OffiI{FsIET UEft[

q$H
ol
a qE qffir< f is <E<T{r{< c<Ni erem f<cSsq:

.{16 M M M M M
cQ CQ CQ cQ CQ
c C c C c
s rl E a s { d { a F rl rt s o F rt n \ o o { 'i { o
Elal 2 2 3 2 2 2 2 5 1 I 2 I I I 3
<rq{Er 2 2 2 4 2 2 2 2 4 2 I I 2 I 1 2
rlEr:r 2 2 2 4 l 1 2 2 5 I t 3 I 3
1166 2 2 2 4 2 2 2 2 5 l I 2 I 1 I
<frfrq 2 2 2 2 4 2 2 2 2 5 2 1 I 2 1 I 4
rrll< I I 4 2 2 2 2 l I
woqr{16
5 3 I 3

Tfrf,t 2 2 2 2 3 i 1 I 5 1 I 1 1 3
Frq]o1i{ 2 2 2 4 2 2 2 5 2 1 3
a5q'{fr1q I 2 2 3 I I 5

ft.u.- qo\o qtrq ffi6 'iawt q-{frs <s {lt q<( to)r cl-r{ {s,-q 6lfli q&{ qs "iqfus "rATI q{6s qs r

o 6ffql< aft' m c-< cqs <tFrs effi{ s.I(g re{ fir{d qrTEi{ E{ El[q-?rr$ ffial|< <tFt <cq r

o cet-< nft: cl {< de rFi6o effi.t c*rs qF s fr"s E-sLTn sc{M*{ E{ Efch-<rs ret< Aft <ra I
o sq cstr{: GrolzlI Erq< E K qfrs q&E{ <<T<-{ qsB F?-d <l-r qR seB< 6s-ds'ffi< wtq. stro & srr< sq
F?-{ (Area Vector) <t {FI Cs-q FB-< I

<9 qsr (gt{: F {F

(< qs -rtrd
q66 s1-56 qFs (sE< <F I

o q<EFr cst<: c{ FA-6fi $fi{ {{ fi-6 wfcre ssB R{< q{qF ffit F<T E{ q-rs q<EF (EA-d <t <r-ct{
ei< <rq r

efr oB< : qF e?-<ie cfi il fuWq< cfr fu <tq qr$<at@rl fir* \e fr rs {is s-{r ccE fr -xcq< p6r aEE
fr"tAs@c{ csi<EE<qfr M{Ffi< r &.e, F + d = f,

Rl /7c
t..,
--{ 1A
t'={
tro' &yu1o
o cs?rr< c{6$Im er&rs $cnq{ = {E FA-< x 1r e?-< c<t & ftrc< wsft, 6skq-zr cos r

o qft Rr<t"flr+< cfts{ csr4 @kE<r c{rk 6aga gfi-q1 6Q 1 ets yv ffi+n-o errw fi+ a<rc< Rg 69
qfuqrr-+lr
0 ffiq&rs{E{ 6s{k<tFt q<(B"fi(,tE{ d-<<lFt r

o 5E cvB-< ctrct< mro aft, n = /er1. q, * ,a*o*

I

N]

-
 qqt{fr'Eld bEI ot'Jt : ulriflE-oi
O @ frmnt< mm sR, n = .F7TE -,pa -. t.ltr ftrr qTrqfr-{
ei,-<
"
c<c'j-< lli'l,€{t Tlt rI
(i) u = 0', R,,,"* = P+Q [(--+ +-.+) =-+]
(ii) c = 180", R-1n = P-Q [(-+ + e) =-+ {$t, P > Q]
(iii) c = e0', R, = VTZT at [(-+ +1) =,,]
(iv) zR'z, = R'z*r* + R2-in
(v)R*lnSRSR-a,
o I e<jB Ft{rfr x,y € z q[EI {Ic3l {ell4crt fl, p € y 6otci bqffi srd = 1qa
v-6{, cos2 c + cos2 p + cos2 y
$$rq clGsirdt EiE 6sH qFf qlFEFr qer<E fi4x +-<t rt{ r

o GTriFIr{I frr{-d qF.F Ftd !I-dt & frr<r rIftE crlGil cArsd qFrir ecI $<rE eqE @A-<D {€{III{I I

o @ o3'-< e<F wtoRr+< tE <tq fu6t +-+rq, qtqrq s{F s66 efr ,{<( q"K s{E vffi6 qga3; qpp-6 I

o Ffistrr{t{: frEft;t< sFnT srfc-{iEM qrEt qfr"(g 1ffiftq TEI&TE q(t {qqfd eFF rE r

o q"l[r<F<: q< q&{FnE q{ qrql Ddrs <t >R6{ <t T'l{T]-:rs I qF ,4F {{r{i rtfiFf Efr-s <l q(r$s I 6{
{nFs eKFt{ Fr<.n Effi q<:EdlFft-r\l-qr.{sFdlFtcam{rqd<r{t{t{<t mtnt qRcr+ft'e <tftd<rMI
CqTT IIIT E'ItrF \T'[(<Ed <TE I

o eE-< Mmrq qffi?: qro V <t v q-fi et$Ft qdl qrr qB qstr
'fififrs
qrtr<iB-d !I ffi-s Ctc"{
FArs-d U"lti fqrqld afua5q ffi1 q5q I

o qsfts-<.r: wgfts-d.l c4 ,{vq q<58 Efu{t ${ {tqrm qsB D-{r€d €'Er && Tcr ffi'Ir{l Trrrrr{{ Et\sft{
qffi{iEkfi4rr'firkr
o cffiq-{<ctqsft-firfrfrqAE sfr{tr qs+sfl*r{t{icqFt(.tfldTqET "tffifi4rc<tqrr
c{Ftfr$-dct:
q<( RFlfr€{c*r ffiro qt
TsE{ "tR<\i{ cnfr-s <'c-< qtffi "Irlfd rFt{,iir ffi<s 'lt€$ {'s I

o csB-* ffi-qq q"ftdd frrq qF\3 ftR sql: (i) .{t q{'ffi m< $fr&{ eFlT qtfr$tit rt-T{ I

(ii) ftfi (r<5 '6sEI'dFrs-{cl oGr{ I (iii) q?qdITN alrfit | (iv) 6g.E, i= d3i
+
dyCi+ *k
(v) {l{E t eir+< xcq e<s Gi-d {{ tqt crfr+ esB TfFk s"R EF{t Tr{ "E-{ e<F <tft< @ +r<
{cn(q r r

o rqqk mq: sr+tcqf (s-q fir{F{t Fdt Rls { Cq+, rerq{ s&E R{r {ftel qsF ds sct Ts c|tr{r I fficrd
clret {(ft? cqso"l {fr 6satl-?r qr, sr< .i mr-+ fsEtl3r cEI <rEr EqEl"f: q-{y, EsEt, frrr-< ffir
ll,l
1ffiavtr<- 0(x, y, z) = xzy + Zxyz + Szx
I
o cst{ (s-q: csGl-?r cf(q c€E dEe.t {fr FB< ss, sr{ & ffiElr-r 6si-d 6EI <16r r 6$r{: (<rt, s&sdlTqi 'l

ffirF(x,y,z) = ar'yi + bxzyzzi + 4zxzk

o 6{rII tffiS]Er: i (& y, z) qTF F?-d ffi qFI crlc{f qf{q 'lcct c++fefiE iEdl-{ f v . aJ r

e{trt ai qFrq "Icqli q$E q\.f a1-d qfu.t il q<( q&{1 Q qq1"i<r -et{-r 331111-< I

I E
 qqMfrsrd lEI rt3[ : o[TIIIT-ot

cdffi:
Ql-< erstt sr< I qrs 6{4Er qc{s-s€ (FI I

o rfr qsF <lTfq-{I{FR 6{6rk (Er q{, \s-r{ io Et-.<t o q< 6dfuKB6f E<Fl"l q<f qs I

srad o = na = (ifi + ira + nfi)o - iff + iff + nff

cqffi<dvsts"ff:
(i) csq]< dftd 6rlffi qsB Ftd m qdte,, q$tr F?-< <tft r

1i1 Ee eB-< <tFt< ffi & e6-q< {fi<

q'ftffs $< qt<< {ffillt {:1fi I

1ii1 cr"rm aft< lffia ts{ R{< q{r$-{ E"rdQ fr6 or< +, 6qft6E q1 {ffi cq"rn-{ <* frftrffi
e"r<effiomr

r+nm q$6 @b-{ csrqd sr+qlq oi-<, V(x, v, z) = iv, + iE + Qv, sr<q V . V 6r & ei<rqm< q&r<rqiT
a5q1flaGsE[5q-
i V = (i*+i*+ Rf). (v,i+ v,i+v,[) ='* ** **
@E-{ FR l-l <t ffi@< utfut<reeer+ estr 64< +Rfi { sFt CIB<mre< r+nnt R1ro $rs{ er1G qm {F r

qi-srqcqrq< c6s q{:

(i) glasEcqq EFr qs?F qENr{ ft-sdFr< d1B s-qf F]s crlrdT ft-T
6o1-6[ * *qflft qcq selt u
Est-t s.r< | . V <t aiv V sRf qqq q:rm 6sfnf E<ar "r,ilr{< q*-W qffiq( <k d<tF
i r

(i, {I{ tr{rqs qr{, s+q qEftd< ql{rsq Ifr "nr; s'{('gr Rl'{ s6 r q{q, v . v =' +' ve
(iiD{F{ 1t6fl-qsqrq, qltEr{-?rftrslFrwt;mg1k tmr sft, V.V ='-'u"
(rv) {FWW{, qI'lE s fifu3nsc-flrqr q{ts, i. V = o

(v) mltril Fftrs@ds1t\r=FrqqW<Eq{8, V.V = o <q, st FAl ffirs >lfr;BrslE (solenoidal) {rE r

.ts4.
-),{- JlL
Y{ f*
re-/>
qapqov-ftwrrw q.nEl-oqfrErqE avuq-w<rwa
o +r(:
+6: mnfffi-oqtrqd-dFFaFI0(x, y,z) =vri+Yrj+ vrt<n,v"1rd<V,a<(Vq<q;{e"1Erss6<6qr
cs[tril @A-{r$c" +F( esB F < Tlft {l & ffirq{ T{rq< cE"t q"1fur oE-< cmrq q<fu q+& frA<
E-afr5q q1 alt{ ffidlffid qrq qG es-s cffi{'r{ qrft F E(q q Ss fr -trs Ff< mre< F,H ETI"I sr< r

ffdq<dgsts"r(:
(i) sF{ ,{sE<tft r er:ill-{ & 6s3<ffirq{ eFs 6Frqd qri rr6ks 6<tf tFdfrcF cil{ I
@b-d

14 cvffi<ft-+ & cFrEFr sol<rqREEtr<t-<r<fr-stomr

1ii1 nr4 e< Tt$m EIs Fffid rt{ 1fi vrm a6a6p ffift6 Gr{< ftaq Esr qdts, i = d x i qre,
V x i = 26qs, C{{rr{ 6 q$F s{ Ffn I

csnm FALT{ sH Tf{ fr 16"r +r< r mtifi fra< ut<fr-co effi ewr<
& GAL<{ 1c< nr{ iot ffi I +m r

c+lrlt FA-( 6srE{ +trd< vttstrmq o r q{R, V . (V x f) = o r

6sm F [Td $I4 o <q r\,+&F q14{ftq s{ r q{t\, i x F = o <rn, F vq4ltr r

 ,tql{frE|il bEI
'EI : EITIIIT-oI

EE csbm{ Efr fi4cr{ {l*fsfro {E:

. F e Q qfr ei< <tft (c<ut, <q, .rd"r, V<{, q|sry', !-{-{ ffi) aq vtrm efr,
f, = F + d q<( Ek< !1a, I
R I = Jp, + a-z +, p
a .". 0, {rFr 0 = F s Q,a< ws(o cO"t r

Efr E,F FEG< rTcel o mFI?sfr 6d6ol, o = t n-,r*e.o.r...]4

q<qd FAre< cftqt g cor.tffi c<rq, B = trr-'o*r.o.r_l!l!L

F s Q e< 1qea Efr-a {F[, R-", = p + e,4<(Tu\i{ ekr llT, Rmin = p-Q
. Et<fqsftrcdc-T{qk,svIr{c1nltsfr-s{@fisk{l.fi{rGrE<EElrF{mq{,IFq€fr-lpofqqrrc-cr<r
csemdfrrstrt:
. F, dqfiRE<ERPEffitsR[T<T{1fi-< e+Fnk e v"r<Efr6sF|q-qFrd.tst-<1 C{{{-
r-----+
q? frFi'If ( - l-)
uN- turF*.ffis(d-d)
?
q"rltl, lF - 0l = JFmt - rpGoso ; [c{?[rq 0 q'r F €d e<{!r{q 6{I"t r]

o}r<-<qtE"ttct:
. xy.o-req-<QE e+E e3-< p <fr x qm<
firq e coq Oqtq s'c< EfqcE,
. x T{K{ E{t(r.lT {FI, Px = Pcosg
qs
. y q5 aa;qa $qBr6 {R, Py = Psin0
ffio qmsror< {aI6 <r<4r csir<< ft-srs.a:
. F= p,i+ pyi+ p,tqcE, lFl =
.ip,., +
pi + p;

eltcl, i, jsR<rmx,ys z vt<-<r{<ffie q-oo d-< q<( p*, p, € pz qrEtq{EK{<l{< F d.r<< S"fl\rRrT{ I

