;-zi

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tn-zf-vrn

qrn-l

t4

-n- lvl

0

- f'n

o

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3

Lt5

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o t/a

x -MXo

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:. tA"** ,/Maru L'- - t2lg a* F,, , nz = 6/s

"-/

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21 of 48

{. (a) (i) LetRdenotetheringofallreal-valuedcontinuousfunctionsontheclosedinterval[0, l].Is(0)aprimeidealof R ? Justity.

$) ln 71(8), the ring of integers modulo 8, is the

maximal?

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ideal generatedby 2=2+(8) a prime ideal ? Is it also

(14)

22 ot 48

4. O) Prove that

I[r5).,.=i(h (12)

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23 ot 48

.1, (c) Find the Taylor's or Laurent's series which represent the function

(i) when I z l< I(ii) when 1<lzl<2(iii)when lzl>2.

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(t+ z')(z+2) ,

24 ot 48

Head Offc 25,8, Old Rejindor Nagar Market, Delhi-60Branch Off,€ 10$106, Top Floor. Mukheri.. Tomr, Mul(heriee Nagar, D€lhi-g

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25 of 48

J.(oAnautomobiIedealerwishestoputfourrepairnentofourdifferentjobs.Theffidifferent kinds of skills and they exhibit differentlevels of efficiency from one job to another. The dealer hasestimated the number of manhours that would be required fbr eachjob-man combination. This is given in the matrix form in adjacent table :Find the optimum assignment that will result in minimum manhours needed.

JotMan

ABCD

I

2

J

4

5328'/92664575778

{12)

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26 of 48

5. (a) Solve(xz-yz)p+(yt-a<) q:z'*xy. SEcTloN - B

ryry q/ohi* -4x = ,4*'- U" A''.*= dh+l+ z)

@'16**- ny'd"*,)

= d. (l'z)Q'4 h+Y+z)

= / a(*'ntT")

@+X+z)fu+d'+? ':Lb 0-pttryrr)

@ttril d ht+y+z), ld(*+y2+zL) * ffirztzt = ct Al

,"\ 6ttt py,rut ryth lu^*,A Ma-d 0-4 y -z

tq*,r'- \ib a^^/, liiD / A/4/,Ah'1/ ^*''t-"^' ;

"-r/

+f a^d+r,z+.2, #1 =/'tst,rnt- f a a^,r'WfM-'|

(10)

U)

= 4* - <.4q=";.%

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27 of 48

5. O) Solve the following partial differential equation

(g'. - +o'o- 1 1D'q" ) z: 4 stn (2x + y). (10)

o + D(D- zD"f<=o' (o'- 4DzD' +vDD'). ,\b

pyrtlAr{r"v ";try*/ :

?; I

D( D2- v Dd ttld')

+"*4-

4)(

6(ze+g) > 2 *b'rn;"0^*tilD

d2. z!

) + = 2 frc* +),tu(2"+y) t T,a; (2"*a )3f ,rywL/*4

(il ou-'fuiD :

4.+zp 4

vsl*/.t *, , 4, o,*t, yy-ffi,/an'y .f,wr,,oh0'"a

I

/-------

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"bri".,(zr+pf = I =*' ,^D T- 2o,)2,e;Ao*d

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s" (c) The area A of a circle of diameter d is given for the following values :

(10)

aqA&

VeAtu veA v94

8D 5n2L

5nq

6r62

85

6v9

6Et

-726

Yo

1o 78

q5 TosB

loo 7S5l

Md NuJn^;2L> lD9.

qo

766

b*Na/vl X,fulnttll-,-wQ)= to5-loo /t

,// atu+D(vi2)e(x) = An * 1"or-yryv2An+

t u(urt) tu+zfii+s) ViN .

--_ q.

vuAn@3l

A(ros) = 1s5v + 766 * Yo I,/

?-/+ 1

i

=>Jntrt, = 8666 | ,/

Calculate the area of a circle of diameter 105.

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28 of 48

d; BO 85 90 95 100

A: 5026 5674 6362 7088 7854

V,/

!a

29 ot 48

5' (O Show that 0 : (* - t)(y - t) represents the velocity potential of an incompressible two dimensional fluid.Show that the stream lines at time t are the curves.(x - t)' - (y- t), : constant

4 (ro)

M44,pr/.?t;

Lofr{r.%

&^ v\dt^ W f n apnnr** 114 ,rr/r0r4 lr{?tr.*ial -W4lrAr, Q kxa fo a,litfiy fi,/. L"fW2!=Q-+l =/#=o /2t=6-t) + L?=o L6i2t-2d",fr*b*du, #-#a-o ) f,f =o *r!?'fu, f ^q

N/xn$ lk ,tu/"r6/y 7tfur.6ffi- an

fue r6\e*'tu4 P 'f '' "" A

I, [A'tfu-t)' ' o"1'Y'Jfa'*'/t

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30 of 48

s. (e)

Abe

Llx

Use Hamilton's equations to find the equation of motion of the simple pendulum.

