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A CONSOLIDATED QUESTION PAPER.CUM.ANSWER BO
Under the guidance o/ K. Venkanna
MAIHETUIAIIGSPAPER - I : ODE, STATICS & DYNAMICS AND VA
rEST COD[: TEST-0 3: lAs ( lvl)/24-lUNE.-2 0 1 B
IIYIsi"(rilsfl TUIE 0r uaIEEuaTrcaL scrDlrcEs
,uNE-2018 TO SEPT.-z018
Time: Three Hours Maxinrum Marks: 2 50
INSTRUCTIONS
This question paper-cum-answer booklet has d 8 pages and has
32 PART/ SUBPART qLEstions. Pl€se €nsue that the caqy d ttE qu€stixl
paper-cum-a,lsr,rer booklet yol have received cortairis all th€ questbns
2- Write your Nane, RdlNLYnber, NaneoftheTesl Centre and Medium in the
appropriate space provided oo the right side.
3. A consolidated Ouestion Paper-cum-Answer Booklet haMng space beb\ /each parusub part of a questiqt shall be provided to thern for writing the
answeG. Candidates shall be required to attempt answer to the parusuu
part cl a question strictly within the pre-delln€d space. Any attempt olsidethe pre-defined space shall not be evaluated- "
4. ArE,v€( must be wilten in the medium specifed in the dmbsbn Certificde
issued to you. which must be stated charly on lhe right side. No marks will
be given foa the answers written in a medium other than that specified in
theAdmissiorl Certifi cate.
5. Candidates slpirld attempt Queslim Nos. 1 and 5. which are cornpulsory'
a.d any THREE oa the remahirE questbns selectilg at leasl ONE q,estbo
frorn each Sectirxr.
6. The number of marks carraed by e&h questbn is indicated at the erd of the
question. Assume suitable data if considered rEcessary and indicate the
same clearly.
7. Syrnbob/rddbos carry tllgr usud nrea*rgs, unbss dtE wise hdcded
8. Allquestbos carry equal marks
9. All ars\./ers must be w,itteo in bludblack ink ooly. Sketch pen, pencil or ink
of any other cdour should not be used.
10- Al ro.rgh t\,qk shoid be dorE h te spee ploviJ€d ad scor€d od fndy'
11. The candidate should respect the instructions given by the invigilatgr'
12. The question papercum-answer booktd must be relurned in rc entirety to
the invigilator before leaving the o€mination hall. Do nol remove any page
from this booklet.
READ INSTRUCTIONS ON THELEFT SIDE OF THIS PAGECAREFULLY
Name
RollNo.
Test Centre
Medium
Do not write your Roll Number or Nameanywhere else in this Question Paper-cum-Answer Booklet.
I have read all the instructions and shall
abile by them
Srgnature of the Candidate
Signature of the invigilator
AR
IMPORTANT NOTE:i;#;l;Gil; beng anernpred, a[ rf, parrs/ suuparts must be aternpled cmtrgLouslv. This rneans lhat befoe rno"ino 11o11.e "'nqrJi- r.G jto"pt"o. can;idates mg5t lini;r atempting att parts/ sut'parts of lhe previous qLrestioo allempted This is to be slictly fdlci"€d
o f"ff Uuni ir't t ,a*r-book ae to be cl€dty sin; crI in nk Any asv/ers fEt follo'v p4es ldt blark rmy nd be gi\en credit'
P'T.O.
MAINS TESTSERIES.IB
1
I have verifed the information filled by the
candidate above
2of50
DO NOT WRITE ONTHIS SPACE
I
I
P.T.O.
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INDEX TABLE
tto
+l
+\
fo
+t \
QUf,S-IlON No PACf, NO. M,$(.MARKS MARKSORIAINED
(a) 0t(b) cr(c) og(d) 0r(c) (98(a) IL(b) rr(c) r{^
2
(d) 0t(u) l3(b) t3
t9(c)
l
(d)
(a)
(b)
(c)
4
(d)
0g(a)
ok(b)
o*(c)
08(d)
o&
l
(c)
(a)
(c)
(d)
(a)
(b)
(c)
(d)
1
l0(a)
lo(b)
og(c)
tr\
$
(d)
P.T.O.
(b)
Total Marks
4.of 50
DO NOT WRITE ONTHIS SPACE
,
P. T. O.
5of50
Jlan Cr 14"*ttALe
-;He-u *7,'u4dl*
SECTION _ A
r. {a) (i) Find L r
{*;ii}
{'*['-i(ii) Find rr)l
Et &l/.r^( k?h. * gt.r): *?
=) f(s)= toS(pr3) _ lo!rp1z)
PtzlP+t)
f '(J)G*z)
,.lc*r O""u"t hptaU
l?*t )
^r *'(') r'( ltcrl) = L (4"; - .-'1
tJe Er\Or! l4,At e-3f-e ^ r-b
:) L-'[,"grffii]
_at-_ _t ( - {")Ltb"l
,.8 (, - a)
LP
ro?(p1-,7 -rogte,)logtr-r)*rT(r+,)
_ zlooe0 gP{ (r)
fl(/+ t
-1
,JOuJ / -0r..r<,\zx
(P-\)
to{k* qr f'6)L'' (+'rt)l = t)'( !-, \Pr + -l ( ve)Ptr ) LL
a'tL-'(4 (r)) -t
t L '( +' ( t)) + e -z) r:'(toX\-tteL)) = -j("t -t -L )+ ?
(1r stSL:t
IIYIsi,'Esmrll $ tllrl]uM& SNllt(Es
BEAO OFFTCE:2s/S, Old R.j o. pr' gslsrsrszs. orlnsszlxr!7. BR i|CH OrFEE: 105-iOS roP froor.i"rL.i- i"--, i.rt'-i;. x.e.,. o.rhi's. REGrcn^t oFFrcE: r_ro_?3?. rr'd Fr'or. Room xo' 202 R'(s x'nch'6' Bru'sapphn. A.hok Naser Byd..:b.d-20. rlobar. No: 096523s'11s2
-ii.t..r-$..."; ll ,,'.ir.o:th..r..'nrns .on tl Eh,ir: '[email protected]''6h p.T, O.
[ 10]
-,
tei fc.r-;
L'' (+t(r)
IDISiItsltlutolfu D iq{, sollEs
o,aL0
1 (b) Find the orthogonal trajectories of the family of cardioids r = a (1 + cos 0)'
[10]g'.ue-o {r"^'t qr *A*^odt
4r176)= { - o( t+ cor0) =0Nod
"tf"rt*g (D o.^.t & uX yt) a{/do *q.coo:o =)Q
d,/ao) coreco -63e-t
P.r u{
=) (ara.s) f'cr, t + ldt/ao) (co.rks.".( o i,n5.1 @6/
'b g"t o^+L"g""*{ E.q t. htr\ [+ao/ar) +1""
- iad+do = od%
Jioo
)
) 4'cr,or -.t"do744 1-n'( do ole.6 rcoto)((o
-\-/ ( de--vJ"^
.,copco-t coto) ;) (corero +cofo)do
= dv^rC.)-a G. trq lAea vx
'x
zr) =J fr+colo t d6 _r\-t;; / -l j=!%/ ae
''':"/,ld
) ,/t
) :'z lo2 Q;a(o1z)) = logr + toge<.I
a[t*)' 6 ,"1")
=) '{ -- A(r- 1916.,rl.lrt ara t
d{ ca-il"
6of50
HEAD OFFEE:251t, Ord R.jind.r [.s.r flr, o.rha-60. Ph.99,9r9762 5. 0ir,456299t7. ARA,lCH Orf rCE
-o
10-237. Xnd Froo'. Room io. 202 R.xxukh.rFo To*." hrh.rj.€ n.9.r. o.thig. REGToiAL oFFtcE: I,sepphn. a.hor [.s., Hyd.r:b.d-20. H.bit. No: 0965235ii52
I wws.im..m.rhr.r.ahinq.c6h Em.ir: [email protected]. T. O.