. F effi<rcr<< <I q<{qrsflcqfu-ffftq qFF cEB-< = = IIPI Pg

-lei+e,"+el
1_
(g?r{<ecFFE:
sb e.Fr
. I. E = AB cos 0, crllr{ e S(qt F?-dKTdrqf{S 6sllr
. d = a*i + Avi + a,Q q<(E = B*i + Byi + B,R<q, f . d = axBx + AyBy + AzBz
. F<<r+<Q q<q?EE[s,t = ecoso q<(b{t$ ef< = Qcoso x (Fc<f+<ffiq qFF@A-d) I

. 0 <<n< F ot"<F<EtqGrmq = P cos 0 q<(Bflt\.| @B-d = p cos 0 x (0 m<lffiq qco d-<) r

. dtc?-{ErEC{, I E=o;
. eTE @B-{ f c< x, y € z qrqrqlcq cTFt {cg6[:r o, g e y qGT, cos n = *, U = .o.v =
fii, H, ffi
F{ ecFt:
. IxE=ABsin0n Grtl6{ e ErqIcgi-<w{-{TffiffcrF[q<( t ccErF -<Et-qrq6-ssrq<s"fi-roes-r
FA<FI{dGq;Fr6{)

. a-
li i kl
= a-i + avi+ A,R .a<( E' = rxi+Byi+B,kErEr,Ax r r=
li: l; t;l

rMm
. PxQ=-QxP
\\\ lffi+*qvrlfrF{qrtr... (I
 rm{friEEl 5Er[g

. 5Fd-<wro<n<q,f xs=oqRI , *=fr=fr

. erF qFflEfrs rfl€<<R&qE a, 6, i s6a,a<qwsq = a. G x 0 = 6. tt x il = i . (a x 6)
sfr Gqtr F < qrsAr q{, E({ Tfis?qssr o),
T,(V;, "h

v=a (6xd)=6.(dxa)=r.(dxQ=lu,. t, t,l

lc, c, c,l
3 lfte GlE qi@fs:
T-{srE\f (c{lqlI&) lrnE:[JI:m
tt
ol,l ,4*
I. I Vn" ,u
}iT{ =
?- qriT nEJ Qf\,- qI{rcFIBT ' .min _dd
' vsinc - iiii6: - i
, ffio6 Eft 6rf = vsinfl = \fir=G
- e ffi1;6qfr c<'l = ,l$l?
c- firqt{& trd<R*ll = .or-, (-}) .- q1N;qqkqlffid<qfr C<"f< ffi.t,
" 0 = tan-1 (r) = tan-'(S), nqm, x {Ffts q<"t r

eefl-frgq{6t, x = u x tmin
cffiqlTffE<9, s=.|d'+l'
cs6-d
eB-< q"rfr<F<.' 7 =
I ,. i s .,'. p 1
i dx 'dy dz
ffif<t (EI e (x, y, z) q< dil&Irt, grad rp = f,1, = iff + iff + t f
ogtr< m V(r y, z) = vri + vri + v:8,a< uliJe-lfiqq, div V = V . V =
* ** *#
. ot<mV(ry,z)=vri+vzi+vsRe<s.6, curtV= V,,V=|*
A *l
60< ftwm frans *T7Hn;T "* E-c< sn << +

/-))r-qtK=u

.,*"*Tz''*

qtcal q:Eoftct a[r+ EIFI 0

slq Ek aF Flt-<< = 90' q<( p > Q P

o= .o'-' (- f) = .o'-, (- @J , lRl = lFz - az

I

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I

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 etrl
'FiI{frsB-tEI oEI

qfr n, F e<qft{ 0, io Q,r<{-c{ e, c+Ft$q"rqor< r

qi ({]-{d
{ld ckrt c$Ft gl<

Qsina
I _/ sl< etrr< i{Emn
Fe( {-c.t c$l.t,ta, e, =_Ffffi
{<clr?trsFt
\ sm{rq c+Fr fitc"a
crfioerl<qr{ {<lird qoR FAIT{ qBrs4
o w{Grcrff: q1-6qfr-r c<a fi4cr{ 6arq ffi qk'rffi qkifr-s
fi{T {{(g q(< eEE C<ri crs El< 6<rft"F FA-d
c<rt
ftmtulcgCoqr<rA€Bq<ef{sFtio e vs {6{, q1qkqft-s C{, ie.. = vs -?a
A q< {ftalrqF B
o Partial qtq {rlq s-{k {I-s Partial differentiation $3IF <tfFql qr\e- fllcr I qrTr"t cq E-{rfd {tc'[rsF F{t eT EIrf
Efgt W$ E-mErEIr$ S?FF {Gtr differentiation S-{FI {R(I E{ | mfi. O = 3x2yz
- yz2 + xy
-* = 6*y, + y [y, z CS {F TC{ SiliF {trd x fi IIC9I($ differenriation S!tr}

a McQ:fr'lio<ET+{r{rmdq$eqfiffi: )
ot. ii+]i+ m[c$Eeos@E-{qc{, m q<rf{{re-? 04. @A{rf 6sEtl<r ilft sffi e.t T-{c{ e"lTET erB @A-{

@+ <u); IDB'zzl
qr<-fuT<cst{E? IRB'221
(a)5ast (b){6{dslE
@; (d) I
(c) Dt"t (d)qlE{{
{qf${: (a); I: +
ii }i+ rnR
Nt{Fr: (b); <rffi q]E = <q ;a qag = f ys= 1f,

Er=
li'
l-+-+m2
os. (ixf) x (ix R) =r lRB',22l

"J4
4 (u) i (b) i (c)[ (d) -i
,FIttlF[ (a); i x | = P.ei r P - -"
+12=1+m2+m2 :1-1 =1.'.-:+1
,
2 -r,
2
smrq, (i x i) x (i x R) = R x -j = i
02. I=zi+zrEi offi y-qrffil {-ral ss cfft6[
ffiqz IDB'221
06. ftcE-d mRF< qtvrq 6{ErK ffit celrs NA-d re
,fl\3{t {l{?
(a) 0" (b) 30' (c) 60" (d) e0' [RB'221[Ans: a]
rFrt{Fr: (b); y-q6F< rkql mT.f, (a) 6nGr{"t (b)s]tqfls-q
(")sH (OftE$r
,, = .,,-, (ffi) =.",-, * =,0. oz. i o<r (i + f) ofr<q[Td l<-r<q 6sn ss?
(a) 0' O) 4s' (c) 60"
tRB'22]
(d) e0.
03. 6{tFI Fartr{nt(51<eet<F [DB,22llAns: cl q':rHR: (b); F?-<ElTd{ry{s Frt,
(DqG-rrsE (i, {(<sctft{ (iii) q1{T+{
fr[D<c{HE{6's?
e = cos -'
ffif0J = cos -' (#-,,)

rEm (a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii

\\\
= co.-'(f) = +s'

qa<6qr@
 em{frsraowqa
N
frru<rslltrr{Ekat8te ISB'22; Ctg.B'221 wt{l{: (b);vF-{&qfiNE{k<, f xE q<fr-s
(a) C-<cf (b)c,{ffi fr[D<ft-rs4q, od q<frror
(")oF( (Og.lqsBrq{T 15. i q<( i cq Effi qqF q \offid a{frrs .4s:F @t<
{:It{Fr: (d); et<sftq<rqq estr 6{Eln
c+rq-d
scqr- IBB'221
<lftr +Er"t cs{ q"ltdr<i (D cftel vt E"ffl{ (a)t(ixi) (b)+(ixR)
{t{rot Ftrd ctct-:r vlEst<rqq +-<t q{ ffi r
(c)+(txi) (d)t(ix[)
09. i s R qhs qFF F?-ds"{ fl sre ir'<@ of srl<
E"r< qt er-r FB-nqffif-
{:rFtFr: (a);+ (ix ) = tt; R r ie<r[ r i
(ixi)
lcts.B'221
(c)(Rxi) (e (ixi) lRl
: Vtz: t
<"1(i*A) 6;
qqtqFr:(a);ixR=i;iri 16. i = xzi-zzi+ yR qre, ai"V =r IBB'221
q<(i+fr; lil =J7=t (a) 2x (b) 3i (c) zxi (d) xi
qqr{ld: (a); divV = (*i +
10. 3 N € 4 N $rfidE<E q<fifr1re q<-,KETsIc< +
13i *R)
frtt q*rE era-d FA-{ e"Frm< rF {r{- lcte.B,2zl (x'zi- zzj+yQ) = zxt o * o = 2x
(a)0N (b) 5N (c)7N (d) l2N tz. ry+srffr{ q{:r{Ir{< q<( fr1AETat FB-<rs ff
:flIqFr: (d); (R, lFl = ar.r q11 |q| = a5 <FI? IBB'22; Din.B'211[Ans: cl
i. opo = eo" . lF = lFl . lQl sin epq (a)Rflfts@?-<
=3x4xsin90.=12N
'dl 1l)fr+sqF?-d
(c)@e?-< (d){1't FA<
tt. d=i+j+RosffiqsB lcts.B'221
(i),{qrci-d ii1a6o oE< (i)frFrr{E (ii)<t-{{E
(iii)q{qllcsB(
{(r{f't {E
(iii)
ffru<mrEc6-{z
fiiarcontrc8rrz
(a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii
ii
(a) i, (b) i, iii (c) ii, iii (d) i, ii, iii
cqt{Fr (c); lF,il=,.PTTTT? = J5 +
... Ie$Ec6'rcaarwo<, lIl =,/-z+t
o
{:rtttFt: (a); f s a EF FA< ScE, I. E'= E' d
.'. f
err$ F?-{T{
tRfirr{q
tz.
r

I. (r +d) = I. B+d dt<t-qWt

ix(ix[) = +sz lsB'22, RB'2ll
I (E d) qrrsr, FRq @A-< \B 6$ffd<tFn
i 6E{l<
(a) -Q (b) 0 (") Q (d)
eql{gtT{r
t 0, t ) = ix i = -[
qmqFl: (u)ri
r:. d = iE = zi + [,i e Eod,c-q<q{Fq@Ff-tss,zz1 ts. erF <e fiffi firq o q<V sG's,{ Fc< EIrs ft
(a)25.72" (b\26.s7" (c)90.67. (d) 180.25. TC6T? UB'22llAns: bl
rFt{Fr: (b); Fi-micr< x<i-<ff mt't, (a) \5KqFI FB-< (b) {trf Fi-{

o=cos-,(ffi)=.".-,{#H} (c) qFr FB-{ (OTEr\rt-{

20. f s E e< FFt F{k c+,rE FA< rT<t-rrr firm<
= .o.-, (.fJ = ."'-. (;J = 26.s7" rsaiin _
IJB'221
14. fEfq A e S' C{ \e-r{ qICE poe oQ vr+< B"f< ntr (0 rE4 q{tr{ Elffirs cn]-giFtslGf qFIlBr <I q{
i x Eqnft<F- [SB'22; CB'2rl I
(ii) d q< qtfifrTr$ e< q,(ktrc qr"Fr $r< I
,{F qtfrfrT eBs E' e< ft{ fr-aro a-rerat Er+ <r
(iii) s q< qtftft-Tro f c< fr(kacs q"rr or< E
,{< frdkT celr+ I q< qtfrfrTrs q-rcrfiI
bl{rq{
r{rE-d cqqD ca's?
1ay6?u<fi"ro 0)00qnfrclo (a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii
(c)I,s<qT|girFr (d)Eq{wts<l{ >r:r*tl+: fi6$ E-sd ffiy; rto Ev< <c< ii

)t
 qliIdfrGGI 58re
21. i = zi + :ic<i E = mi - zir m qT {FI Suqr{, 27. T@i{,{sE- [Din.B'221
FA-<!-{ q?? lJB',22l
"K-i< (a) qFT 6sa-d (b)w[Gril@?-{
(a) -3 o) -; c); (d) s
(c)cn-sATFBr (d) qsF Fb-d
TfirfFI: (d); FE-{E{ "i<-ldE?E6{, i.E = o
+ (zi+ ai). (mi-zi) = 0 + 2m-6 = 0.'. m = 3
r{Kfd:(d):
l$l =
ffi = .E= r=,
zz. <fr <p (xy,z) c{E {r<{:qft{ ffilllt ewt qT vr< . sffi erF qs-s Fi-<r
ErrA<rq? [JB'221[Ans: c] 28. I e E et"< smr e+tr ffitgkm{ Fftr fiffi s6E,
(a) \{"[rfid (b)F6 ql{fsRrs-d cFqEE c+fqB qr<? pir.r'221[Ans: bl
(c) 6d&-rs+ (OsE-\,1inqq (")I.E 6yildxEl c)iE.El (a) lixal
29. F e Q @ oA< aFm ttr<ff 6rl"1 1z0.qn EIrnrr
qR-<ql{- IDin.B'22]
lcB'221 (a)P+Q a, + Pa-
O) !F,+
(c)P-Q (d) JP, + a, - Pa

oz <]f€:rFr-?rU9fl$Ts? {qtrlq: (d); ;F + dl = JFz+ at+ ,Pacos 1ro"

(a) 10.30N (b) 12.01N (c) 17.14N (d)33.28N
T5[qFI: (a)i OZ <irFr-?r <FIit Bt(t,
=
JP,
+ Q, + zPQ x (-:) = r/ez + qz - pq

30. WFar{dcsrq- IMB'22llAns: cl

Foz = F cos(90' 31') -
= 20 x cos 59' = 10.30 N (DWFA(r<$ffifr{6t
24. lRl= ldl+ lBl= o<cr- ICB'22I, (ii)Tff FIr{< cF@y<q qro
(a){qF?<EEwqTl{ e Rqffqt (iiD.FJ FBL<< vtft fu e ew fu uoi fr1rs rrtr+

O) {T 6etd1Fqcffi \e cnTQf ffru-d6{HEcEs?