+h-e ?""*@ ,CWArr**r,k ' +n bc ''rnAaa '

ry

>n _>

30

= ?*9,W Y'lL e;

l-

t/:

\/=

-tb7W 1nlr,^*iul *tt, .mgt (t- ,"-tp---* iO

t-= T- o^gt(t-ca/) -Qi!

I

*/-J4!---

h.A---JJ*C'dL2

,l,w?12 02

tu)

ir=

rs+^l t<^i.thc %",

be, k= ',*(b

Ht,rr;,li$^^O,t^/ H =

*

ft

2

- |us_r'/-:../-

Prg''

, mtgl /J';^9 O

L

-Pr-t't12

+ rn60.t t- .@e)

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h- ttl-+55?$rr. a9999197t25 I I Emair: imsdmszrl0@smair.com, ww.im4n.b!.e'P.T.O.

(a) Find the general integral of the partial differential equation (2xy - l) p + (z - 2xz) q: 2(x - yz) and also theparticular integral which passes through the line x : 1, y - 0. (10)

-u)a(x zdL I %dL

rt z-2a2 z (r-7 z) - (x-{r) - Lz- Ln')

,&,rn"* ttlU-4 = - ( zdz* ad*) + ytz*--_9-* 8t, .

--tt''

: - Q filnk[,wlk*l,J/1/ 6; a'4 a^b/ht"?

y+z xJ >o

or'1, V' o

n.k, lfz=c, ) z=cz /"-u ) '' t*cz=Lt

b2 U trzxL = tt+U2+z'

31 of 48

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- *rrll- E-*.-.

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33 of 48

6. (c) Form the partial differential equation by eliminating(az-1):x+ay+f.

d^W t91,t ^ '*Q)

,-t'\Mh /'

the arbitrary constants a and b from log(10)

-J- ^2,= a +,2=az-t 2d!' ,rr'ry

q a, *( #.,)ry o a^^nt'cib:

u*@r)#. = iffT

_J_ a+= t

az-l d4

ip +LJ. "4/'""d l#''l

Az-l +

At)

ez,= %+ldd

It0 ,-

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6. (O The deflection of a vibrating string o{leneth 1 ir eo'initial velocity is zerg. The initial displacement is fiven by

v(x,o) ={;':;;;,','r' !*. , Herey. = o,y / ot,

ard y** = Ay' / Ax2 .

Find the deflection of the string at any instant of time. e0)

[*=cLXzx4 %=c'4-A.iL 2rt>

(t) ffk,il = Xh) T &) ---(ib

Wruav,,rltir'+H = +-1ifi

cLT d*z

Stv,,* Zrl ant ,.&/r^^r(/^/,

J- #=L4= .k-(c*"*o,.tS -1igCt dlz' X drz

&dr'

futhyrall;

cosT:

Gr*q:

xti(i

n'," Wl:Anry/Jtn t (

ir,.Aliala$+z' n;r

ll.ro:

k>o

/ = Atz+ az,

b^.bt krP2 \= %e'

*:;f^^T, Af € +AE' lri

Tt 4, arrF*t4p^f,t,-tt"

--a'fwty Tu)+o v+7

4:\;o *'4 x= q@P'+a/o&Pe)

rt y.b,t)'o * XO) Tk) * o D lto).p/.*fu>n. g;,;tAr.,tv . xf il- {

71,,fi 6N 7 qS' *' D " q' aZ'= ^1

huonl p e y,e

oo

:.y(*,+)= z unn Y(0,

@+8hI

Ml ;D z-:o/w -') #lo*, * o

&;^ nnof \0./v

n7(eT4B

rfrr@L 1'"

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aTt+ t

35 of 48

Oo

u(r,,+r, z, An t6Trrory.@

t' An-- f [, fftu0) e'- Td"Ltz r

= Al 3e,r,gdrutt I Ua';-gdx-'

. 4 w*Tl;,.T* ,,-bP-"frw'-r" :f o-oJrr* ryt 1-')* 'cn ff d^J

TpLrg 'fn- -t tlh h,

ar=t.t rrAnfr+ t.l- ahA 'llz1FL;.z,v3T+!- #/,"-ry lo )

+ q r--L *-D,f * *"^-?\;, 1'i-L nn /--

. -&. rna * +-L ,o*,(*t j r lP'^no - ef )n.t L '#'- lZ/ ffi t

q

^2tr..

*

A= -L Ai""E\ --h- rf* 12 t

'' fr(*,+)

-- *n P;

,tn

\_- ("iy*@w)

D^! oq, nT'! --/ -Y'.-/

? (niP

&Lnl

;o

END OF THE EXAMINATIONHeAd Ofilce;2rq ol{r Rtr hF EI" t5.-BEoch Of6@: ,oStOG, Tot, Floq E-F Tc ErF E!_ fa*a

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7. (a)

h+f)zj

htt)N.L

Solve the following equations by Gauss Seidal method.