-o
./
-o
(d'la.J
-[Q-ru* +cotols
u
I (c) Two equal rods, , pj44-AC, each of length ?b are freely jointed at A and rest
on a smooth verticE-circle of radius a. Il 26 be the angle between the' rods
then hnd the relat-i6-n between b, a and 0 using the principle of virtual work.I10l
I "Xa ttrc aod+
b,",gth,
,4Bt
zb;Ac
LOAt.. ,r-O
.ict! 1f
bq
t, -fii o va cr uoltt(I.l?
oe
,,qdi&1 q Cr-W OC
Let FrGr k ^,Ap,^trall ABtk( *.Wrfr*6;n" dV g,*4* t"o d,.t lp o" j 1-,.c17o.^1 4, ra)
^JOul AF= Abt -- b / LDAO --O
oH= oA- AH q coJcco - bco.l0
-x\rtq^od,
tD L< *AI" wlg"n ) v,iluo)'.,cdl<de"* bil g^*q o ('. g
xdo 1^ ud". t-et -e^glt-/y-t^ro
'-LDt(on) = 6
d1 lf-)) -u) 6(oco.,..o - 6ro.ro) o
) f ocorturcote +asr",o) J^o = o
) bs;",o a cojol.^1O
4
}lil, d"h"" v&*r1 qlbrg
ale:.<" aJ< .o"rlt-:-
b -r;asg = o coj O
7of50
r.,t.r. D.lhr-50 Ph 999'
Srprliir. Alhol x'9'rcErgt9t$flnllO tntEllltI(r Is"Iltl l,i,;"'il'.'*1,"2-.," rhr'im2oroocm'ir'con P.T. O.
g
) 1L-.,
E- t
1 (d) A particle is moving with S.H.M. and while making an excursion from oneposition of rest to the othffi distances from trr. -iaat. pffits path atthree conssgglEye seconds are observed to be x, x2, x3; !-:6?e that the time ofa complete oscillations t" :
2,r7.o" '( "* " )' 1.2*, J tlol
t;otr q porh.h r, 14 J.H.r,
) ngl^ohs. ,,L 1ot!" ',, Six{a bA rt= Ccot(,,p| _O
Orrr,r.t.ng O-L *__o; X
No LJ
)
:) NOLJ
z)
il*tu
a
t3=
0rryr
3 o^a(r,
l^
/ trtl' r/ ttlL
'L
=)
? Ao.^rof &,1e
t\' coJ{ (
I\
lnru.4tPo^t-'c{4 01.
G",nu+)dt
a, , ,r, ,j
acor(xb,q cor( ih +r)a cor( tae + tv)
1t:"9_@-o
2'i7-
!"r tt-r)
( co-.r latr+ t4) ,q{x)('.-.,@)t\
J,,nc fri"J + J,H.d
(
27r
CaJ
X. RtJ,dt
I
Tiit n4r.'od =
srM
8of50
1i6:i.9li'j","?i3il""H"-1",i1'."*.."rj';1"^.;
P. T. O.
..D [ ", rm..n.rhi.r..hii c.coD I EEn. rD 16.20rqEe6.ll..or
/
-oI
tr *,{3L1L
z
1 (e) The position of a point e!_!gg t is given by the formulas x = er cos t, y : et sin t.(i) show that a = zu - zV--- '-----:>
(ii) Show that the angle between the radius vector r and the accelerationvector a is constant, and find this angle.
.ftfl=:==:-
9@ ^adnr4 vadffi et colt 1 * etsr.rf J
Nou) / W*""fo Q.^r"r.t t stc.J< I-4
.l t ( ut.o.t - et ( etrt.,6 * ot-,, i!'i,t r .l-
(et rb 1+^166)
CO
a
etr,at7 i) + f -et ,+e 1. orflJ+ (-ets,,r11
NOl.J a3o.-4 " 6t..rrFi) o**PLg@ ..l.,r.ti-u
4o
ytd,l ( l-?)dl
UFdt f ( -.t -ll.rt i -ebco.t? +ctcorLl_etrr(7)
-)v+(-i
( etr;ot I +"t lrctl ) /t(e(oJti +z's;<tj)_t- r7_ 7)
a
r): z (_ets.rf i +zrort])
NOr,J --)o.
1'{
L(-etL; atl l .tcor5'). ("e.orel* zr-rnrl )o
a^r-.f& "^d" bfi*r4 d -1
1 \4 Cc,^4 +o'!t^^d ..',-to
( 6o-il^
*.b fr"'zu.-tqlt
a Z-r - ^-)L"1
rMs",!$mD 0l lllllllltl(ll scEllls
HEAO OFFTCE:2s,!, Ord R.j O'rhi6o Ph'999919'525' Orl'456299t7' BRA|{CH O-trlcE: '05_105
Too troo"iiiii.i.. ii-i'. i.r"-i:. N:s.,. o.rhFs. aecr6iei orrrce I r-'o-23?' rrnd rroo'' R'om xo 202 Rx's K'nch'm' aru'
s.ppr'ri. e'r'.r rob . t{o . 0q552r5r152*'II-r."r*'r'..""'"- tt r".,m..n.rh..r..mrn! com ll EEII: in..rm'20r0@ei'ir-'on P.T.O.
9of50
d?
)'l
-o
r 90
i)
10 of 50
2 (a) Solve and examine for singular solution of the equation(1 + p)" = 127 /8al (x + y) (1 - p)'. lr 5I
9i,P4 Q"pi ("rU t ^*r, "r{^4&- p-_ dv/d.,t1, - p)3
trt ''(ty-- v / .(_y=q
=) dvd 'a P) d{= '^&1-
P
'z)dv7i
(/)
HrJ
CD ule yt(d'4..;3=
8o,'=)
d v/ilt = ,/3a ) dv
,/'/3zo'/j - _: ,du
L av3A^ ,fr7.d,* , ,ae 7t+9. ,/
L
z/?3q +C
20 t/3
) ? ( t+ylu3 -2'. 3 qC rt-g ) +( t.l.g v^ bt\ td,g)
*fu,*-rt'+gfic"_,)-) (74; r"of
LaYs
'1(",- y8cr
2)- +.?
L) tL y@!labt",cil) O
c"t "hfig ) Q*j' Gnt- -D_(t-*g+(P) go
rrYIsItBmr't o utlEltflIrr 9cEIc$
HC^O OFFrCE.25[. Ord R.irnd., N.o.r r r.r D.rhi60 ph.9999r9r5?5, Olt-a5529917. BnAXCH OFFTCE: 1Or106. Top Flod,urh.rjo. row.,, urh.,r.. x.o... o.nr-s uoo-u-r--orrit, i-r;;!;. x; ;;;;. ;;;.-ni."jo, ".x.ss'pphr.
^.hor. x.!.. Hyd.,.r.;.to. xob,r. xo. oe55z3stl52wr.rnrrD.th..Eoa ll ywjB.m.ln..t.:r.hs.com ll Eh..r: ih..rm.2oro@sn:it.con
P.T.O.
L1 ,
o
11 of 50
hE O OFFTCE:2516, Ord R:j,.d.r x.gar Ph.9999197625,0r1_4562993r' BRANCH OrrlcE: r05'106. IoP Froo'Xurh.,i.. Ior.,, [urh.rj; N.s.,. O.lhi-g REGTO|AI OfFICE:1_10-237, rlnd tl@r' R6h xo.202 Rxs K:nch:m' Aru'
.L+'( ?) - g( r +P) * +r
tie) ( r_p)L$-o
--i-,/ -ff tttv) -(c
6* ca,o
r"19+,-hfig
,.j.r[o.t ,oL.f;.., oa -tg =o-:+-.