-
(c){TFi-{EE:i-{l{€c{{$ (a) i, ii iii (c) i, iii (d) i, ii, iii
(b) ii,

(d) T{ FE-dEFqqlq e fuA-s{$ fi tu-d Eft"fiT{ qtrEtrs qesl E-sq qt,s

"rr<6 EQE :

{:rt{Fr: (d); erswryBfrql{tr r Er$ ii,tffiqrs@tdMtqr<r

"lRI = ldl + lE'l = o" €< <!r-6{ "R = I+E = d" :t. i. (R x i)
=asz IMB',22l
(a) I (c) 0
fo3mrc<mqF+E-s<Erq d + E = d, d = -i (b) -1 (d) R
j)
wt{t{: (b); i. (R x = -i.i = -1 [.i = 1]
2s. A= ei+ oio<t"r<mconFz lcB'221
O) 3i B[ E-s cirft,qB{ft q+FfuWwgq<tq e€qrq
(a) 3i - 6i (c) (d) 6i
T'n(Fr (c)i n. (eR) = (ai+ oi). (aR) = o+o + ffit <rm Erqrq crr mtcrt ttB oBr-<< q{r{fr
o=o...Bt,Aq<G"ErE{r 6+Ft sE? IMB'221

26. trrq q<sF Ft< 6Frqd stt\r-lir6q-q cq"ilc;il

(a) 30' (b) 60' (c) e0' (d) 120'

qg\ffr41r$frr5-d66ffiq6ru
Erdrt

[Din.B,22]
I
:Tct{Fr: (c); Fb-d ffi
fr-s(w &q <R fil6t n]re
fr'Yq{6sEr{rf r sr< mTnmqFFerir{{r<fr
i. i = o; i. R = 0 q<qR. i = 0
cstcl90" I TR.t,
33. 4 qs?F € 5 qF$ {ftrrd (E Fi-d aflrql fr-arc 60.
c{'llctft{tfrEq6E6SEIlltaqrFEqtrI- tDB'211
(a) 0 €F<F (b) 1 q<nF
(a)Vxi=0 (b)v.n = o (c) 10 €sF (d) 100 q<r<F
(c)V.v=+ve (<l)V.i=-ve
rFIl{Fr: (c); qts\,_(iK effidct $rE, stfuRffiq crRF: G); ldl = *, lBl = s c-{k ect{iq,

A B= lil. lBl.o.oo' =+x s xi= 10

IE
{-dtE?F I
 cmr{fr'cm lEI ot-J[ : qF$I{-ot
:+. F s Q q< qft< Fffi{ {I{ 6ol{E? IDB'2I llAns: dl +:. qsF F= (zi+3i-R)N <q atnT'f
<sqm E"t<r

r"i lFl+ ldl <u)FxQ +-<moqF< i = (i + | + t) m x<"t qFr F s i e<

(") F'd (d) lFl- ldl tqr+SroFlTE? ICtg.B'211
(a) 48.19" O) 51.88' (c) 81.84" (d) 84.53"
:s. (i+t)xR:r lDB',21l
(a) 1 ,i:IttfFI: (b);E+ = cose - 0 = 51.88"
(b) i (c) i (d) k
Irrlln
qqKF:(b);ixfr+8xR=i ,- -,2 ,- -.2
44. lAx Bl +lA.Bl q< TF- ICtg.B'211
o**,ra<fr EF qek Esr Efs
firu-* E6a5au
p=zi+i+Rq<c0=i+zi-zk
:
la-l lil'z+
'- .,2
lBl' 1t) lIl'.lEI'
.-
(c) lA + Bl (d) lA - -.2
Bl
36. Fi-<E1T{sR-dql{TE? IDB'2ll r- --:t2
(a) Vrs O) 6 (c) e (d) 1e T{lfi: (b); x -t2
lA Bl + lA Bl
c$qFr: tal; lF + dl = ltrB sin 012 * IAB cos 01'z

= A2B2 sin2 o + A2B2 cosz 0

= JOT'$ +G +-rP+G -zP = tre
37. F e d e< ER< rrlqdtE qs-<r Ftr{ mtqB? IDB'2U = A282(sin2 o * cos2o) = ArB, = lAl' lB l'
I. 1. 1^ , 3. 3. 1 . 45. FA< fu6lT ffi{t(.lF< crfrfrT e eFfu <fr
{a)-i+-i--k
' 2 2' 6 Itr).:i*.:i-..:k
V19 119' Vr9
frcr {|s slcrq Eirs ft et< <cqz [SB'211[Ans:d]
(c):i+:i_:k (d):i+:i--:-k }rln
''3 3', 9 t9 19' 19 (a) tlftq
(t) (c) cT.t(d) t{l
3i*3i-R q{ql{ @3< erdr.fir ffirq<l{qEqT-
' '' i4 =
fi$qlq: rut:
lP+Ql v1e

38. F=i+ zi- aRffi vzsmqFsE?

11i=xi+yi ISB'21][Ans:b]
tRB'211 (ii) ai = (xri + yri) - lxri + y.i)
(u) JE (u) Jm (c) r45 (d) Jl? (iii)f=xi+yi+zR
aqlqF; 1i1; JZzTJz = i13 frIu<6sHF{t{r?
39. ffi+ {rq Fs Q {Eqrts<1q Fb-dErq- BB'ztl (a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii
(fF0=o 1ii) FxQ=o [Ans:c] 47. FE eir-<< $q wll{ r fr[Ed c+ll w<qn qrr< Ek
,iii, P*
'a*Qya, - -
Pv P' qSF Fb.r<< {ttrK r/5 e"f <<U [SB,2l][Ans: c]
(a) 0' (b) 4s' (c) 60" (d) e0"
frru-d6sHFq6{t? 48. F Q@mrn-+< cEB-< "t-+-"r<fr"t?sftFftlr
re
(a) i, ii (b) i, iii iii (d) i, ii, iii
(c) ii, o<cqr e?-< EE< Ek E e F q< {{I{S c$I"t
6str{I F?r{d {fqfi-T e frdk{ ,ari c[E m 6sEE? tsB'2l1[Ans: d]
e?,-*ro <rs- [RB'21; JB'21, MB'2ll[Ans: c] (a) e0' (b) 60" (c) 45' (d) 0'
(a) aq5a{t F < 0) qFf F?-d 49. firr< 6st{E cr.qk a"lscq< {{? tsB'211[Ans:b, d]
ret<
(c) rFT (d) qTql{ F -d
(a) qat F?-{rs E!3t{ frcE ecirrEl W q{
41. {:rl{ @Acr< f<FiBI'- [crg.B'2r][Ans: d] (b) "K*K q? (E drffi e"Fm t+r S{

filmqtqr<tft (ir{FrwFr (c)'j-?rBl< {:rr$lFr Etr cstl.ffi Eq}q w E{

(iiD ft-s q4Qftr<t (d),{st csa-<rs EE-{< frn q{l<ff cH W s{
50. ,{sF TlE F?6f,r qF ss sc<? [BB,2rllAns: d]
fttu-d6+tiBc6€?
(a) 1 (b) -1 (c) +1 (d) 0
ii
(a) i, (b) i, iii (c) ii, iii (d) i, ii, iii
51. ixf =7 [BB'21][Ans: b]
i=si-si+st;E=1si+mi+eR; m 4r {FI
aEcTE T il dcc<?
(a)0 O)R (c)ixR (oixft
ICtg.B'211 sz. ld + El = ld - El <m,I e i e< rl:-{S rFr qc<
(a) 15 (b) -1s (c) 16 (d) -16
IBB'21; Din.B'l7llAns: cl
qIT{r: (b);
* =# =I = m= -15 (a) o' (b) 30' (c) e0" (d) 180'

\o ,rffi+*qomfrS""ialwfi...
E \\\ e
 qn{frcrf,
'ti!t : ulrsT{-ot
aEI

Etr F?-d {tFk 6frdI( e.FF{ V18 q<( FA-d e"Fr{ '^l./ toN q< (TE rrEr?F Er E fl(rt
1G r e?-<crr{ r{I{fr 6s,FI qr<- [BB,2ll[Ans: dl FrstMs s{r{ oY q< Tld 6\5?
l,4o*
(a\ 720" (b) e0'
(c) 60'
(d) 30.
o? x lcB'2lllAns: bl
{TtqFr: (d); i.
B = eBcoso = Jra-......... fU (a) 4N (b) eN 12N (c) (d) 14N
i " B = ABsino = y'6.........(iD rF[{Fr: (b);
ldil = +; l0?l =,[W=T = r64lt = stl
(ii) + (i) = t"n0 =* ;0 = 30. d s d e< q$<q 6ef"t o ,s<( i qr frr+ esE
#=*=
54. for<mntr<frofiffimz ,{FF F?-i a qrq f q< B"f< E e< ru qGrF1
[BB,zl;cB'171
(a) w:rFr Fi-n (b) frflfti0 ei-< Inns: ol Erqt- [CB'211[Ans: d]
(c)ul:rs<Frrt{ (OW@B-{ (i) Acos0 (ii) B cos O (iii)E.a
55. ftru< cst{fr"v-ffi2 Fffi (*-i1 e?cmm s firu-*c+HBc6'r?
fr+ffit+r<u (u) i (b) ii (c) i, ii (d) ii, iii
[JB'21][Ans: al
? 'I{IttFt: (b); ABcos0 = I.e = ncore = i.i
(a) ,* b\rk + Bcosg =s-.a II=
TAJ
al

G-dI{<fiR fu- [Din.B'2|; MB,21][Ans: al

(c)
-Z*7u
Y
,0,flu
Y
(a)

EE
slq (b)TE (c)R6 (d) cfrn-s s{(< t
csictrER<rllqr66 << {fi q&-dtllr$
Y Fm,f OU e< qsE <EIfs' "l-<*'K 6fFt- [Din.B'2lllAns: al
+D (a) 0' 4s'
t6N ,/', Ef,\)-Ir< EF U"[$t oi re oY skl
(b) 60' (d) 180. (c)

#-* L,'10N
fictf.t +<t crqlr 6i
UB'2lllAns:
€< TH
cl
3i- 4i + rzft effi< {rq- [Din.B'2tl[Ans: b]
(a) 9

f
qF?F O) 13 qTiF (c) 18
= si+zf +ft;E: oi-mi++ft,
{sF
m
(d) 36
{{ {Fr sE
qFF
(a) 4N O) 6N (c) 8N (d) 16N
qrq, Fi<E-{ ffi {6{?
q:rFtFI: (c); o; l[Xl = r/r oz: oz = aru
loYl =
, [Din.B'211[Ans: b]
(a) e (b) 11 (c) L2 (d) 13
57. @A-d F <-+rqs x-qs <<F< q<@r q{' esF
>FIt{Fr: O); i E = ra -2m+ 4
reB< I e{'{-sk< q<@ m< F x Q qr :rlt W q{ r
+0=22-2m+m=11
glCFTdq(g"l1r{-
UB'2lllAns: bI <fr i = i+ i + ft q<(B = zi + zi+ zR qssr< E
(a) i (b) -2i (c) aR <al (i + R) <<les f erq? qGffi{Tsz
58. d = si s d = i iqrq, eA-*N- [JB'2tl[Ans: b] (")+ (b)V3 @\z\h (o#
(i)cEIco?,-d (iDfrer$qFA-d (iii) *r9:{d?
rr:rt{Fr: (b); i <'<f<< i q<q{qB-rsfl,
fiio-<coFtrc6sz
ii ii (c) ii, iii Acoso =I!=g=-!-=.,q
la-l ,/6'+aF+arF - z"/t - ""
(a) (b) i, (d) i, ii, iii
59. i c<i j c< Ero q{Qs cqt.s-ffid E"f< q? qqa
d = si s6{ q< frEqq FB< furr rffitr?
csA-dEtr{t- [CB'211[Ans: b!
[MB,2lllAns: b]
(ixR) py (ixj) 1cy([xi) (ixR)
-si (b):i
(a)
60. d .E = o qrq,fto< mFGqEcfu
1a;

ICB'2tl [Ans:c]
(a) r") si (o*
t0 qsiF qlcim q+E o3-< @ rt S"ttrrt fr'su
(a) 1ty. d. Rt s-{F q4Fd{l{ 8 er-sqr{qekFdqFl sEz.
A
lMB,2lllAns: cl
(") I ,? (d)---..-.---.}
E' (a) 2 €$<F (b) 4 q++ (c) 6 4T<F (d) 8 qF?F

I E
-..-.--->
\\\ -,rfur€esr6nft6?,FtErTi...
 ,ril{freEl }EIrt3[

70. i^= zi+xi-4R€B =yi+oi-etr xsy(<

{lq +s <e I s E "r+*r< c{M<IE sc{? tDB,lgl
(a)x=3,y=4
(c)x= 0,y=2
(b)x=4,y=3
(d)x=12, y=1
A* Av Az
"-z\" R
T:tT{rn: (a); E e E l.{rg,-<le EF.
"''B* - By- Bz Fqc{r+c+rlFq6s? ISB'l9llAns: cl
=2_y6-8 =! =1= x = 3. v=4 (a)F+Q+f,=o tuli=0+F
(c)F=R+0 (d)Q=P+R
i,/
71. Y ;Frq1A€Ecs?ii 79. I.s d cn]-g{lE Eaa €tq< {{r<ft cl"r"r< Tl{ sE?
"B IBB'l9llAns: al
(DdEtRFr{IENr{Drr (a) 0" (b) e0" 180" (d) 270"
(c)
IDB,lgllAns: bl 80. csI[{I FALT{ ult\r-krqq <q1- [BB' l9l [Ans:b]
(i0 4;r ecl{ frfrrlr {4 c+rq urE
(a) rei< cs-q (b) c[cr]3r csE
(iiD @rr eci-{ ftRm {E c{C;r Er{ qI
(c) eEobr<<r&$ql (Oq1f{q-{
firu-d c{HF{6's?
(a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii 8t. ri = i1):i Ecd,
lAxB I
-n q'Trd ss Er<? IJB.rqllAns: al
72. EtrFArsd@cqqaqqerfqEl{F- IRB,lgl . . s-xl .-. A^B-
(b):- lu..^A..l
{al;- ' (d)1441
(D 4rT sfi'aT (sF Ftd<fft lAns: al lA./Bl lB,Al A^B