Xr*x,*xl*4xn:-6.

= u- xy)' xy) - 1)(1s)

5

,{;''L ^y'- 4,

4'" , -9 - n6*')' af+tl - 1'

Cn+,1

32

6

-C- ,,y#'-eftl -,,raktud, Gl ,xy,g,q )= (o, ,Pr/--'

ll-1 ,1")

4") =

= Q-8, 1.6 ) - l'2b?) - t'742)

(l.o*s, l, qq./- l' ov'1, - z' o ol)@', ^t', '-P .

4

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te

),

x1

,)

i)/U)

(4', "Y,

rf ,"f') = ( l .otgr z,oo6r - l, ooz/ - 2 .o0y)-----'-'--

0)

)

(,, r.oot,

o

6.r ( t, L, -t,

l/ ?2t2, n3'-?Dr y i

qB -l / - ,)_-_--

*--<"-_ z) = G;u, ?f), ?f)

.{ z;

6)\,7y )

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(c) UsingRunge-Kuttamethodoffourthordel solve *=**withy(0): I atx: 0.2,0.4.

{(*y): # @,Vo)={o,D o,--@o-t

T,=ffi l1l)Q : 4"{(u, y+) = O'l

KL, t* 4 ( u+ lvr,'Yr+\9/D v o' oqqq

ry, A f (x^)Alz, h+ku/e) :o'ooto$

ry = ht(n +'l-' h+kg) = o'ootEb

Vt, tn * | ( 4 +xLrzk..+\) = t' oq6e

/fu1 Ut= Ub't)

Kt-- 2"$bqtfi) =

v2= A lh4+ Nz,V1+Yt til =

Ke-- ln I h+Az t frt kuD =

ql, ht irutln, h +Kil =

'ta,-h+*U\+2*rf&+\) =/

T

\t I'dkz'Y)

\, h { Ur+ blo'' [z+ k''

'x9ral @z+b4z'"t'r

K

Y+ -. Al (%-+l^, /iY>)

o. ool\45

o,a9zz

o' oQ 22^

o, obqT

19) = l, 2g"r_-___{/t3-- y* + [ (*n 4,*4

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40 of 48

h{ (ae,X; o. abql

o'087 o

o. o87o

0,0819

hl k;j+412, /tf h /z)Af (zt+4r, fu*b/DA{ k, y7", h+ttj)

ut, Y3 + ! &*4r dtY+ \ ) = t, b72z

7" (d) Simplifythefollowing :

(D (*+y).*'.y' (ii) xvy^yv znyvz,

(iii) x v V" [(*,. y,), y]'(r0)

(i) @ry). *'. y' = a(.x'. V, + y, *,. U,

ii) 6, il"Qr4*(yvz) = r (s%)

x.

C

C

zvU I tn, tu-' I r

C

o

t

I

I

,w

= L'no,[&nu)

:V=7 equ;,azl &i'

furr,*t. tr

o

o

I

I

o

o

I

o

I

a

I

o

I

o

o

I

I

o

oI

I

,

I

D

I

I

I

I

I

I

o

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Fvl

$=A'-o<

\z

Zo

cI

I

I

I

I

o

\

oo

o

I

I

I

l

0

t il, (*nv')oo

o

o

8. (a) Determine the motion, of a spherical pendulum, by using Hamilton's equations. (16)

Heed Offie:298, OId Rajind€r Xagar Ma*eL De hi-60B.anch Otfie: 10t106, Top Flc Muhri* Tffir. MukleD* Nagar, Oolhi-g

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41 of 48

,Cii,

42 ot 48

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8. O) A uniform straight rod of length 2a is freely movable about its centre and a particle of mass one-third that of the

rod is attached by a light inextensible string oflength a to one end ofthe rod; show that one period ofprincipal

oscillation i' ({s + r)n{(a / g). (16)

44of48

8' (c) when a pair of equal and opposite rectilinear vortices are siruated in a long circular cylinder at equal disancefrom its axis, show that path ofeach vortex is given by the equation.

(r' sin' e - b') (f - 6:1:: 4arbrr2 sin, o

Q being measured from the line through the centre -l- lar to the join of the vortices. (lg)

H.# O6€:2tE, Old Rajind€r Nagar Martet, D€thi_60Br.rE r Ofie: lotioq Top Ftoor, Mukheiee Tow€r, Mukho{€e Nagar, Dethl-gft, lr€a2!sr, 099091',025 ll e*il, i*a*zoro6,smail.@m, ww.ins,kneths.cm

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ROUGH SPACE

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H.ad Om@:2418. otd Rajindor taagil tile*ot, Osthi_60Brahch Omc€: 10t106. Top FId. Iukhe4@ Towef, Mukheriee Nagar, Dolhi_g

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