!r<- tW .+tl:* @ "otrW, a_g,ve^ d;46rr"ta/ "qAot', O
)+--<o- llq urril,t
(b) Find the length of an endless chain which will hang over a circular pulley ofradius a so as to be in contact with the two thirds of the circumference of thepulley. [ 181
!i*n c,^,r.Jnr *& - ,tqdri-ra q
@ Covel-,
!a"d c,nc,r.-1,"o^ k- dV t|" yUtoag.1 tr< ,"ge&ol,
t"dJw A,' c ho-.r't
rcat 'tL-,9
L^y4a ,t^.; o"e.t pl/gf
li- tr, ( rrra)' =
sqwo3:-
k
^JOrJt lrd
l'*\
rlS cr* Lfi +4" ,U;
.r,'."g",1u Jotutu,'x +y = o
Xk41pwe)
I^"9r3
IlYIS"Estml! U Ullttltlalt SllElgl9
S.pphn. A.hot x:s,' Hyd.r.b.d-20 Mohil. llo : 09552351152
P.T.O.
L0+ P)
Q-P)-
*yt ,@
<.-<.--
UJa l<.,\OLr
LAog=- zt(3 =t2:,
-) /-AoL-- r43 ) PA= 6oo ("gt" 4*tr+ *A lakoo odJ,*B_o!bg*h aB = 2qsio 60. aG
= (-rpo.,,r dV tL\zc (g(reca,1 + t!" w)
+{*
1r {1"
9 G{i= z c tog (z+ {3) ", hx tl-i,
Ce*afu^L-
.&"ga C= oJ32 lo3( ztc)
tarlFrOt^) t-gt.. * * d"ar1 A"€n J: cto4 Pr
oa "t3-'. $rU rr_g.U.r t ^ilg
atT (z+6)
2-J 3g ,l"g (z +Ji)logc z
aL.1A tror t-g+t + d. "AL,n ,fo;- v
3l*u 4< ru^*lt
9rro ^3',logc r*G)
3q
12 of 50
riEAD OFfTCE:2514, Otd R:iind, x.s:, r,k . D.rhi-60. ph.9999197625. Oi-a552!B!7. BRAllcX OfFrCtj t05.i05, Top Ftoq,urh.ri.. To*.r, Mukfi€ri.. ll,g.r, o.rnig. REGToNA! orrlcE: t-to.23r, thd rloor, Room xo.202 R.x,s x.n.h;,. Bt".s.ppirr. arhor N.0.r Hrd.r.b.d.20. nobit. Io: 09552t51152$tmtl u trllltlitcrl lcEtco!Ir}IS
lt *wr.imtm:th..r..mins.com I Emair: im.im,[email protected].
13 ot 50
rl2. (c) IfA =x2yzi-2x23 j+a2zk,B=2zi +yj-x2k, find the value of -" 14.g1 21
:- ===-
(x'ry
(1, o, - 2). Io4
gt^,"4 A= .r"yzi -..# l* a., iB 2'L, + 5r - -tL !<
ArtS - I J
-rxlfK
*t-z- -LZL- *5t")a ?-
L-L U -rt ,-*ti. -.*f) j*
,.rOt^)& o,.
( *t', + r, rze)i
a*Oy ( (-t,*3-;1C ah 1",.1*,tf + 6qr 7_ K
6o+q 70 tu4*)
-zL + q43rl*r,\JZ ,<
!o5^-d t( aro; o* (, ,o r- t-1D,t0 I
X+^qtJ*^-.rJt
8 J
iliil;.il';lT?Xil -"1#",,i:','if-'"rj"i'.t":",f.'i1f:I: .h..rB2or ooci.ir ..n P.T.o.
*i3.Lii !5-li,l'"?ill# ""':"::%:,il;*l.Bi}ii*.',i'";rrlSj*i*:l :*',r:*l*:-:li:'j.L:,;,:::";"';l " i -
LIo
C'b
t
-1; -
/
14 of 50
UE D OfflcE:15,l. Ord R.iind.. nrs:, x.rr.r, D.rhi50 Ph.t999r97625, 0t1-!s529937. ARANCH OFrrcE: loti05, rop Ftoor,Mokh.ri!. row.r. irukh.rir Nrer, o.rhi-e. REGrolaL oFFtcE : 1.10,237.
2. (d) rr, =1,{-,ff),.1,*- -'*)t.l'*-,*).'prove that(i) F = r, vI (ii) F'r = 0, (iii) F v/= 0' [101
c\) AirJ{4 E-\) ( Jzl - zt4 I. a_Z ,_S )i* ( -zt* -=5-.t
*!t 13 * l.^dt-A.L.t,\ 'b*
v{= ol i+ a(?y ) F K?+
4̂,
'b'z-
L-t<"Li + g j +
l'rv1 t J Kv ltlt,- - z atVo
,I
t+1a,r o?&z-
:}long- F
cii) F.'r (*rt).v ?1 ,(urr)o
i ,r, ) tr.vf - (oro+).r+
(v. (vfi v&) ( x
ot a.J
-tk'acs-'A< xurt6
,
rx*.ftr
II}IS'$ImD O llr!!!.tIlcrL 9CTETCES
s.pr6k. a.hor .g.r tryd.'.b.d.?0. x.brr. xo i 0965235rts2ll *l?.iE.hdh..r..ni.e..om tt EDn: [email protected] com
P.T.O.
3. (a) Solve xld2 y /dx2l - (dyldx) + (1 - xly = x2 g*.
gnle/, &tt*"ta, .'*tr;oo,t(#) -(deld*) +(r-r)J= *ae-t
I15I
ta
LDE b'"d @,?tuq-|
tot"l r"L,he,
,oU^f*., an amptnt +=dr.64
Let9= vq
rr"fo*g q/"d- O ",ctA "rts^dqrd[ ,Xfrb J.,fpy+q(.^Jt
-l-s-t J=e
s ?-, t O-- r -r -x-Le
",brd,rfu O->
qrt
d2,( P* t S;'zr* =drtr +
-) d3(6l^ u
t( ,( + ') ut* . -2LnL
>) kt d v/6n = Q. L,u/d, ,dL
-d ,Lt (**)t =r"-Irt
.D^h7.d,"{.f"Jrf
4q Wutta))
>) "JG:**,-1..
2-
2-.L
s.(o+) = lrf
e,(
'=l ? a -1-*+-'a * c. ''t.e-z{
Jo,\
d u/d* = Ito-.*,.+ cr ?( .,.) r,
4.t = -?d -LL L"e {-t cr rCtt -L't-L I 7-- t Lz-
(-{
ed,u'a) roL*"9
alue 14&zut
u=ea
-@c,rrfa -c,e-at * Cz- e 7*>-2- v
-te-Y= -*
15 of 50
H€AD OFTICE:25/3, Old Raiind.r x.g.'l,urhrj.. 1o,.r- llnrh,i.. N.s.r, o.lhi'9 REGToNAL
5r
t't
J
OrFIC€: l.l0-217, rhH OfrlCE: 105.106, Top Floor,
RooB o. 202 R.x S x.n.h.m'. Aru.IIUfs)'-Esfitll! Ot tltlDlttru smtE
S.pphr€ Ashot xaga, Hyd.r.bad-20. itloblt. o : 09652151152
P.T.O.
b<-
-l
o'rJt
K
9.
' <tt a- +c,
b'^tl"r- lfrfd,b",:)
-cr Q
16 of 50
HEAD OFaICE:25r1, Ord R.jind.r tr.s., ..r.t, D.tha-60. ph.999919?G25.011-t56299t7 ARANCH OFFTCE: 105-iO6, ro, Ftoor,irurh.rFo roe€r, i ur(h.rjoe Nas.r. o.tha-g. REGroxaL oFFrcE : t-10,237, nnd Froor. Room tro. 202 R.x.s Krnch.m,. sru.