(ii)4q qcl+rs:Ifi-s F?<ET R qT\9-r{ EEr 4{ $rFr< az. et< cv< V v$rftrE6, RrE< rFF qe$?
(iiD @q aqTar frfrrT IE ffir{ Er{ IJB'191[Ans: bj
frro-<mfficfi-sz (a)V.V=0 61 ixV=o
(a) i, ii (b) ii, iii (c) i, iii (d) i, ii, iii (c)V.V + o 1a1 ixV+o

73. EB
qT{ e?-< I 'g
E sfr qst cnm qst fr"gcs 83. esfr EFF vr 6<rf "I{ ft-$ q<( q$E rlir vc ffia'l
fuqtoc<sqq- IRB'rel "lFb{ frr<r
qBfi-q ccE ftm-{ ct'("tffi FIiT{
qfclfur(<{ss?
(DI.B=o (ii)dxE=o [JB'l9l[Ans; al
(iii)lfl+lBl=a+e (a) (v1 * v5) (b) (v1 - v6)
fu:rcconfiq6-oz (c) (vc - vr) rdr lr
(a) i (b) i, ii (c) ii, iii (d) i, iii S+. I e d _eT T.-^-fr cr1t1 e Erq A _q< \e"K E _q<
qctql"r: (c);
v{ @i-d qr4,
{{r<6 cott 0' etq&6$?i- IcB,rgllAns: dl
.. I.E + q<(ldl
o + lnl = a1 s (a) Asin0 (b)Acos0 (c) Bsin0 (d)Bcos0
C
74. 6s'[rql 6sB-r !c<( qs:F Cg?r{< w<ff c$,I[*r E1-{
sE? [Ctg.B'l9l[Ans; dl
85. lDin.B'l9lIAns: bl
(a) 180' (b) e0' (c) 45" (d) 0'
75. (i + f) .t -o<m<<- ICtg.B'tei[Ans: cl
(a) i (b) i (c) o (d) 1 ffic[56< aq66-q3 ccqrq cffiF q6oz
;NT{I{: (c); (i+l).t=i.R+ j [=o+o = o (a)EB-Ad=Bt O)AB-Ee=aE
76. f=zi+si-hd=pi-i+aR, p-e< {Fr ss (c) eE + Ad = Et (O Ad-AE = eB
<4, @6-<iBI 1-?t-'K E? Sa<? lCtg.B'l9l ao. firr<ffir d e< s"r< F- q<q&rt.lsu?
(a) -4 (b) -1 (c) 4 (d) s lDin.B'19llAns: al
rTl{Fr: (c); ret<sT "K-"K e" qr< {fr,
i. B = o + (zi+ si-i)
(pi- j+sr) = o
+2p-5-3=0+p=4
77. F=i-i-tqo,F-qa{Fse? ISB,lel
(a) 3 (b) J5 (c) 1 (d) -1
(a)T- .. . F,a
tnt- ..
tct-
F.d ... F'd
t.|t-
{:r{t{: (b); lFl =,1 12 + (-i)r + (:1F = V5 .APP
 qn{frl$El bEI
'tir : u[rfl{-ol
Ms fr'eE< rt6s.o-rry< E"f< qt e6E FA-d-
[Att 8,181[Ans: al
r"rffi rtrffi (c)#+ (d)+# lCtg.B'171

88. F'Q = -PqE6{- tDB'171 Str<-< $q<qrctrc+ | I +r l =z

(DF rg d 'Errq{{qfsitfE (a) 15.81N (b) 14N (c) 13.23N (d) 11.23N
liiyFs Q ffierfr{?ETft ,l'fl(Fr: (c); i.g i q<{sr<ffffi.t
(iiilFed q<{ry<ff6oFto'
firrdffi1-lFr6-rz = (180. - 60) = 120.
(a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii lA + B
| = ,'lA2 +82 * 2AB cos 120'
wr[$r:(a;;F.Q=-pq
=r PQcoso = -PQ + coso: -1 +0=180"
=@
= 13.228 N
89. xz {IEFT 3i + 5i + 4[ FArf,{?qfTE qs-sZ
(a) s O) Ve4- tDB'171
96. SrE EF FA< f e E c< Erc-rrfir:rfl s q$-sl
(c) V+r (d) t?. Efcq3r{(lsk crFI 60. I I d - El fi{T<5irr [sB,t7l
Jl:ftflir: (a); xz {qErE csffi" E4t\-f = gi + +[
:.ffi=t@ +4 =S
*,--N IDB'171

f
bft"i(srqirEnF cslqBrts? (a) o 0) s qFlF
=o
(a)i+N-M 6yi+M+fr =o (c) 7.07 €FF (d) 8.66 ,4T?F
(c)i+M-il=o (OM'+N-i=o
c-jlt{Ft (O;@qTq<o,N +M i = Tct{Fr: (b);
+ M+il-i = o
r. llxdl =frcu<6$r-{E? IRB'171
(a) A,B, - 2(d.B')z 61 tzez - zi.i lI - El = JAt + B, - 2A-B cos 60'
(c) A2B2 + 2AB sin 0 (d) A2B2 4 2AB cos 0
crt{Fr: F6'sEErot); ld, El = VAzB4mio = Vsz + _7. S.5cos60. = s
57

= GzBrG - cos, o) = JArB-r:lAB cos o), s7. i= (px+y)i+ (y-22)i+ (x+32)[effi

qfr$g]qEmfip =2 IBB'l7l
(:a) 2 O)4 (c)3 (d)-4
92. p € Q c< {-a1qT (3,-2,1) q<( (3,-4, s), pQ qi
{FI TE?
:nilqq: rdl; i. f = o = (ioa + irl + R-3)
IRB'l7l
(") ,,lzo $).'lE (c) Jso (d) 6J5 (tn*+r)i+ (y -zz)i+ (x+ az)fi) =o
T{I{FI: (a); +p+1+3=0+p=-4
Pa=@=vzo (ixR)x(ix[)=z IBB'l7l
93. cstdiT'iFcE-{Ttul{Cr{? [ctg.B,l7][Ans:a]
i.V
(u) 0 O)i (")i (d) [
1a; = o Q)ixV=d' {d{Fr: (d); (i x R) x (i x R) = (-i) x i
1c1 iV=o (d)V.V+o
q+. FsQ e?-<wq?E€rtr16 crFrB? =-ixi=-(-a)=fr
[Ctg.B'17][Ans: a]
qs. qst qfiR-T frFt? s?<q{q[-{ft <Fr? IBB't7l
(") F'0: o (b)F.d= r (a) lroftt ei-< (b) T:IGFtr ei-{ IAns: cl
(c)Fxd=d (a)FxQ=i (c)w-aFBo rei-< (a)firRqFAF
IMq qfui-l-* esrcr hoil qqEil...
 ,Fil{frsordl }E et3[

100. {julr+d EF qk< {l+ Elffid m corr{ ssE<

<FFr I 03 . ABCD :tElgR-SF< 6EITFdI FE? uB'171
wcft<r4<+ffirqJTft 6al"lsEZ IBB'l7l (a) 0.5 <tf qF-s (b) 1.0 <tl qF:F
(a) 28.90" (b) 4L.40" (c) 138.6' (d) 151.04' (c) 1.5 <tl qF-o (d) 2.0 <t1 ss?F
rutrr: tat: ! = VFi + PzT2 + +. cos 0
= P2 = 4 (2P2 * 2P2 cos 0)
+P2=8P2*8P2cos0 + 0 = 151.04" qqfql*: (a); A
101. clslls' rftrs q(Fx qqrs e"m< m(9 @l-N< ckal
fr-str{ d-sl DIT{T a-ircs Er<? UB'l7llAns: cl
B
(a) 45' (b) 60' (c) e0' (d) 120"
ABCD = A 6SEI ADC * A CSEI ABC
(StG-?t ffiEIrFdI
frrFd firgr cIrqrs "r<-<ff lfi Eet< Es'< EIE :

D =lxACxOD+1xACx0B
22
= lxAC x (OD + OB) = lxAC x BD
= 1x
2
Ix I = o.s <r{qss
ro+. d = -2BEcE, is reB<nE- ICB'I7l[Ans: bl
E
(i)cT'f (ii)frcT-i (iii)irrtr{t
frrr-< ronB c6-sz
(a) i, ii (b) ii, iii (c) i, iii iii
(d) i, ii,
o4r* <q ec- = i e ao = i
fmo 105. y- q[Erwat i = 4i- 4R cs?tr< Esffi @l"t {c<-
lo2. [E @Ard< c8s {"t csl{F? lJ B',17 | lDin.B'l7l
. i+i
ta) , i-i
(br; . i+;
(c); (d)!
,,2 (u) 0" 4s" (c) e0" (d) 180'
(b)
q
i- cqlqFt:(c); i.j= trl cos0 = 4r2cos 0 : 0
'22AO = ::: = l.
{:fl(FI: {b): _-L
OB =
22-
-BD =+0=90'
Ai = Ad-l oB = (?)
a MCQ: fr&q cfir$-{ Et<tc{K w{ffi< E{ s {sTl{rt +
EE FAr<< {{r{q 6sl"t, a = 180' qr<, cvE-<el
eF9rfr1AEftrsfuTIq<r<r
01. 6{l{E6@A<dft?
aI{F F}<s(ia qgi56 csl"t 90'qcq Etffi<r qk ({
(a)'tG{fu, c<'t (b)E&.qfrs<, V<ct
q$Efic{< r/2 eq qm
(c) 6ffi{ft y<"1, st"ffi (O EBrFq, ({ 6SFr @artrn r

c$tII{: (d); E&\ Rs<, 'lB-{G', ETeE]qT qerET

04. qft fr(T qA-< \o-Efir{ {ktT frm +r* 6$Icrt fu
Erdfl-nl E{ E{{ 6fF <rq-{ FErc"t d-fi qFnq slTz
mark<fFir
12 qs?F q<(T(frT${ 1a) a1o<r+<wpfu+s"tt1t [Arrs. dl
EEft-silFi<Tr{to'fiT 2
qTrF I <tFlsI-IK {1T frfte< (b) STsEffi{S4?B"[Cf
r
IAns. bI
(a) 6 SS?F s 8 q<tiF (b) 7 qs.iF \3 5 q$?F (c) q&&-$<rffiBqrU{$
(c) 9 qs?F €$?F s 10 (d) 7 qs.iF s l1 qs?F (d) qfufu<r+<w{pfroS"rtrt
03. <fr @ w*a FAL<< Ek qm{ c<-cstm qsF< 0s. FB-{q1Tc<i{e(El{R$tl?<fi3]<mrq- lans. trl
qnt{ q{vr< Fi-{ccil<ryr{q (D r{qrd <lFR 6elGrr+ q$E Fi-d rtFl
6s.Ft <<-
(a) 0" (b) 180' (c) 90' (d) 120" (ii) 6$Ic{I ei< cqrc< vtl-sKrqq w EC{ (gffi
{qI${: (d); Etr eil<< rlrq 6rFl, a = 0" {c{, qq4{ftd
et<sr{{ffi{<<r (iii) caffi< FH,h d,r-<< $r fio{.r or<
6s1-(nl

EF clt{ eit.<< q$-{tr (st t, q = 120' Cc{, EF frrFdffihficfr-€?

FALTdqk ercr6 cstr{{c{Fqr< (a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii

rNtffi
I

t8 \
qffimevrcrRs"'FrE-{... (I
 qqMfrcrd DEIcti![ : s[<IIT-ol
06. I=zi+i-a[ e E=+i-R cs?-rErr< ffi-{R l 5 . qFtr Fb<rs {r{t6' Tntr B4ffi \e-Ft s<t {tT?
Q.lqq ss? (a) t, (b)frffi [Ans.dl
(a) 3 (b) 7 (c) e (d) 11 (c)wE (d)qq($
{r[{Fr: O); i.E sr)
= (zi + i - (+j - R) 16. ftrD-d cs,FrE qr-o eb-<ffit<5.m? [Ans. d]
=(4+3)=7. (a)a== (b)a=r
07. i=xi+yi+zRqmV.i+sz
(a) 1 (b) 2 (c) 3 (d) 4
(cra=-T (o)a=-A
wtqFl: (c); (i) . *(;) * *s (i) 17. f=oi+:i-sR q<( E=-4i-si+zt E6{,
d*s

=1+1+1=3.
i - B =z [Ans. bl

frrr< r+ntr x qrw'< qqfsBrFr?