3 (b) A particle starts {P!q rest at the cusp 9f a,smooth cy' vertical and vertex t6-wnwards. Prove thatiih"en it has fal
distance measured along the arc to the vertex, two-thirds of the time of descent
will have elapsed. =-=-- --- [151
--=:->gi.!t,n o qgdo.d _
6to.l t-g+t, {f4gc
arc b< Bq
cloid whose axis islen through half the
2q qaB
o
C
'-) ogsB- 1,.
,**t t^-" Ao.2t et A
s?-.trb"at t--o', ,t.s1 5-rq
(.4aoJqr;:g Fy.rl
2(d t lat) o.d .!"b?^ot^t
oUf;*. 4r oe
dlJTtT ' lv
\q
h .9.lou.,(s'4"'t)
s
bu il,y _$\J qa
)
zl
z)
?
a
^IXVWov
I 2dasEE
. d2lp:
-9 r'*l'{a ( :' al t.-o, ds
&
&
(o'lat) "== u __o)
) r..b.,ttrt-A
t(fr) ('Gro-ll s'J
(g/rq) (eqt - s'l
t:o)
/ "Pq@E,( ^@,,
-l.-n eq q-{r,<L
ll9
t
/dJ . ?-
L:F' =
dtldt =
.@ -0,q detrc.a,aiq )i:-9
IIltrStlsflt1lI o uiDllt[u sctltaEs
S:rphn. l.hor x.s.r Hyd.r.5.d-20. xobit. No : O9652aEli52rw,.ih.rmth..coh [ f,**.ih.4mdh..r.r,nrn..com EDir: im.1.msro1o@sn:ir..omP.T.O.
A
oy <otien
I*:x
ct4 5'r,l
\
17 of 50
) d-r 'E .dtUva('10) -JL
No": -O"Q."h"l r"+r.,
s;nj(r/qq)= -Frot +cs cq coi
r',Au
2)
=)
qq ) .tr=
'.' J= qe af t-o )t^x qp{- c- wZ,
tOeJ t-c {oro.n lf, 'h{q,...g
{Lro *| 1,,,,
b\Ll+
GqVTh( .. Ao aJ=
tq 6;'1.1 .fl
tr (iq2-V g
---'&c 4rd-** u-J)
ABA=re)
.< - ( at-1rt-,1
GL
1T
3
-z
)
3
:)
"H:4ruJP-a {* a-ul*(c) (i) Veri[., Green's lheorem in the plane for $.t'.r',lr+1.1 ' 9]),.h where C is the
boundarSr of the region enclosed by the circles x' + y2 = 4, x2 + yz = 16.
{ii) (a) If E = r r, is there a function 0 such that E = - VO? If so, find it
(b) Evaluate $, t ar lf C is a simple close{ curve. [12 +.O8 = 20]
C,) g'r. $ ,tlg dt t03- *gr)4 ovo(.
&.gr$D e^rluaA 11 ciJt"C,= 111y1: q =6
(cr t J.q,lrue J"tq\"] c 1= rt1+g?-15 ;o C.
Aerq rt = iL.L
LeLl +qk{
rL = ^r COt&
O-tl.. xag',o.
J ).u-- g -Ly --)
roh toold.otz !)-- "r,rtg )
3
.r:r tot 7.o O GII
t z tL.
IIUfS'-t$Tmtt o rllEltl(& slEtll!
H€ADOfFrE,?5r3.OrdS.jind.'N.q.rx.rr.l,D.lhi'60Ph.9999197525,011_45529907'ARINCHOFT|cE:105r05looFloor'iiiii.,r.. i"-l-. n"rr'-ll. x.e.,. o.rhi.e. REGror{a! oFtrcE Froor. R@m xo' 202 R x s x'nch'n' aru'
irob'r. xo : 09552J51152
-,ii.r-.r-rr..-; tt rw 'n.n.rh..l.r,nrns.com lt Enrir: rmraim.2olo@em'l'con p.T.O.
ffi.)
t 'lL, ott ,: ( .
/-4! -a^r , .-'-'(-a-,. \;) =Z-=5'-,'
o
18 of 50
HEAD OFfTCE:25n, Ord R.iind.' x.e.r ll.rk.r, D.rhi,60. ph.9999r97625, Orr-156299rr. SRAiCH OFFTCE: r05-r06. ro, Froor.urh j.. row.r, urh.rj6. N.s.r, D.lhi,g. REGtor{aL oFFrcE;1,ro-21?, xo.202 R.X.S Xatrch.;! Bru.S:prhr. A'hor tl:O.r Uyd.r.b:d-20. rrtobl. xo : O9E5235l.t52
-11t
3
C )),e)
( "ltf +.L! 3
)-1T.
=) JJ-c""**)a"^4 =') J -,,]) ('aor,)Y=oL o:o
Y 2rro
rO -(z-rr) /eo1 - | zoTT
cal L.""
$,rat LL: ."L4t-6 --O -2. et:- .i CO)O
I tr\d,t +1u3*.r, )q __rn
y = r{ Jr,rO
o: ob zz_
- 'J t"") ( .nu'fuI ,,-6 ,,,\,\4 ra" r.-*odo.
Co.rlg l,rz,9 *(o) cor :,'.tOJ d
o -8.1? IL '* . -.-rz:
2-
Oo?f Cr: .tt+gt-.{ =o
=JrL
=) -1( L CO.Ja) y=l"r< ./-tJun-,o a. +- t93-rgrJa,,
Z);aO
O'-a6 z.+
( t-l 1 ,-r;.
=l Lrtr-**rxrl/.-l .r.LL' >
J,+5a^* - 167T ,r- L}lt/\e- bc+r,) (f^
fur ,yagtro
eat'4t - \Ect.? i lfes La
---l-b,,,*av1,)Lu<
l[\).t I (E) e* (". D =q=-) E \,); &rctats.,r".l
1YY + , (v*7)
o9-,"
t,t F- -v 9= '.7; aa FI 3
*L+yt+,Atgrtrc"x-6f
zL-'L
.JL'Lh l; fi" N4lnwl
J.,J(
-L28rr
- 8tr
de. a,<_ -O,.{q 5,,n& Jo,/A o
* | LoTf
IrvlSItst[t1t o trlE!,lllcu scE$t! lt "r.iBlmni..r.:hine..om tt Etrir: [email protected]
I .u1.,.1
P. T. O.
(a) (i) The number of bacteria in a yeast culture grows at a rate which isproportional to the number present. If the population of a colony of yeastbacteria triples in t hour, find the number of bacteria which will be presentat the end of 5 hours.
(ii) Solve (D, + 1) y = x2 sin 2x. I18l
I
t
19 of 50
HEAD OFFTCE:2srt. Old R.iind.' N.s, Ph.9999197625,011'a552991r' BRANCH OFFICE: 105_i05 Iop Froo'.xukh.ri.. lor.r. raukh.ri;. Nag.r. O.rhi-g REGIOI{AL OrFlcE: t-lo_237, lhd rror' R6m xo.2O2 Rxs xrn'h'm'r alu'S:ppirr. A.hor x:e:r Hyd.r.b:d-20. rtubil. No:[email protected] ll ,r*.i6rh{h..r..rnint.com ll Em.ir: [email protected]'com p.T.O,
4.
20 of 50
HEAD OfFTCE:25/r. Otd R.jind.r [.0.r ra,,t.r. D.thi,60. ph 9999t97525, otl-15629937. BRAXCH OFFTCE: tOs-t05, rop floo,.u*h.,j.. ror.r, ri!kh.4.. x.s.,, D.rhie. REcrotat oFFlcE: r-10-237. No.2o2 R.x,ss:pphn. A.hot r{.s.r Hyd.r.h:d,2o rbbx. xo : 0965235rr52
4 (b) A particle starts frojgl'elt at a distance q from the centre of force whichattracts inversely .Tffi-" di"t^.,"e. pr-oifiEat the time of arriving at thecentre is a"lpr1zy,1. 1l7l
Ittcrto 4r bs* t(potr& .of p oP -_ o. O P
[fl,rr (r< //a(
a4 * '4 arl-to(Iure I--
Itsfltlll 0l tllfNrnlr! scrllllsIltlS
ll x,'jm.mdh..r.....ns..oh ll EDir: im..iD.2olo@sm. ..6nP.T. O.