(a)1oi-Bi+2[ O)1oi+8i-78
1a; (ixi) xi Ol (i+D xfr
(c)si-7i-st (d)zi-Bi+zR
(c) (ixi) xi (a) (Rxi) xI 18. 7 kg g(T{ ,4<5F eas <o {Siir e"r< i = (zi - zi +
6R)N Erq TgE ss y{ol EIS <<?
,FTIlFr: (c); (ix j) = X
(a) 1.0 ms-2 (b) 1.4 ms-2
lixi;xi=txi= -ia1 x q(ffi mtsrfq ft-g (c) r.) / ms ' (d) 7.0 ms-2
fra?qtr
{{I{FI: (a); m=7kg; F=
09. (E rnT{ oB< catrs T{r FBr 6{cu q6q qrrd
q${S = ",149 = 7N ... a =: = 1=
1 ms-2.
cstq qro cr<- fn7
(a) 0" (b) 45' (c) e0' (d) 180. 19. I s B "F-"td q{ETFr E{ E?l{-
wt{Fr: (d); R2 = x2 + x2 + 2x2. cos 180' 1a1I.E = r 61 dxE=o
= 2x2 + (2x2)(-7) = 0 ;. R = o 1c1I.E = o 1a1 ldlx lnl = t
10. ffi d= f x E e<(d = E x d q{vrsm d q<(d- c$rrlit: (b); I x E = AB sin 0' = sin 0' = 0.
q<T${Sc{Ftsg? q+frffircfrrc< cfrfutB<rq<frnO et< sT-<T
(a) e0" (b) 0' (c) 180" (d) 4s. ffilgEtsc<e<ffi-q'€{- [Ans. d]
q:rl{Fr: (c);d=IxE (a) FtdltrdRFlstE<cTFr
d:Exr= -GxE) = -d c) 69?-<EE<Rrflaero<crl{
It. f +E = f -El-"rdBc<:H.rdq8s? [Ans. bl (c) csB-<1fr< rtnt-:r EqIF.ffid{{FI
(")I=q 1r)E=o (c)I=E' (d)I=-d (d) FA-d Etrd FE< sts6{r {{Fr
tz. A.B= 0{6E{ittrfs- [Ans. cl 21. r+ffiq{tddr$
(a)I=o iu)E=o (a) sin0 (b) J- (c) los (d)
*
1c1 E s E cr+ v"tr<-< S"f< nt rFTttfFt: (a); q'ttrfd
(operator): q'ffr<iB-{ q<fr
1oydsE"t-*-mwfo-*fr tr<&.mr qn q&qlq'isqq1 q6sr 6a, 4,aig6,
13. {E d-{ ftqAE ft,$ frfl<s aIFrE eR-d qlq ss <t F,I{-$lT$'r fr-E fiE'ki<Et{t{<-4tE(<- q'ft&r
E6{? [Ans. c] sF $ItrEr efo-{ <I qg$\9- | q< frq{ 6slr{t ${ 6{tl
(a) qr{to (b)T{F q{
cT{ <,f 12), (3), +{1r (.f); sin,log ffi r vm
(c)+{Rs (d)cFffitm q${rFr qdr cff(;rt <Fr< cfcQl {sEr\,_ll{ qsF Fffi
14. *11q qtsffidBElqr"t-
{F<qqoc< r

(a) Fle o)F6 22. 6$rrqt qgfts-d.fi'{FiT ei-< qr"fsro< er{ crdll- .

(c) <EI (d) rfiftsE-{r< t (a)fxV (b)Vv [Ans. al

T{T${: (a)i W = i.? = Fscos 0. (c) v.v (ov. i
I M
 mdfrsraaaom
23. @tr{I q's+s<.l(i1'r Ffu qr{t,m< s]EEr<@q :2. qsFrr+ERrs-deFqEfu aq F = i- j + t q<(
Q= i+ j- R ffit <r< srq fi{rcfrrcd (E+-{
1alVxV @)V.V lcyVv (OV. i ffi<-<r
z+. cot({t q-sftfiqlsFir 6Ft<cmsr$< 6qffi Er{t-
(a) v. V (b)vq @)24 (b) I (d.'12 @)2
T:IIfi{: (a): 6Aqa-a = |F x Q|
(c)VxV (ov.@
25. p € Q qr qms G,-2,7) q<( (3, -4, s) I PQ q< ri i Rr
Fr0=h -r rl
{l{ TE? [Ans. b] lr r -rl
@).,1b (b)./20 (c) r/so (O 6!€ = i(1 - 1) -i(-r - r) + R(r + D = zj + zk
26. fr{E oE-<I,Ee d qtq c{ IxE+Exd+ .'. (+EFFE = l71T = rE = zVZ e+.sr
dx f = 0; @t1 &{F <6E'f- [Ans. b]
:r. IxE=r lAns. cl
(a) qsc-!ft{ 6ye+vfta (a) AB cos 0 (b) AB sin 0
(c) qsg,ft{ iT (d) csft-{lfrEnr
(c)-ExI (QrxI
27. ii -) u<r 1i - fr)- sF r{I<6 crFt Fs?
(a) 30' (b) 60' (c) e0" (d) 180'
Ti.{f(Ffi s eE@q5q6pl3 Eq-46E'a6q-

{qI{]T: (b); qFr<l qlfr, i . ? = AB cos 0 (a)e--uuFl €FFls (b)F6

(i
/"=""""'= /-
-i).(i - [) = Jt'z + 12 Jl2 + 12.cos o
(c)dftsvfi<'t loyu+utE<<rm(
vnrkt+: (b); , ?l = .p.1n s
i6,
= L =.J2..,l2coso - I =.o.0 1.1 = l?
+cosoo=1+0=600
2
Ieie<frffiEoBr.T{sfr<El-q- [Ans.a]
zi + :i effi x-smrqlratFE 6sfc{qFE2 (a)0 (b) 1 (c)A (d) 2A

(a) tan-1J (b., cos-"2

';13: Fedeb-<sr"K-i{s?<E@+0) x 1F-Q; =7
12
(c)sln '- (d) cos-1J (a) (FxQl (b) -2(0 x F)
I':rHFr: (b); a= cos-|#= cos-r (c)dxF (O 2(0 x P)
rft-
29. Efr qFs @A[Td rrB erE q$-s @A-{r I,{Kf{: (d); (F + d) x (F - Q)
.4I[d frr{FFF'{ SE? = (F"D - (F"d) + (QxF) - (d"O
@),,12 &) J3 (c) V4 (d) J7 =o+(QxF)+ (QxF)-o = zGxF)
wt${: (b); qF{t q1[q, Etr qF-{ FEc-c< qqB esB (d + E) s (I - E) Gb"dsrq<*"rrEr<{-
qs-s @3-d qrq s?Fr fl = 120'qr{
(a)A = 2B (b)A:B (c)2A: B (d)A= 38
I

fr6qFfq{ q6 ffirq ErFI a = 60.

{{ttfFr: &); (A + B). (A - B) = 0 =+ A2 - 82 = 0
R, = (1), + (t)z + zx 1 x 1 cos60"
+R2=3*R=16. =A2=82+A=8.
30. d=I-B Fffi< ${ (A+B)r I q<(E e?d Iv E@oE-<- [Ans. c]

5tr<xrffr+t+ [Ans. c] lqIxd= ExI liiyf +E=E+f

(a) o C) ; (c) n (d) 2n (iii) d.E = E.r
31. (a.D'z = a2b2 qgdId e6 e?< {E< c{I{tr ffi.t firu-d 6sFFc6'{?
(a) i, ii (b) i, iii
ss? lAns. dl
(a) 90' (b) 60' (c) 4s' (d) 0' (c) ii, iii (d) i, ii, iii

I M \\\ qffi-.nqwqBw+iqt..
 qqT{frgFl5Nre
09. @EsT c<[tk flffi{ ,2 e"t qIfl{ 6{I'l rto-t{ cstt
01. frrF{ r$l{furrBRFlT{6 mc{EFr{I?
,qFq{ IjEIf 4ft{ qq< "firg cctqE& ft(T c{qe r

[Ans. bl
(a-) eB<ffi<6 efi' a${
(b) csB-{<tFk@{ +ft-<ftrr<wqd-ow< mcaR c+Ftfi4ro< r

(c) F?-{<IFK ml
(d)B"ltTd 6{lcrtFt+* (a) 120' (b) 130" (c) 135" (d) 6s'
JFIT{|{: (c); u = u,v = JZu,0 = 90"
02. @?-dI,d s<(e q<${{atf@({ 12,s q<\13 q<(
^
tanu=-vsincr
I + E = d r e?< I e E e<{sn{S 6fi"tsg? u+vcoss
IAns. bl <T.coso= : = -l-
-l= - '12! q= 13s..
n 1o)i
(a) (c)W (o; ,12

10. <fr I,e s e f6 -7u <tFr q<q e = I x E e{,

03. csi"< I =zi- j+ 2ft (< crtq<rq €F6 Fi-< -
. -2. 1: 2. sfqrq d s< frs CC<- lAns. cl
{al-1--r--k G):i-:i + it (a) A <iFl3t (b) B <-<Fr-rr
c):i - ii+:a (d)i-i+[
1cy I s d Uvrr< ry <fl<<
Fs*{In' (h): lAl = G+t+4=3 (d) I € EBs(.T<cfis<l{Trl?rd
^ A zi-i+zi z,' ). : ,_k 2.
"-lAt - I -3' 3J's 11. G;{tr Fi<, d,6 ig d {rr< qll setrstl 4, 3 q<i 5,
04. 5N e<( loN nlr4-< 5E <e c+E rqt< B"r< e{6 6qH $<m Tri S{ qats d+6+d=0 el{cE
<q foT< col{ <qE F"fitr< B"K Ek Er "m< ql?
(a)sN (b) 10N (c) 15N (d)20N Ir x (a x 6)lq<{FqC61T-
(a\ 12 (b) 60 (c) 25 (d) ls
Wqf{Il: (d); R.", = (10 + 5)N = 15N
:l:[t{Fr: (b); x (a x 6)l = l[ x (ab sineo"ff)l
R-1,=(10-5)N =5N ld
...ek-<ql{ (s-ts) q<{c(IEr< r = labcsin90'fl =abc=4x3x5 = 60.
05. c$lc{I <EI SKI grc-Oq, W = F.S r 5F qa cffiq p 12. EE F?< i oi "r<-r< x:ttsrlrE frrt +<rqr
sr(SWIIq(E\BTEslqWr q Crf+ qFKI {4as <<-
(sA-<sT
lAns. tll
{R (i)rq631*6u (ii):f[saT(sir
1ay F,s<rS e<fr+ <+t (iii) eK-'Kfrflftscs,<
1u) F e<(SfttA-s{ft FrDTcrFtrc8s?
1c; i e<i S "r+'"tr+< E"r< <t (a) i, ii (b) i, ii (c) i, iii iii
(d) i, ii,
(d) F,4<(S "K-'KTfig-fl{ t:. Fs Q Etr F?r cslrll R1ru fttt r<m onr<
C{${: (c); W = F.3 = Fs coso = 0.'. cos0 = 0 qfu<ff{{(g "ffn- lAns. dl
06. {l{ W T{ q (s{ esE csb-{rs El-< ql{ fi-cT \,-14
(i) JFz+ Qz (iD
s-<Gr fr "ll\3n,l {l{? lAns. al
(iii) JP, + Q2 * ZPQcosa
(a) €+F FB-< ol q!-sfi-{ ei-d
(c) slr{Fl ct{ fuiu-aconBq6oz
(d) {r{ @B-i
oi. [) q<{Rss?
(a) i (b) i, iii (c) ii, iii (d) i, ii, iii
-
i. (2i 3i + [Ans. dl
14. mlr{t et< n r+ <fr @
(a) -2 (b) 3 (c) 1 (d) -3 "t<*t< E? E"11(rt

08. EG fr-$ ilFKErsl-$E< l0 q{sr qrrd

fi{ qkl ffiss-dlq{qqcqR q<qlcet- lAns. al
{F 1 oaD q$-s qc{ q-cq3r Ttrr+6 cotcl $\o?
(i) n c$,k6t U{l(ffi qFI x= R cos 0
(a) 0' (b) 60' (c) e0' (d) 120' (ii) (90' - a) 6rk"t Sql(rfi fi{ y = R sin c
qEltfl-{: (c); = (10),+(10), (iiD P cslriB9iI(IR{F Y = Rsin p
1to..11)'
*2.10.10 cos a frru<ronBc6oz
=0:
rslil 200cosa + cosrr = 0.'. a = 90". (a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii
{ffiqrEE rrftoa'ietFqr...
 rm{frgfA rE[ : Ulttfl{-ot
N DEf

ts. F ie Q ei<lFceno "flt- [Ans. dl 09. F = qV x E, {fi-fd"tqrsFqe- [Ans. cl

1i;FxQ+QxF tOF+d=8+F (a) tsgrq i q<{ftets? (b) E{{lE EEfi q?
rrl F.0 = I.i 1c1 n s EEvca-< B"nE{ (<l) q e ES"r<
qt
firu-d6snFc6s? i0. qqfs<fE I e E csFl s-{l Er{, efr q(<-
'lit--'K
(a) i O) i, ii (c) ii, iii (d) i, ii, iii
tu) d q< fte"t 61 E o<fre"t
ro. F s d qaqrr<ft{fts ei-<e@- [Ans. cl
(i)P+Q=0
(c)i e<c{lq<l{ (a) I .c<frrAE
tiit F.Q = o qqt{F[: (c);Icls<]4 ffi{cqsk ffiSGF
(n)Fxd=d
S"[g.m:Wfg*m r
firu-dcsHFrtr{r?
(u) i (d) i, ii, iii t t. q$t cT,.9-r{ &{{l{ c-i{ fifiT${F @A[Td mFt
+EWcc{?
(a)2 (b) 3 (c) 4 (d) 5
q${l;r: (b)i qE{ cs@ B-s{ GE-d rrr{ <qA sft
ot. ixR=z lAns. dl
{+i r 9t +{hx ffi qq{Fr FEr{{ flFfs-q xf{i I

(a) i (b) 0 (c) 1 (d) -i 12. <fr I xE=Ex f sr< em-<qrr<ff cqq- 14n..r,
02. 6s?fl(T FA-d EkI ect $-<I(dI ecf+q q{ [Ans. c]
(a) 6{alR (b) Fa<
(a) n 0); c); (d);
(c) s:Fr€ 6{qk, F{{€ Fs-d 13. qFF {Fr c9lN i. x wffi< q-{lqs ftrs< {1?I
C-\rrE-{

(d) &FrF 45' 6<'Ft $S"EI S[3t ERr{ d q< * e y s9llsrfi

o:. qfrdxE=dq*de?-<- q;r- [Ans. b]
(a) ts{r.{<)Ilrat4{ 6; e1E e<xrc<m (a)Ax=d"=1 ftrA"=A.=l
(c) Q{E q< {Icat
Tirs{lE (c)Ax=A"=1 (d)Ax=AY=JZ
1oy f s E Esr-r< nnt
urt
r+. c6sEsdBfi4To<r [Ans. a]
04. c6'st-sr rstlF ?
mq e <q csfd<tFt
(i) E{6{'1, Y+t,
(a)Rxi=i 6yRxj=r 1iD sw, w, t-6,.tfu,4<(stq 6{{fd<IFt
(c)Rxi=-i ialkxi=o
05. firo<cst{trFAdrA? [Ans. a]
(iiO dfrsF f,vfi<'i,DE-sGq @B<dFt
(a)?<EI&Fslffir G)?<Eifusfr-s<
ftr<c+nBl6-+z
(c)w (u) i (b) ii (c) iii (d) i, ii, iii
1a)frnrs er<R
qmsqe$tr- [Ans. a]
ls. qsF e?-< [ q< B,tt$FFg Ax qr( Ay q<(
(a)rwF<lft 1u1 csi-<<fi oi-<E +x qr$-t mcq 0 coFt Bs"fi +<ro- 1ans. cl
(c) <Ft oi-< (i) cos 0 = 4 (ii)A=A.+A,
(d) F:FI€ GqR +!Fr€ Fad ]e?rel
07. (f -E) e<{r<tETFT6? (nI)tan0=A-r
(a)A-B o)A firur6stqFq6-{?
(c)A+B (OJaz+Bz (a) i (b) i, ii (c) i, iii (d) i, ii, iii
08. F1 € F2 {dI 1F {6d rE lR Ecs Ewt[< fr$ F<(q, 16. cflq F L<T "iIiRT e ff,ffit qst Erq Fffi
etr{Efr(qqr<- {r{- [Ans. dl
(a) F1 * F2 (b) F1 - F2 (i){{rd{6gi< (ii)WFb<
@1^[{TFP, @\^lq- (iiO 6Etd
:rqtrtFt: (c); R=
JE +F + ZFlFrcosg0" "lft{
ftFd6stlFq6'.{?

[ru =.,IFTTFT+o:,{fTrF,.
\t
(a) i (b) ii (c) iii (d) i, ii, iii
 ,ffifrgtd lEI 'E[ : u[$El-ot e
17. ffiqs <K{T ll,lqs x, y s z sct<I fr6$
qFTs 21. A.B q<{];I - lAns. cl
TFrqs {qr6n i,j s I eoo F?-de[ER-$ qtas (a) -3 (b) 3 (c) l8 (d) 3e
e{s @?-< TCE r [Ans. bl d x E e<n{- lAns. cl
i, j s R er+ oE-m< aq-S qa1; (a) 0 (b) 23.62 (c) 34.73 (d) 39
(i)ixj=t (iftx8=d (iiDi.i=o F15:r Eft"rrs< q1rqr6 "r<-<fi EF
q6rr< Es< qt'e:

frru-< c+tq6c$-sz erF erc<ltlcsd w

10 kgr ,a< S"r< frrF< &.G<
(a) i (b) i, ii (c) ii, iii (d) i, ii, iii {iln TE Ar:rFi q<t <rrq I g = 9.8 m/sz
18. la + <ro-
6l = la - 6l lAns. bl
(i)a=6 (l)ard (iiDall6
fitrdmnBc$-s?
(u) i O) ii (c) i, ii(d) i, ii, in
fuu< Eften-o-< qrnm "r<-<ff EF
qnr< E-qd Ets r

f=oi-:je<(E=6i-2jq6d
19. I q<Ttl - [Ans. al
23. E"rc<EEB{qI"llu sw1fr-< "iR{I"t- lAns.al
(a) 3\6 O) sV5 @)27 (d) 4s (a) s0 N (b) 86.6 N (c) 98 N (d) 148 N

20. ld+dl=z [Ans. cl

24. {ft <|gl< iTftal {
E{[TfffiE< rs|{ q{q altrs \e-C<

@) 2.'ls 6)5 (c) 13 (d) 25 E-{1-<14RB{ cFrq frr5< (T'FE q6s? [Ans. bl

frru;< Eft"rco< strqlm 1-tr<ffE0 Er,rrd E-trd q'N r

(a) EI'flE sqn EFT
qlr{ G) {{({C't E{F
i = :i - :Q e<(E = oi + zjEtr e?<<tft r
(c)Y{qIfr flIr{ (d)Yd6iqFt{rcr{

a MCQ: >rsl-<TlTqI ES € q5fi{l{:

fficTt qNsRrF< Em qRfo <tq sfr Etr @arffi \l{ t3

Lll-"'-
01. N
e fr+ffit+r<sEa & clEERar< cE+.Ecc<- ->-
,Ir\r{: (c): l___ r--e-.r.-,. ._---_ro : cos-rg
(a) F?-<EEr ffil'l$ffi< rTF IAns: dl
12 M '...q^
O) FE-{ Efr-< frGl'iTle< crF
(c) e?-<1fr<vBeq\t_rffiwf{
07. c$I{F m-q]-{ <fFi? [Ans: al
(d) @A-{ Efr{ qc ecFErdr{fir{< q'{I{ (a)s&.sfts< (b)s&.qEKsl'
(c) CE{ (d)E{6{'t
02. d = -d qrq, i x d e< {F FE? lAns: a}
(a) 0 (b) 1 (c) -A' (d) -B' 08 tr,
03. EE qnrRffi FA-{ e[-s .Ili (gE< c'r(s qrq-<
q(t{ff 6slq Sr<- lAns: dl RX

(a) 0" (b) 4s" (c) e0" (d) 180' q<{Fr 0'qcs 99",|ds1frs{t{c{- tAns:
e- al
(a) R* o:t6{, Ry {Vr< G) Rx {9r{, Ry TsIr{
05. fi F.0 = tF, dt qq, \iR'ffi ei-<a:a-< q<i{6
(c) R* <lgr<, Ry <IV(< (d) Rx sTtrI, Ry S{r<
6$lcf $E?
o 09. foT{ 6ol{ csffi Y-qas< qTl-g<rFr?
(u) 0); @: (d) n
(i+i) xi
wt{FI: (b); F.d = lFx 8l = PQcos0 = PQsin0
(a) 61(ixi)xfr
(c)(ixj)xi (a) (Rxj)xt<
+ cos0 = sinQ = tan0 = f .r 0 =i wt{Fr: (d); (Ax I x t j
= -i x R =
06. {fr FER LM q<(N qr{F{q@cq s, 12 <<( 13 10. j+frei< JAns: dl
qs?F q<(i + M = Nqt, sm M e<(il q<r<:{6 (i) q< qF{ 12 q{EFIsGr
(ii) YZ g'cq
cdc"l-< {t{ csl{E?
(iii) x q6q5 ftq 90' 6$l6t bS'F Frd
frcr{ctrtrq6-r?
cos-1f (c) cos*1 13 (d) cos-1

rstffi
(a) cos-1rr1 61
f (a) i, ii (b) ii, iii (c) i, iii (d) i,ii, iii
\ iffiffidqvrc{frurrqql.. C
 m{Fulf, lEI etiit : qrillT-ol
tt. IxE=z [Ans: c] 20. [(i x j) x ([ x i)]. i c< rlr +-sz [Ans: bl
(a) fAB cos 0 (b) AB sin 0 (a) +1 (b) -1 (c) 0 (d) -i
(c)-Bxd 1d;Exd 21. I s E ffitsRcrr 1QE F{ <rn qn m'.q{ <<-
12. I = zi- zfs s = zRq.E-
[Ans: bl
ialIs E"r<-mwfu<rr O)f € E"K-"tdEt
x El<{,.roo
1c;IsEo<wr+Srott+s"
1a; ld (b)
i lI x El<'f ooo
(d)f \e B q<{{r{ffr+ttoo. 1c1Ix E<dcc+ 1a;jIx a=<{<o-+
Tqtfi: (b); f . B = (zi - z).(zR) = o 22. i=:ijlyiazftem,V.r=r
13. EBe?<l11<tr"ffIs.1=e- [Ans: al (a) | (b) 2 (c) 3 (d) 4
(DcrqffiT (ii) ft-s.s€ (iii) TFr qF cfi${: (b); i. i = (*"i * *i + ul [). (,i + ri+ zR)
firurmnEc6-$z
(a) i, ii (b) ii, iii (c) i, iii (d) i, ii, iii = *<i ** (i +fr121= L+t+o = 2
14. Ccq I,Ed,D s E affi ei-<r csi-{srdfl-d ErR-< z:. I.(n x d) = o {cE I,E e d e< {r{T q-"1-5 fi
qFres?
Er<? [Ans: c]
1r; q:ft1ft$ (b) qir-'REt
lcl ltrr-ftt (o{cqlq-{
(u) E O) 2E (c) 3E (d) sE
q:nqn:(c);f +E=E; d+D
=E;d+E+d+D+E=sE
15. F?rftq{nrc< fficrq({tqr'- [Ans: dl
oF+d = 0+F 6ir@+Q) +f, -- F+ @+E)
1iitl m(F + Q) = -F + -Q
Gru-<c+nEc8sz 6r<<q-c{tmIFQ| =suz
(a) i, ii O) i, iii (c) ii, iii (d) i, ii, iii @^lB (d)./r3
16. i = zi - 3i + 4R Err, f en RflaE FE-<TEsr<?
[Ans: a]
(a) zi-q[
si- 6y:i+ai-+8
(c)3i-2i+4[ (d)2i-3i-4[ dE 20N T{ l<rqR 70" c+tc"t frvl rct r qfr e s
17. I, X \Trffi {Lq 30. Crk6t ffiq, V q5F <irp-ir mtr+,astr <rq-d q&@ fficGEr cFI F-\9- qr<?
B"llir.]-?r {l"t 3,{s<r qre x qiF <ir{{ O"ttct< qrq- (a) 30' (b) 35' (c) 4s" (d) 50.
@)i O)3qsii (c)3V5qFs (d)6q+o wlqFr: (b); ffi wn s6d sR qs-,?'g; Gfiqrs
yl{frYl&EFrdt
EI$T: (c);Asine = 3;... A€os0 = (Asin0) x cot0
= 3 x cot30" = 3r/3 o-<rqqf TfoE G1rs<r 6tl 4kmh-1 r $Rt
t a. 1fr eB'r<-< oi-< e"Frr- 10 kmh-1 6{c{ qsB furs qlrsr {l[q ris &B
[Ans: d]
tD q$E Fi<<fi (ii)Erq<ftctE{ r$iz'l.. EiS-i a-{-Fi qgrqrfu9-fr< +r=r <o
"rf< "n-+r<Z
(iii) WqrE @t-dEBqrls{q (a) 98.5' o) 113.5.
frro"crHE{6s? (c) I l6.so confrQm
1a)
(a) i, ii (b) i, iii (c) ii, iii (d) i, ii iii
19. {ft d = -eBq{sEr{d,Eoa-<qsqr{- [ans: c] w${: (b); e = co'-'(-}) = 665-r (--1) = 113.5"
(i)cT.tea-d (ii)l:rl<fr{AuftrofrqTsr< 27. i= zi+ 3i- s[ q<(
= mi+ 2i+ 1oR q< m
B
1iii)arctff*ei'< TIT $S <r4 NE<q'{ ei<r-'E<3r B'EI4? qr{?
frrr-d GrF F{ts? (a)20 (c)21 (d) 25
@)22
ii (b) i, iii iii
IE
(a) i, (c) ii, (d) i, ii, iii {:nqFr: O); 2m+3.2 -5'10= 0+m=ZZ

rfirdrfi EE rr frga rqprr...