21 of 50
H€AD OfFTCE:2sl3, Ord R.tlnd.r f,.sar H'rr'r. u'1hi60 Ph 9999197525 011''56299t7 6R41{CH O-FFICE: r05-105 I'p Froor'''ii,ii-].. i"-.' r,rr'-r:. N.r., D.rnre. neci'iiiri <iii[i ' r'ro-rrr' rr'a rroo" i@n xo 202 Rxs K'nch:m' aru'
S.rrr'"'. a"r'"r xJg,r HYd.r,b.d_20-JJ-,-..-'r'..-; tt *" rd1h,h..r.:nrne''on ll EBrr: in'4rm.z010@!m'rr'con P.T.O.stm
22 ot 50
4 (c) Use divergence theorem to evaluate the surface integral lJF nds where S isS
the surface of the solid in the first octant bounded by the co-ordinate planes,
the cytinder x' + y' = 4 and the plane z = 4, and.
F= {6x2 + 2xy) t + l2y +x'zli + 4x'zt'k. I15l
,
IDISi HEAD OFftCE:25rt, Oad R:ii.d.r t.., x.rr.r Ddhr-50. ph.999919r62s,0rt,a56299r7. BRAXCH OfFTCE: 105-105. T.p,toor,xurh,r.. ror.,. Murh.,i.. ,{rr:,. , r.ro.ni. ,ii-iiiii iiJ-.i; ;", ^
x's x.nch:m.. a,u.s:prhrr. A.hor N.oI xyd.r.b.&20. I.b,r. iao: o96s2t5ii52'*r.rm..mrh..con ll yrl.r .D.$..r..mlhs.com I Emfl: im..rm2olo@oo. .con
P.T.O.
23 of 50
P.T. O.
HE O OFFTcE:25,0. Old R.iind.. N.sar M r . D.rti_50' Ph'9999197625. oll_'562993' BRAxcH OtFlcE: l0tl06'iiiii.iii i.i.-i..;rr,.,r;. xrsr, D.,hie. REGToNAL orrtcE: r-10-23,. rhd rroor. Rooh no 202Rxsx'nch
Nao., Hvd.rdb.d-20 i.ob'r. No ' 0965235115'JJ.iii.-,r''-; rt,e*,B.n.rh..r.:,ni.e.om ll td'r'm'rir'201oe,e'rr''om
( -r.-t--v( *-) t<
SECTION - B5. (a) Solve l2x + y - 3) dy = (x + 2y 3)dx.
9;rr., dy d+2.r -?2{+Y - 3
u-t
[ 101
d.rr -o( tn,q lr<
+"-L,
Qoaa^x l-edL.D.nOgL4g'4
l:"a-<a1l^.ffio,)
h+ZKa_3:O=
?.tr1 0x+ 2y
2h .t- K -3 *-)
2r*)
9 dyr-Jt ?,1-
r+
d= x+ l-r
r -vL
3= Yt K
dydx
' g+J
2xt)Le* y= Xv,,ut^,Lfr," O,) v+ xd2^ =
dx vt Xd,r7^n -O
L2+v
:)
.k
lii) i,t
L+y-) (z+
a1 hlt{r U",)-dv( _v-r-l zvd y
G,!o'+(#; = *o9{ g ,u3[ (r-",):.
tonx + ta
x
n
=)
J I*;JQ - rt,)
!,)' i,, -- ^*
) /r+nt r- v
,L {(x-y) =lK ) (rtx) = (**t)r.x-
44<^,qiAxl Afu"=)
h^ X =
Y-- Y'tX= a
24 of 50
r.95' r.'r.t, o.rhi.60. ph.9999197625, Ot1-a562993?. BnAXCH OEF rcE 10t106. Iop Fro.r.asa', D.rh|9. REGTONAL OFFiCE:1-10-217. nnd fioor 202 i.x s (.ncnrn,. aru.r Hyd.r:b.d-20. xobitr No : 0955235tr52IDISilrsfirD 0r frflDtrlfir! lcEras ,w'..m..mth..com I wwl.iB.mdh..l.:mrns.con Em:it: im..im.zoto@sm. _com
P. T. O.
a-4 o ule
o
) xdvEF
dvdx
(+r)..--.---(-r 'n=
3
5 (b) A sphere of we]gh'[_W and radius a lies within a fixed spherical shell of radiusb, and a particleSt weight w is EEI to the upper end of the vertical diameter
prove that the equilibrium is stable if W ,b-2o. IloIty ct
$''^xo o 'P'*^o
dL **!- &d2 rodir.r q
hc,l a "x,ghf LO cf+ad"d qt.f, 6P
L,t. b? t< +L l"**+d* ce-F;.J c,^b.
ArpLqt|.l.) 16rL xad"ia 6
?6 srob;QC n &"o-l - "nol.r: Ona
(
.-r[E#.a-r)I -t)h q -o( b -r ak4^ .^rdl
ot-4 -.1,p-,--r,n1,
* '=1,*.) h< ob
t"g
b- o.ftrgtf
* coc6 aqJ
'l il/;
Uq +u, (eqLJ t r.J
dtov,) a o
4(r* L
L.r + ur ( b-ot w-)
+ zu:(b-a) <6w+a^r1, ( b _r{) z_ q.N -)
J-q -ti* *"u1L
,
> b-zqa.
ut,
25 of 50
hra r ur:la ,A
-" L a:; rJ
IIYIS'-Ilstml, u t{!!umu sqfii$
}IEAD OFFICE:2514. Old R.iind.r lr:s ra.'r.r. D.rhr_60. Ph.9999197625, Oli''562993' BRANCH OFFIcE: 105 r05 Io' troor'ii;i;.;:; i;;;. n;',;.;t:. N:s,.-o.rhis. REGrona! oFFcE:1-ro'237, rr,,d 202 RK's x'nch'm' ar!'
lro.' Hvd.r.b.d_20 xobil. No ' 096523s115?
-'ii.r..r-rt'..-; ll w.-.rd.n.rhr.r..hino.Gom ll EDir:.m..in.20r0@oi'ir''on P.T.O.
sn:^d_ .,$,o
a
-_------.-_.....
5. (c) A particle is thrown over a triangle from one end of a horizontal base andgrazing over the vertex falls on the--otfrr end of-tEibase. If A, B be the base
angles of the triangle and a the angle of projection, prove that tana = tan A +
tan B. [1ol
h'. B/ ) yasu +trc.^gr.,
vorh.* C G,rd, ta^a at "r"t^ g
LCA6- AB
, c(\ ,*)
2 L cG*=
A ho R-\ 3
4f fr-y"f.-"-.1?6*5Le* cD _- h /
a.{ [e .l".ta]
-)
(
y -_ .cto,.,a _ -LL
+ tL- f<Dje&hton*= k/t, , -:l.oB
-_
WR_
u) O71
q
t
IKa----,
R L2 c,. COJ,( J r.'
a9c
1A.D[^J
=)
a
.TL
.LL
u 1colh{ lal c( h,ar,:,
(f) =
E 6Lu 2co
t oo4
torrq r9
= ucoJ,( fk -_ trr...,.rt -{gtz
h -.l,na
coJ<
1-KR-h = feia a t to..B
b-t^ -,U .\
q rco-F"(
to...a
R
2 (ucor,t -r,aa;
I - h-l?Jt
) K
h (e-nrto4q k
*'"@ 4. Ra^))t
\
t
qrtt-6tanq -- +04A+ t
26 of 50
HEAD OFFTCE:25lA, Ord R.rind.r r{.o.r r..,k.r. O.lhi-60. Phxukh.ri.. ror.r, r.urh.ri.. f,esar. D.rhi,9 REG|ONAT OFF8yd...b:d,20. gobfl. xo : 09652351152
011-45629907 aRAiCH OrrrcE r05.r05. ro, Ftoo,,No 202 R.K S X.hch.m.. At!.IIVIS
llslmll o $m!t/r cu sctEtils w'.imdmti..c.m lt 'rr.in.rm.rh..r..mine.co6 [ Emit: i6...,[email protected].