 'tn{frgrd oEI erGI : s[$I{-ot
28. q$F <,frsrE'd EEE ir4 {a[-+r{ I ,{<( E qr{ 3s. EE e?'-{,I = +i+ 3i+Q; E= zi+i+ sR o+tr
<ftsrE{creI'{qcq? [Ans: bl qNlEfrm< EB F4 fr#"i {<r{, {rlsR-$tr{
1a; li x El 0)ilnxBl cwFF-{ qr<-

ra,E4 (a) 11.4s <r{ ,4sT (b) 22.9 <{ qs?F

ic; ldlx 1d1 ' 'lAlxlBl (c) 5.1<rf q4s (d) 27.9 <4 qs?F
ficu-< Eftdm-d qlrqtm'r*<ff EQE qra-< E's-< Et€ :
ffit{Ft: (a); csqFq=; x li x El = lr.ns
,{sF ffis'R GrMr.qffFT 6rI lokmh-1 q<(
afrTcq F{ 6 Prr-t ficrd G@d qtrcnF "r<-<fi EF qrn< E-w Ets :
29. frsRF tss?
(a) 8 kmh-1 (b) 7 kmh-1
(c) 10 kmh-1 (d) 4 kmh-l
T{I{l{: (a); v+u = 10; = 6.'. v = 8 v-u
30. frflF qlF< 6(rld cftrl TE 6t[c.t EEim El 6{tql
q"K {frs c{qtr{2
(a) 120' 240' (c) 110.2' (d) 104.5'
(b) 39. 0e F-{csFfi? [Ans: a]
cI{t(Fr: (d); o = .o'-'(-;) = .or- (-i) = roa.s" (a;i+i (u)i+[ (c)i+R 1a;i+i+ft
40. OABC (STE-{ (EFFE_
31. freff{Gac-Tdfr-s- [Ans: bl
(a)a?fi-rf (b) qrlftrf layf <{oo-o (b) i7<{qqs
(c)Rqftsftm (Odfrsftm (c) 2 <{ qT-<F
1a) z./7 <uf eo-o
32. I=i-si+sk e<( d=mi+oi-roR qm, m r':rt{Fr: (b); 0I = [ ' oi=i+i
,a< 6sF{rffi<q{r ei-<qr
"r<*r<
q'flgff{z ri i kr _i+i
(") 2 (b) -2 (c) 3 (d) -3 o,q-xoC- = lo o rl =
qfi{Fr: (b); = = L rol
* * -.I " - = -, l6axoel=J.ry+7: =.,t-2
33. EF<r.r<TqsqEfr 1oN € TEs{sk 4N; <eEF
41. ixR=Ss? IAns: cl
i-:r,'rffi {rcal 90" GI[.t fr-$ q{r{ sR-< {l{ ssz
(a) r (b) 0 (c) -i 1o) i
(a) 7 O) .,68 (c) 8 (d) ./68-
,l'cT{l{: (b); 2Rl : Pz * *z .n
42. BftfiFBmafim4{q*iGf ellMi- [Ans: c]
+
2R2r = 702 + 42 :. Rr = {58-
",.
r,,,1
ffi fiEt{ c'(seF stft-+ "wM frunq qRI't Tc<i
11

34. 4ms-1 6{[Ef (fiw <tsK {q{ ffi 6ms-1 6{rut

(iDY+t\e{qffi{llrEEfuI{ffi${
q?Etr{ {&E IE{ cE.fr{ cm E-6rrd {Ical 4s (iiD{g{ E<cEu, Flq s EI't R{Fr F{l {Er
6SI(61qFrl {!tN- {C<? fi[E-dcsffic8r?
(a) 30' (b) 33.7' (c) 37.3" (d) 43' (a) iii (b) ii, iii (c) i, ii (d) i, iii
I trn-'/1) 43. e?-<qry3f66a267r-
>F rfFr: (bt: = tan-1 r6l{il6{'l = ::.2"
\ 1l,.{ ca" / = \5.,1
(i) sFi qoD ceB< <rft
35. c{tTtc.t qsB fr--Ws ffi-e Efr c{lT <(E{ Efr<
(ii)cdffiqrFFE-<<IFI
cFrRrsF[q<:E<ffid-
(a) 2 ect !u aq (c) qrl"r (d) V5 eq
(b)
(iii) 6st6[ cs?-d cs.@d FF(,4< sli]e-Ecqq W
FcErc$FE{6s?
{:It${: (b); Pz = y'Pzl Pz =e R. = ,../JP (a) i, ii iii (c) i, iii (d) i, ii, iii
(b) ii,
fi{ w {{ q {${ .{+tr F?"{rs qir {Fr fr6 sFt TtlI{FI: (d); , i.(VxQ=o
o<rq A "[\e$ dT? [Ans: a]
44. Tll-qTsT{q{llr< 6<rr-dc(E.l csnFr [Ans: b]
(a) aGF F3-< 1U)+rsfi'*eE-*
(c)q<{lq ot< '
(o{Fr@E-{ (a)v=9rs o)v= a.s (c)v =9r (d)v=*9

rElil 9) \\\ qffi-r-*esnsRw'Fffi... (f

 o1ql{frsrabarftf
45. -
vE-\r.]Trwq-?f ffifE 46. p €n{Fsg(r{i = (x+ 3y)i+ (py-ii
(i) emt{qrs{qtw c+mfrT({{Yel{ +(x - 2z)R cfr{rseqR?
"it-rr
qfr<-+tssfl{ (a) 1 (b) -1 (c) -z (d) 3
qqr{ra: (u); v-.
\ox +:i+:R)
i = f:i oy' nz ,/
(iD i$d ff{ W q[E qt'ts rg frufu *rg xrF s{ .
{{* + sy)i + (py - z)i + k(x - zz)}
tilO tq FALTI $r qirlfi fut Tr< 0=1+p-2...p=t
ftcr{6$1-qEq6s? 41. Vx v=!
(a) d (b) 26 (c) 36 (d) 46
(a) i, ii (b) i, iii (c) ii, iii (d) i, ii, iii {:ll{l{: (b); a1-( qq qt{rc{ sTs @ffi< ffi $-{
JNI{F[: (a);FH (Curtl co?r<-< 1(T trBf ffit +r< w$< clc"i"rs cdft<r F[$r frQ6i {s I

a CQ: f iE <ETc'{r€ r<16 "ra+f< Eli{q.s St s qT{t E-tr< #

01. {5It{@fda? IDB'221 08. Ftrd ffi,6il-d uE)e-|{rq-q F|6p 15q?tCB'22; CB' I 9l
EEil cst ftT ftFlT<E EF qnqler csir|s{ {l{ Wtr d< 6sG< vlt\e-Kcsq qcEI qoB 66at< {lT,
{{r{ qr{, skq{trF qrFt-d <1 wF Ft{ <F r
- sI G?{ cs@{ fiffi ftare g'rga aF frd.f sn,
02. (Ei-dmlc'ldfryqa<fi cn-<t IRB'221
r
09 lq.{Fa-{Sfrir<rq? IDin.B'221
EEE.ft c+mr frgcw @<tq e+{on frro rr fi!f, crqrqT 1Q <t .o-rrc-tfi-s o?< sft e<t firo
$r{ EE FE< $F[T fr[If"r Fr{ Etqrq fr'y-cs< y6r ft$ trr r:-r< strr*ro cf'I @A-{ {ffi r

<TEE frr?-'rw{ c?<Eca< lo. Ftdqffi<ff? IMB'221

"rk fir,ilt o<m r

03. sba.f{A? Etr cE< otldrc-ftT ur<as q"rt5afs qgq6o reE-<

lCtg.B'221 - E"ffi<(rqr
@!t em @A-<<IFR q.Irr ecFF6r eFE cr<l<<lFr
qrq qI e$nT6sEk<16 e.|{<r{r ll. <16r<tlsA? lDB',2l I

04. Ttrftr TcEIEI€ EEE c+rc+t <e u <r+< frff+trffi e"rs'dirs &
I ISB'22, S B'17: BB'22]
ilrE.?TE]g<FTI
EEE +F{qq qrtr F?-<<Fr, {<ltl-r6{ crtrqt F}<
6et&-isiFtr<Fqrq? IDB'21 ; DB'l7l
ccFc< fiffi fr{re $ca-< rrt s fi-o vrm <m
ftfi
r

corurrt*et eFF Fi-<(Fslqft-s<lB<

V(x, y, z) qsE Fi-rcrc Qrq, Curl V = i x V
rr6ks1fr< qR els,Hs[-Tr
05. q{qF@t{Ftm<re? t:. qEs eFs Gf{TIriF<rq?
ISB'22; RB'21; Ctg.B'2t,17; SB'21; BB'21;
IDB'21; Ctg.B'19; BB'19; JB'l6l
JB'21; Din.B'21;l
fifift reX ie ffi-+ nrc6ftrtqc<r+qrrffi
Eil qc.n +tStcar<
1t Rw crc"rrs crfiff fr1< ffiqs x, y s z q{F <irRd c fu{F qFF Ftd
w{EFr G CgB.rffi cfqmt fi{n <t ftr+t E<t Ss,
frrss-qIsdl s$, EI q-ls qrir @td r

$row<{qe?-<<r4r t4. ;rFt cstr{< mEI Ers I

IRB'21 : BB' I 7: \t B'2 r I
06. B'l7l
"Ift{cs}-da?
IBB'22; J
iEE a csEr<< {Fr:tir <I tlir 6sIHr fiffi ft-o
EEtr com (gA< rrf,,r< ffikT coRllr q[< q !ft -ellrs {, Elrs 4Fr <I1ll eB-< <mr q< {qfrt €
-
AcqfiFT qqr F-{t {-{, vr< cqt Ft-<r+ {r?H fr-ffi-tqs?E{r
oi<Ecqr 15. s{F?dFIr<F<rq? lsB,2l l
07. ,{FA Cst-{ FtrS <r{? EEE qr<rmr sm< s"r< sfro s&q{ r<r+< e$F
IJB'22; CB'22,21: CB'2I; RB'171 ei-< <m qE .9-{B< (EFFFK wtFr, orr+ ,i rorr<
EEEtr cs @Arc< tld -oo ug?F EItrF qir6 d< <m saf @f{ (Area Vector) iT 6$.€ (sBi <m
-
r
r

1fr<6-+<e.mft6iqqrqt- e
 cmdfrsralxqa
16. riFtcsfdfi? [BB'21; Ctg.B'221 19. qtmft-f c{ri irtGF <rE? IBB'r9l
!!!! f+nm <f r-?m 'rcE 661int fiffi qss"fn EEil aO usflq <E{ .{TF{ w"irs qREd mtm
v&-4to 69 eFFl-"FF @A-{rs {<ct F?-d <Fr q<( q16e1fu6qa5a1
q< fr-r s{ qfrR-T crs emfrA< frcs I
20. fu<hFt-dft? IJB,lel
17. csi-{ftstq-{A? [JB'2r; SB'161 EE ffi Eh[s, c{ F6r*{
cs @a-<< qnE r<rt
fr't c.<lr{I <tsql Et fu<6 F -n
EEE q$E Ga-d <tFtro Et <r Erstfrs csir.<
r
"rlfr
21. g-Fq&q;FBnFfr1&mr lcB'l7l
fi-ss +'-fi< a&$rs <rs FB-d fuw{
F?-m< tr'r-cq q6E snqrfu qro
r

tt. frqqq@t{Fhio<[?r? iEE nqf

IRB'r9l
Silst< ql FI TGI g?N @:E-d clire' fr-6T @?r<<
EEE! fO mB-$* FArr< {Fr esF q"RFd e.flqs firr qus< r+nqXflre qE m ftr+ v+r< <c<, cr
fr"t?-rcrfteq"r qcq stcr< frqq"r @i-d <rE r
ffi <qt o?<qrg< qq q"fqrffi ft-s 1

a CQ:>EFriwlqF$SSsla*tt-sT t
01. ffi"rEr tq4rFIRFiIrdI?
T A
Wtr EE CgBn-< cq e"fn q+F 6TER<tFf "lr€{I{tsstm 6$EII< E$r<Frr qr qF <Ift EFd $ffid e"lsr6rd
qrcq ionc< xqFq 6sl.gcld c{|{lRTffi lcosine) ecl.FFK {:tFt I

a ce:friE<$c{rq{Gt6{Asl{qT(lfi{T$EftsrT{IE-sd >
0 r. €td qffiT 6qqt{ <lFlle Ftd <tFks a"ts< +m-vHil v< r IDB'221
EEE! cfi.+.s-4n<rftdE"F Fb<wtd< lrwwtr<F<, V 1<r<qkq<rq ertr FAl<Ft ,rt..B{tst{r eEro
(Fr<dfi<catffi<-qlE{r qtr\,.E-{rtFrFqsBfrFE@E-d6srqciffi<<tFitr*qfumckffi.Isr<r
q 6fird dfifi{ FA< sot T{, <<( Fi-d crq cs-{R rtFtFd qfr+ie-fi Et< FrArs eb-* <fir ,{glr€ Fir
q"ttrd< m-qlT <tFlrs eil< <lFtcs m"nq< or< r

oz. Efr c<rtt*qs'qs{t<Ir|l.dIR{tA? [RB'22; Din.B'21 ; CB'16]

EEE Ea<rf iaqrqrs4flqr<RtsE € pfr<rvr<6 cot"tuu+ar<r
q{Efus E"IirtRT fi{ <tgr< qn E-ffi Sti.t +'qrlt
qr\9=
r

qEE Ts Eitl 0 Es S':t I q{R, cos O Es <V qr< qk F cos 0 €i {FI

<tgr<r Fcose q< qH q{q, qpfi-5 $ag1 {vre Eh <ffi iFtt
T{qs{Er{l
w
(tl{l{clT)
03. ffi@A3-dqfrF.fiWcfl lRB',221
U
EEEI
ffi FbL<( sfr t{r E{ {1q 6s1I{I ft-vcsTd ffi <rq,{aq-s({ ffi
eE-<ffir+r<r
"lIrR G@ fr'W ABC qr Eft cc<, AE + B-c + eI = o
I E]
 OEI
'FiI{frgEI 'fiT
04. fiq.r 6dFnr
c{rqld dqtqsq6lffi
crqt 9c1q't uHt ns'
iqq mr? 1jl1fl $d r
as.t, iit{rlTir r
[sB,22l
!!fl c<wr<< sw W € qsffid sirr E{@ <q F qr{ F <n cildrs{ o frqrv qryfts< wlcat 0 6{rtq fr r$ft{ |