l-og^rolrre t fq"j
9 k= htao< _
-
27 ot 50
5. (d) A rocket leaves the pointI unit in a straight line
.--=+(i) position vector R,'
(i i) velocity v,
(iii) unit tangent vector T,(iv)acceleration a,
(v) curvature r,
?.ocrzt stotb ot
errb Ol
(1, .2, 3) at time t - 0 arrd travels wilh constant soeedtoWffil the poinl (3.0.0]. Find, as funFti6E55Flthe
[ 10]
tY(t r -2j +fk
)
llttt
fi;
tdT.ldtl)
onOvcd-e^g tt
lgrr0 ti^{
d,^4164k
t\o-I ui ^o- I =
BF-' G
cii) "cbc,13 ,efir( J =
ttt ) J_o"k
*+*+3L fi]{C
tt) ?srtn, ,u&rq, ? =
T
-Ja+ t
( zi*.:Rdt
( ['-a )I s'- a-
( .,. trrl.-d l, )l.^^t,?2-J+:r) ++ f
-\:{n zi +L3 -s?)dr'
+a4y.^f .td d d/d+
IAR
o<olr{oti!,dv-
(21*
)-3e:
v€
)-3x)
odr
I dT/dslldfza.t o
I r- ,,,",t \9e,;. ,ufu so,, 't',\D
,,l!:?
orrlcE: 10s-106. Io, rloor,2o2 RKs x:nch.n. alu.IMS"
I,stml! 0l t[!!tltt(tl 9cEllls
HEAD OtrrCE:25/!, Ord R.iind.r n.0.. I.rk.r, O.rhi'60 Ph 9999t9'625. o'r_'55299t' SRANCH
xurh.'i.. r.r.r. xtrrh.'i;. .0rr. o.rhl-g REGtOx^! oFrlcE : 1_lo_23' llnd rr@r' Room No
S.pphn. A.hot Nasar yd.r:bad_2o. rrobil. No : 096523s1152."ii r-.r*rt'.."- tt
- w,r.rn..m.rh..r..'nr.c.con Il Emir: ihs.h.201o@sn'ir'c'm P.T.O.
/
28 of 50
HEAO OFaTCE:25/3, Ord R.jind.r :, r.rr.t. o.rhi,6o ph.9999t9762s, Otl-.55299!7. ARAXCH Offukh6ri.6 To-.r, xurh.ri.. x.o.r. D.lhn9. REGTONAI orFEE : rr 0-237. rnd Froo'. Rooh ro
P. T. O.
(e) Find the work _!bs_hr the force F = -4xJ1 r:^314i + 2k as lhe pointglelplicationmoves along lhe parabola y - x,. z = 1 from A(0, 0J) to B(2,4,1)l.-- [fO]
9;^x4 ?ito- p= _rn a , JK%, 417,-
1.., c, L)' +8y J + ZK
(L/ett)
,-10u-D, obg ?oxq6oh
y=aL) ) .^:t7.1
60101 ')
J=tL ) z =lF"- -1zA(o ) ,ri+9ir_t
f?+':-4.^. J -tK,A
= l.r-.rl, -L)
,ot,) ) t =o6 B t-,v ,,1
:
dl??
:) f-- :)
"i dl K JDr^4- bd +L-bd ------14
LrJ -- I? A7 J I '/dt.dr
I
rr}Istlsflttt! 0t tfiEllltcll scEtog
s.pphir. aih6t N.0ar Hyd.r.br;.20. rob . No:096523s1l *,,.imt4m.th..t.arninc.con tl Emir: [email protected]
5.
29 of 50
HCAO OrtrE:236, Old R,|r.d.r lr:9.r l.,r.r, o.rhi.5o Ph9999191525, AiAICX OTFICC: r0tio6, Io, Flod.lurh.4.. t.t.r, hrh.ri.. i.9.r, D.lhr-t. REGrOraAL OFFrE : l-lo-2!t, lrld rler. Roon x.. 202 R.K S K..ch.r'. 4l{.S.prhn. A.hor x:o., Hyd.r.6.d.20. fobiL l{o : 096323511s2
ll IIr.rm.n.ttr.r....rnc..oo ll EDil: in..haoi0io9h.il.G.6P.T.O.
9.J= (-',t'i+ st"j .t-ze)1 l+z ,j) . dtt o
'J ( - l*=* ,et3) dt J ru 1J dft :o t :o
1l 2
3'(2'1)q q8 lbulr,]o
Jr4qFQ .-:dkdrl,tr ., -o"-9 o(eg porohh f-rL .4
7t- tr {L. .r.r*gb
6. (a) (i) Solve x(l - x'?) dy + l2x2y - yldx = ax:rdx.
(ii) Solve lxy'* 2xT,l dx + (x,y - x3y2) dy = g.
16+6=L2l
ti, fo.lra
30 of 50
HEAD OFFTCE:25ll. Old R.jrnd., x.t.r t-t.t, O.rha-60. Ph-lrglrgt9r625.0rt.a55299t7. BRAxCx OrtrCE: lott0a. r.r trod.xukhni.. ro'.,, rukh.4.. x.0.'. D.rhi,g REcror{al oFfE€: r-r0-23r, lrrd floo.. Roon io.202 i.x3 x.'.hD.. !to.9.pphlr. A.hor x:c.r Hy&r.b.d2o. IoUr. rro : 0965235rt52
ll "'.rm.m.rh..r..rnh9.com tt EDrr: [email protected].
lltu!!l
31 of 50
6. (b) Solve ap'?+ py - x = 0. [ 10]
t
HEAD OFFICE:25r!, Ord R.iind.r n:q:' rilrr[.r. o.,hi_60 Ph 999919?525, oll_'r56299t,. SRAxCH OtfrCE] 105'106 rop Fr'o''lurh.rj.. Ior.r. n,rr.'ii. lao-. D.rhig. REGToNAL oFFlcE,l_lo_23r. ll.d Fl.o" R@n no 2o2 RxsS.pphn. Alhok t{.C.r Hyd.r:bad-20. Mobil. N.: 09652151152
ll *,w.im!.hnh..r..hins.con lr En.ir: [email protected] p.T.o.
32 of 50
6, (c) Apply the method of variation of parameters to solve the equation (x + 2l y,(2x + 5) y, + 2y = (x + 1)e'. [141
t
HEAD OfflCE:25/1, Ord R.iind.r x.0., x.'r.t D.rhi-50. ph.9999tt?625. Ort-a56293!r. aRA Cx OFFTC€: tOS-tOB. roD Fro.r.Mukh.ri.. Tot.r, urh.rj.. x.e.r, O.rhi-g. REcrot^L OFFTCC : t-t0.23?, lhd FrDor, Ro.m llo.2O2 R.xS x.nch.n.. Blu.S:pphn. A.not .s.r Hrd.r.b.d-20. loblr. xo i 096523511s2*,*.im.rm.rh..con tt rlr..6..B.th..t..hinc.con lt Emit: ii..im.2oro@em:it..om
P.T. O.
33 of 50
(d) By using l,aplace transformation solve (D' + m2)x = a sin nt' t > 0' where x' Dx'
"qr,,rt to x" and x, when t = 0, n + m. I14I
OfalCE: lO9l05, IoP Fl.d,202 a.X's l(t.n-. 6l!.
P.T.O.fi:.:::?:i'i:":lili:'j",,'j,:i";:;:'""1i,?iil;T J'"'i,"l"liii:i.ll"i'"1??331 ""H""il;;;;;ii; ^;;.i
i+., xvi.-ua'm. 'bir' xo 'q6r?rsr152#il;;:.;;;l;;; rr r.-, 'i.rh1'r'!'nrng 'oE tt tE ' ih"'hr20lqasmrrr''oh
6.
34 of 50
ttEAo oFFrcE:25,l Otd r.c.. c.rr.r, o.rht4o.,{.s.,, D.rh. a REGroxaLHyd.r.b..r.2o. tobit. xo :
1.456299a7. ARAXCH OFFICE:
rwr.i6i.m.thi.t..rnihc..om tl EDlr: im.atr2olO@qr.|r.o6
P. T. O.
7, (a) A heavy elastic string, whose natural length is 2ttct, is placed round a smoothcone whose axis is vertical and whose semivertical angle is a. If W be theweight and 2 the modulus of elasticity of the string, prove that it will be inequilibrium when in the form of a circle whose radius is
1* !-"o1o2)"n l15l
35 of 50
HEAD OrFrCE,25r3. Old R.ti.d.r l.e.r 0lr-4s629947. BRAICH OFFTCE1 ,05-105. Iop Floor.hrh.ri.. Ior... rarrh.rim x.qrr. D.rhi-g. REGTOIAL OFfrE: l-10-23r, lrnd Fl@r. R@m No 20? R.xS x:nch:n. Bru.IlYIS,- s.pphn. Ashor x.s.' Hyd.rab.d-20. Mobil. l,lo : 09652351152
It l,w.im..b.lh..r..rnrno..oo ll Emil: im..iB20lo€eo:ir.GorP.T.O.
36 of 50
HEAD OrFrE:25[, Ord R:ii,d., X.e:, r.rr.i, O.rhr-50. Pn,9999t9752s,011-155291]!r. BRAXCH OfFTCE: 105,106, To9 tro.,.ratrrh.ri.. 16*.!, [irh.{,. N.9 , oorhi-g. REG|oNAL oFaEE : 1-10-237,S.pphrr. A.h.r i.s.r ,rrd.,.b.d.20. Iobir. xo : 09652351152*wr..o.l,Drh..com ll ,'r..m.h.rh:.r.:hins..om ll ENir: [email protected]'
P.T.O.
7 (b) A body, consisting of a cone and a hemisphere on the same base, rests on arough horizontal table the hemisphere being in contact with the table, showthat the greatest height of the cone so that the equilibrium may be stable, is
.r[ times the radius of the hemisphere. I15]
37 of 50
HEAD OfFICE:?5/4. Old R.iind.r xas k.r, D.rhl-60. Ph.9999197625. 011-,15529907. BRAtlcHxukh.n.. Tor.r. Irrh.ri.. x:e.r, O.rhr-9. R€GlOll l OFrEE:1r0-23r,S.pphn. A.hor x.sar Hrd.r.b.d-20. Mobil. t{o; 09552351152rsr.im..tulh..co6 ll wrw.lm.m:rh..r.:hrns..on ll EDlr: .n!a,[email protected] .oD
OtrlcE: 105-106, To, floor,202 t.X S X..cll6. llu.
P.T.O.
,
7. (c) A particle moves with a central acceleration plr+a'frt1 being projected from
an apse at a distance a'with a velocity Za.tp Prove that it describes the
curve 12 (2+cos .60 ) = 3a'z. I2Ol
i
38 of 50
HE D OfFTCE:25/'. Ord R.iind, N:s.r X:,r.r, O.rhr-5o. rh.9999r97625, BRAXCH OfFtCE: 105-105, Top Ftoo,.Mukh€ri.6 10*6r. rurh..j.. x.e.', oFFlcE : rr0,237, ,ro. 202 R.x.sS:rphr. A.hor N.s.r Hyd.':b:d.20. x6bir. no:0965235trs2
r,'.im.mdh..r.:hine com I EE[: in..im.201o@sm:ir..omP.T. O.
39 of 50
8 (a) (i) If F is a conservative field, prove thal curlF-- V x F= O{i e'Fii irrotational}'
(ii) Conversely, if v-,. F = 0, (i'e' F is irrotatioiliffiJ'e that F is conservative'
[13]
(i) -t_.,oJU"^/( .U
()o15 :) frot &t1r.dfiaj
=)a f (.r.a,lna U".il"") -c.^c.\ .l+.n-t
'-1F
a
.-rtn#i"gi)'-::-a.tOt.) O"(l F vxF
;-;4 t" 11t-
ah*
?ta*
j%3
ahtt ( to o9 \
EF/?Z ?\d
MS"$ uln!!{lll! tllliqE
It!$mll
ilii:;.?l',lT?iiil -"1#",I "if,""';l3i:1";"1""L' 5r:
It: r6..r6.2or0iocr.rr''oh P.T. O,$'jliflitlti'1,**i:i,",.':,i:':l**lri:
vp
Kah'7.-
alh-
a
I
40 of 50
at-rrt " urlf F =9 a . .r- I
\ttOlnfiDr\ilFt4
t''
) $**, etF =o c;qdato,ol)
t |^o/( @*t6.J*
n ou, Ce-"dlt L:.
P^ ^to 6BrKt = J ,.al
(rB-qhrp
G)o^5 t,)t
Q,
A
KL =
8l r,al (o1o^g w)
fu 'Pl&N"l ,je to,q (it d7I J c" ve), a-i
c,-(r i, "@t ,r:^*-lCt-L>
o
-)l .-' vrFl-_ 9ct / i.a.,- - oCu
4JV.av - J F.a7 9
--[^.
b"q .!,^t^ot nc L
d'r7.<^d"\ +r4". .)b f '. ,.c^otctir.o/
f+o-, F.+, (M] ,ilrr7+e",. {Le
^o."*---
t
#i1",'i1iil;,91 i,lii'.il:l -":#i,:, .,1'.Titi:.".;L,".li::I Emn: [email protected]
HEAD-OirrCE 2r! -Ord R.t,.d.r ri.C.r
x:o.r H,d.'.b.d.20'rr .__.i^r-ar,.a._i,"c."-
P. T. O.
a,
= bn.,ztu+r-
(i) Find the angle of intersection at (4, -3,2) of spheres x2+y2+22=29 u.;16P+y2 tz,+4x-6y-8247 = O.
(ii) In what direction from the point (1,3,2) is the directional derivative ofi = 2xz - y2 a maximum? What is the magnitude of this maximum ? [12]
(/ ) eJ;,c-, Jr: ".'*f* *- t1 = O -c)
.lptut
rp[rrr
8.(b)
ad.\ol
p&"noJ "edd
A)
.rd
%J1 'L v o rz ' ( r 21 ,a* t_L- t_11
h;
^az'(r+rl jn lzive(to{ol (u,-s,21
i,'( F+
+3
*7 ,lK
J + 2F
I
/tg ( .' l- :i*zi),4r t- ,
P ( rt+ Ut*
rzy^6) ?+ tn-Dp(tt
v1_
ld""n6,f
",ohot ,e&d.
( tzt -rr i
a\r+
-qa,_3j3
K
eEV4r J,h*.tD-, 5(s?-sl e)* lrr,-7-; r. *ghrz +9_z)
I;)
e,1,t"n h"."
z$ e) coj 4. = r
t2-
&"r.]"
Lq xiq IEfrE
(
&"" o = #_u'of c'pe"f n vl = Z-L; -zyi
(- COJ
41 of 50
irc^o ofFrc€:251t. ord R.jindir Nas.r uari.r, o.rhi5o. Ph.9999191525, 01
l.!rh.rl.. Ior.r. Lolh..i.. x.9.r, o.rhig REGTOi|A! OffrCE : 1.10:Sapphn. Arhok N,sar Hydsr .b.d-20. ilotil. llo ; 09652351
OFFICa: l0s-106. Top Floo,.202 R.X'S x.nch.h. aru.
P.T.O.II}ISi"
rlmflr 0t ttl:0unclt srlDtgls ll ,wr iB1mrh.d..dins. Emn: ih.1im'[email protected]
@
lt- = {.+yL+ .. *,.1 r _ ay _ gz _V.? =o
o* q,( *6y -az _v)
c)
42 ot 50
j
a1v 1
-1= or
LJa Fv\Or! lLOtr - -l--d^leutr-tr) O rl
d;r.f,teno/ drx;t&w Jp P afo"g
v t.t$E .a ,na.o,n,r.rn LJr.!-,n Vj1._a Pg
d^ref;a.,^ol"nol "ectd
olr"^g
l&"nal ,.to( of (,13,t;
9 io +.e a;lcfrea JIV
rogn4,t^d.{ +1-rpl*r lql
dk *o"rr cl"$,,,ot_.- ;
6 d;+t-o"a &t,4a*',>.\:=
J".ulL laadJt
8, (c) Prove that curl [r" (a x r)] = (n+2] rna - nrn-2 (r'a) r, whre a is a constantvector. tlOl
be" h"d curr ( ra6q1y))
(vrn) r rqx?) 1n ( cu.r1 6q
-6i+zil
f -)
'D
i
I
, o'(lr*) t 7x (d.,.7)) +'*n o * r:j)o" -,v)
ll
t=o) (3
-!ia)(d'"1/(,\
( ) - (v.dt) itl
Ls))
n ( '','' d - (7,d )7 ) -(dr)) _ oq t^ Co^rQf veobrr
-L'l +< -G5
(( -1 -l
ql-e i *"tzi-s3 r I ){,9
= e, i+otj +os ilet il.,' 'ctttL(Sa1
=vQr* +
Fcqt A
'.v)A +(v,B)A -c*tr( Ar b) =(e
(v,A)Bt 4-? )B-
IltrSi'ttsflnill rrt1oifitcr,scE$E!
q
.9
IIEAO OFrlcE:25/t. Ord R.jhd.' x.s.r x.rr.l, O.lhi-60. ph 99t919r625, 0ir.a55299t7. ARAllCH OfFtCE: rO5-106. To, Froo,.xrrh.rj.. Tor€r, Mukh"j.. l{.s.r, o.rhi9. REGTof,AL oFFEE I1-10-2!7, lnd Froo,. Room xo.2o2 R.x,s x.nch.m,. Blu.S.pphir. A.hol 1i.9., hrd.r.b.d-20. taobfl. xo : O965235ttszll I}w.ihr.m.th..r..'nins.com,l Eh..t: imrrim:[email protected]
P.T.O.
h"", O
r:t'qt
Z=tt-t*tryZ)
4" h.,-"dar^7
obo."<<v daru-+-
dU {t{
J"6, s,1=z I (Ar,.z)f:i6@o cl.v)fl. (y,dl).1=o
f '.' n- (o^Alo\E]
4o,1 ,a 4,1i- 2
f( r.o) r. L O.. t
/t . cr. {- noa.r-lt
8. (d) By converting into a line integral evaluate
ff tv ,. r't. nd.JJS'
wherel'=(x2+y-4) i + 3xy J + {2xy + z,l k and S is the surface of theparaboloid z = 4 - lx2 + y'?) above the xy-plane. I15l
fu"re- .l.ralpU I L JI
Jr i,.t Jr' nLa"+S -l'- o
J.rr,lAO{ = "r
L+.lLr -Lt
= O i z=o
J
(
JJ
(v1r; 'n d.r
$ r.aiJl
b JU )kor.lk-du46
I
o-Lc? ((.*" ) n' 4t 1,4) i
-ts
43 of 50
HEIO OFfrCEi25rt, Ord R.iind., x.e.r l.rr.r. O.lhr'60 Ph 9999191525. 0tl_a562991r. AR xCH OrFrCa: 105_106. IoP Floor.Iort.4.. ior.r, gurh.ri;. t{.t.r. o.rnr.9. REGIOT{AI OFFTCE: l_10_237. rLd Fr@.. R-6 io ?Ol ixS x...hD'. Br,.il r\1
llstmn $ t$!!t{!lcr! tcElllsS,, S.rphr.. A.hol X.C.r Hyd.r.b.d-20.
'',.rm.a rh..on lt r* i8.6:thr.l..hino.com lt €mil: iD..im2010€cm.rl.c.iP.T. O.
Vo !^-d .
44 of 50
dhoe- _[, : *"+yL,{ -o i ?-=o
-,, *-- zcoso ) g -- 2Jl4 O
z --oO ttorn o to zrr
' 4 ?--
!*- F: =
z coro ? .r- ,A,4OJ
( *'+ 9 -y) a +3a uJ + (z<yn zL )t( ,r.oJt.s + 2_Jl4c, _ Lt)
Li+ rz co)o-!he i+ 8coros,.,o
"j t '€'1do) ,aoo
= "jI qcorLo + ,-Jrao- q) i+ 3 rz coroo
0j +f cos-l:no ?J
-l'f e
f-zr,.,o? *r.oro3 ] an
,corlo Ji.,O + q J;4Lo | 8 J;aOo ( =o)+ 2q corto J.g
t o.,laot =o)
1
1 € v-.r;',r-6,, ao
- 1:rt t(- 2)
J (,i *r,n) doo
- \ i-i-
dl""tt rt, kaa*
t
lJrvxtr) . ndr- vTI
II}rS HEAD OFfEE:25r4. Ord R.iind.r n.s.' r.'r.r. o.rrl.a0. ph 9999197625. oti.a56it99ar. aiaxcH oFFtc€: totroS. ToD Fr@.,Iurh.,].. Tor.r. rurh.,i.. x.g,r. o.rhl-g. iacroral oFfrcE r.t-t0-217, rhd Ftoor. iem rro.2o2 R.xs x..ch;t Bt".S.prhlr. A.ior ta.g.r Hrir.r.b.&2o. I.btr. xo: Ota52!!tr52
tl rw jr.nnn..r..mhe.coD Edfl: i'..iD.20rooci.lt..o,P.T.O.
Nou)
/
END OF THE EXAMINATION
/
T
45 of 50
ROUGH SPACE
w.)l
tG I
1
P.T.O.
H€AD OfftCE:2sr!. Ord R.t D.lhi-5o. Pn.999919'525 011_'55299!7 ARAlrCrl OfFrEliiii-i.. i"-:-. i"ri-il. x.s... D.rhr.e. iEcloxat otrrcE : rno'2r7. trld cr@'. Roon io 2o2 R x
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i
iiiS^9,Ijf !..1-TroiL !:1,.*, ".r:. .lr]l,j.t _o-"n,60. prr.e!e!r!1625,ofl,.562!'9rr. aR^icr.r oFFEE: r05-ioc. rop Ilod.rurn-.4..- Io{.r,.ruh.4.. i.e.,, o.rh{.g. REGtoriaL oFfEC : i-10-23r, rhd fter, imn io. 202 R.x.s xrn.h.;.r Btuai.c., ar-'.0..!20. rob,t. xo: o!532r5tt52rtrjB.hrh..r..rnhs.co6 ll ED : ia..rB2olo(!en.l,..oiP.T. O.
INDIA,S No. T I N SIITU TE FOR IAS/I Fo s ExAMINATIoNS
I1\{S"trxSTlYUrE Oa MATIEUATtA! SC|EICES]
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O Ph.:0ll-45529982 9999197525 f]) wwwjmsemaths.com @e-Mail: ims4matE@gmail'com
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HEAD OFFtCE.2t8, Old Ralinde, Nagar M.rtet, Delhi-60. Ph.(m9197625, 011-45629987. BRAI{CH OFFICE:.10t106,Top Fl6r. Muhherree Iower, Mukheri€e Nagar. Delhi'g. REGIOIIAL OFFICE : 1'10_237, llnd Floor' Room No 202
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HEAD OFFTCE:ztE, Otd Rajinder .gar tlarrei Dethi-60. ph.9999197625, 01,-45629987. ARANCB OFFTCE: .t0t106,
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