#q|< 6s@, 6Frl-r+d qt{E €q{, B-qr<ffi@

W'= W* Rsin0
Erq< mm, c<tEfmd qt{s €q{,
cos0

-B
*1*4r,,
W"=W-Fsin0

fi$ {rcq, E|+R ffilq (w" < w') ieq-{ ffi

qc{sl sr r <fr EFt e 6qF ffirq esq <q e c+t r<{c"t F-{| q{,
s6q {at {|* EF{t q[eist &qr* rgql 2F sin 0 c<Fl r sjt qq mrqr< dqt qmsf EFI Tqq r

os. TH cstr<rd fr-s <rFEt qr r

IBB'22; SB'2rl
!@ e csE,r-<< ltr'W EtFF{Fr @A-d <t gl et< <rrr ,lsB Fbl<-d {ftat \lkfrqfts (Et-{ c{t't TL<<r EE
qql{ @a-d fr-c{t'f {rd 4FT FA-d ,ftrg$ {Inr {FT FArtrd {rqfr-T e fr'ffiq e+t fr'Trg c{r EtQ rlE cstr-T{
c+tcqt $ffi fro oQ I {q Gi-{r{ ffiRqs o fiI{ q-$I"t F{f Ss r

oo. fr+r* s.t bFrl-* {c{ c& lg q?t ET ffi+t

so u;s urq 6qq? <rlrfil s3r lC ts.B' 22 ; CB' 22 i JB' 2 ll I

[@ert<nQrcq{tEmvrve+:rgq"i{c<r<q{Efrrr<clrerqdrqqGyfi<+tqrdfrctl{r<r0,4rrFr
s-q qcE cos 0 ql Tr{ ffiqr{ I qtqfrs S"lI( t r cos g e aFt Er< r ers ffist TcAq 6c{ 6\stTf {ttrt I

07. EBqqrl{ cvil<<nft1+<o "ttm* 6{Er? $l{rf€d IJB'221[Din.B'22]

r

@ qE qqrF flqt0{ F [-<n sfr< c{ET xrq e?<cm-< frrrFtqffid r{Fr qrslE @tds'sqrThT|csTk r

qrh< qR. $qr{R ql

W <s "ll{c< I

09. crql{ €?r qrt ftck6 Ftrd aiR t €d r

[CB'22; SB'21; Din.B' l7]
@ aw ofmm 1o R'6 lrlrr csmt frT{ q{SF m re?r<-< cRrmi fr4T <t Frfiq F-dt q{ ucF <rr
c<qH F?nr ftTRh FAL<< fiftT fiffi rErt "ilqRA-< q-{StT qfr-<6q o"<t qE atr w<rqf{
ell"r<rr FtK(
mrqre flthft-{ sw stStc${ {sfr'{N fffi efrf I E1t w{qlr et-<,{$E ffTF6 Fi-d r

I0. EtrrdqR+ Fi-dfiPrt mfrf< frfi <lrCrl<,a r

[Din.B'22]
[!fi EiG< qfiro qoqffr{ cst.c<-{ sR fi4c{ <qgq {q <l:rqu q{r qst qtfr{ EQ er qft{ F3? rFR
Rrqrl< 6s@ e?<arqk qisB< q{R"{< Em q'r<F< ffifiT 6rrrqstqn qt&I-T qsrT r*,jra< fiifrt
c{rs el{ FbL, { fidR-t 6{Ft $irr{ 6{ nwq flt\srt {t{ vli eF <'fqB fr"tis-erq eB-<s(ER eft-d qF s frs
FnfI+r<r
q(r$R, F=0n,0=EB;E=BC q<(S=aD q<(i=60 q<t
qTfti oB-<r F c<frdR-T A re Q e< tnftA ccc+,4stq'cl a-<ertl
D
ct&rs oenco <wq qlFq +iil Etrffr Efqre <EVffi< eF <tg oD
Rt?rs-em si<erqm eR-{ ql{ \e frr Fl(6] <'rd r

qds,0D =6f+AE+ee+ad
i = F+Q+R+3
.'
I 1. qftrErfr qrGTtftE<dr? <ir{Bi<rd r
IDB'2ll
qrscJ6$f54f q<rfr-T<ostEI EFsr"
[@ei, vftt<-.teoBqcrrstft{T{r a ce,nqt"ii"arqRl< & Rnrs
frr< <m p=m1+ rt q+t T{ EHI Frq qfi s qgls R{< Bfi fr5< +r< qf , lG"nqt< E+< F-6 +r<
qs r r

"I4E@ drb stq W E{ lI "tfu-d fivw p.e Er<rq, Er ll I v$t <a nr+ p-+rw tiiFhk {sr {r I
q<( <tfu{

l-m qc?r qrTKs.rftq q(6rr ?<FB qem q&t <e cFF smc<s"fft{ {E r

\\\
qfur+tqurmfts?rqrqr... (I
 eFil{frisEl oEqE

12. @trtrd \9lqvldcsl(ol( C<l'18, ft

Csa[dd vltv-Rrqrqd?<FBr $I/
+l A? [DB'21|
E[!t ee< qfulnqq es6 ct-ttn ffi <t skl F?-<rs@{ rsgrl ft-Wo *trs< s-{C ffi:rwo31 q1-a1a1-s ,

et< slt\,-Iilqrq< ?<Ft?n{q fixfl: (i) vltvT<rqq qRl qr?F qFs(;r cstrfi frT <}Frd mb sEirf pEI c$,Ir{I
fi"5 q&Tt <t q"mft {E st ef+Irt +m r i . i { aiu V Er*t E{a "t itr{r tlV qk+nr< qtd g'flr r

1r1 i v e< rR (+)ve ar{ vEtu{ fR <I {{y em {<Il{ r 1ii1 V . V qr {l{ (,)ve qcE ql{E;r q(rffFr <I {{g
.

1fr $m r (iv) Divergence : 0 q5a qns re ft{s 3ng q{]"I ET I qrs, q FA{rErf qfr{ss|{ {F

#
I

divergence
&
(+)ve (-)ve divergence Zero divergence (Solenoidal)
13. rfrof* e.tffid (rr@t<tt< qr{tqArsIff? {rMq{r IDB'211
qlEl
fr+r+ n Tc{ Em sr{ q< Beff(.i F cos 0 d$Eo ql{c{ qFIm Fsin0

frcs {R, qk F sin 0 B'ff(1 fi-a6 ryv{ fi6q El6{ r € F sin 0

rs E&w c<rs tEbr Ecfl-q;r E{r t{tl frr{ {s (<l{r Bt(.t
fifrg srd ffiirf q!-cr< fr 56 qftrt {ls I

14. .qFE sF$ cstrfirfi-+Bf8 <qe qo-+ otr+<ft-+fiffiT{r,-<rtrBl4trr IRB'211

EE m eE< <tB< {r{ qr q<5-<r, Et qs:F 6r,i-{r 6sft{t Fi-<r+ Ek {Fr ftrr sFt s-?rq s+F
Fi-{ 'Nerl TF q<( cct qF,-s cs?LTd ft-+ qr & car<r{ frrs <I Etir {fl-sBrtr{ t fr-g ffi--s qHIs y
<rT{T ffi fdl-qt qlF <irl< G m qos csa< fu F-{t qs, EIcqTrF qlno qFF FB-< 1
<[Et ,{t qr{Is <]T{l{ x-q66 q-dfl\qs fr-s ffir<-< qs?F ei-< i, y-q6q< rqrq+ ft-o <<r+< qs-s
2--
oB-<is z-wm <r+1-q+ fi-+ <-<r+< qr-s F -< fr.i,iIq{qITE qFo 6!-6I q qs-s csA-dqFr\s z
fr -cqrils <r<qT vt-6LT{ ft ro r q+ft ro, e+o @?r+d fr -s c{rfirfl ft ro qro st fi fit +r r
"ttr<,
15. "fficq1 F?r<rdft"tftE oi-< q<(fre$q cs?ra-{q$<fr cst t{ftt qrtq{r,-<llrlttTdr IRB'21|
EEElnf cgErs{ ekT{r{ q{q fr-g qe{'.I frffig EceI Et[q-{ro frqag FE< {tr{r c{:Fr: F ,g -F fi*,i<
Rqfts csb-< r q-{fu.F, Em
q$s<t{ oBr+< estrd ql+ qr<tr< frqfft qrE qrq{ro frEq{ e?< <Fr r c{:Fr:
si \e 1i
"K€R frEq"l e?-< r cqql sm, Rflfts etc-<< fr-+ p' obr-r< frqAs ft. R-g Rerqq eiu<< fro {l
e?r-q< q+t frr<r r q{rq, cokfi FAr<< fr"rAs oE?-< e frer$q e?-< ,e< frs {frI Rqfts frr+ r Eptt {lr<6
r<{"1)rfu 180. qs r

1 6. wr+ csft, T{s?rrq rei-* fiidr <iHrr iF-d r

ICtg.B'2ll
WE SE @3<cnT{qmvl[h-< ql<-$tr{[ e+tqq<Iwrsirltq{r @wn cdr<<ffi is ,fiq R-TqG{qrt
SF wr+ oim< vtfr s ffifu e{ ft-E
qzq-< $<?FGF[ qFt r w{rem, qfi{ Fir{< fi{ e fi-o qot qesR
qrrd tls$'r+{tc{fg<fqq{r qffirEt, {5rFI cs?-{Etrc{q{|E Fi-{.s gr
17. i e i o< r{r<6 rO"t 90o qT cErrT5t q€$ rrsir .t'{-ail{It iF{ I IBB'21l
qs?F o?< r i <q y-qs <!rFK qFF Fi-< r x-qr q<( y-qs q< q${6 6st t e0"
WEE i <ra *-*+ T<FK r

oRi e<( i q<fil<q n;Irt 90'q<F{R€$)rg<+sr

18. qE e&-<-* re?a eGFrRfiT{Itr frr*ETET;tt-<iHfi<<r lBB',2r I
ffi EE FAr<< F?< e"rq frfivt {E {tr* qt r ct+t A x E = AB sin 0 q{ft{, i, I x d €s ft-E fut a61
t I 1

f x E q<i E x i,.{'( ffi cnn ft-g fr-s R1A-rc{t r q{N, I x E = -E x I r ... e?-< e"F{ frfiT{ Tq flrt lt

IN]
r

"\ qfi<6+*eqcsfrw{arFf... (I
 ,tEl{fruld !8't3! : O[t8IT-ot

le. ir x k qrtrTla Fi-d r-<Il$t Fd r UB'211

[@ R = k . k sin 0" i = 0t;,. lR x tl = o r R x R e?m<xlq a*r r vtQ R x R eoBqlq Ftrd

x R r

zo. e€ ErE ffi{ Etr cst<r$ sGr ffirr firE .{<( R"IAg firo cqKlm ER-d ft-{
qrt cr< fu <iNit
Flr lcB,2ll
E!fl csQ srt ffi{ EE d,K(,q$ftT +Ek ftro e<( frlfts fr'r<t mr<lm
a cofc'r E&--d ER-d ${ llr*
q5as ft-5 q{]-q E[< l| r sf<"r sfr rfr cfirqt 66rrd 4ral 0 @t"r tsR sc{ vr<,
q&{TtKfrts c$ittE, trn e, = + e, = trr-'A}"ili;
oSffi ; tan er =
*r:ry
!&< sft< ftlAE frr+ mt<trE, trn e, = ffi * e, = t n-' #.."1a ." 0r * 0z

qdn, ER-dfro q+tqc<qIl

21. ET{t${ir {rdrqr F!-d dtFr* oM-q s,t{it <'it r lDin.B'211

fi!fi e<E e?< afir+ ap1-6 Es EE B"ttrrt frsst €t qrt Ekh< EtIi"I <FI I

e+E ei-<F co@erB"ttc.t F q<(0 e frs@F<tqcqtr F e<(8ErqR q<ffiI$tr

e(F< F -n
cqlq{, f, =F+d= Rcosai + Rsinai

'nft evr* erm ei-* 6{q{ftG <rt$'tF-d I IMB'211

wEl
- ssm
ffi Tffi cl{rcfr-r {4 mr{ Em r e s n ft-a @ tFr< sFrr{ Ers ffi't s'r{ r

Er'{vI-dIEFftrr "tlRF{atEFm F1 s F2 <EI qcIMTGt e<(ftq?Efrm E&frrl

<EI R1 € R, q{E< T-[!t r R1 q<( R2 e< Ek R {FlBro <Ers
qfrffi <Ic.f r

2 3. 1l{eRR Tt qtqrE\3 erq-{ qlrdlQ iloftTd

{tr{'-{ {ttt q{w rr{{ c{t{? <il!]t rFiI I IRB'191

EEqnr+ro w"tct qsn:r< q1qfr-s (<{ R[-,Tl{ qIrs I qttr€ E{l<r

uEf< {Ts <I{s-{R{t elFErE€ qnmfr<
:r-q{ qti?€{ cfrrs {tr, qr6rt{ Ffcq ffi q <B E1-{ frqaE fim <nq @ <l{ er{sor[s Q< qrq r qt
frqAs ftto <E{ qRqfr+ 6('F slTc.f qtr<tQ <tstrc-d <tf61 c-{s< T.Gr r

24. vFr {B qi fr{cffi {EIr{ c{lsffid T3 c{lqt 1f <|[ iF-dt ${- {il{rt Fd lCtg.B'191
I

[!f, ut+<te e fiRT q{tlt.< c$triT qro s&< <ftr< fr {As ftro q<lriT Eror q; <ltc{; fira qFr< q<< c(fiB
F3( <tFk fi's << agq6 3ftrt r ut< $r+< ft-o rQ< <frt< q4G< ftr+ ccs' c$E Fi-{ <tftd frT s {6<
".
w <<f<< wo{ftr Flsrqn $t call-ET <I <I[ on]< ffittlB ,{mq sB-< nftF frrAE frr<t $ $e e !'&3 stt{
frrs cqE,r"rd {.I <{ q{ r qdtq, vFrqt& q fiscT{ mlaj c<tErE-d {rt 6{l{I <I T{ Ffi ${t r

zs. qi:tq.r{fiF{fftdrffi F rtrdsfr WCN'ftrr- <Jt{rl$ir I IBB'191

EE q+tqrr F,d e Rffi{ ffi

e?m-< nk a+l <r< <ft om< B q,,
""n, qft< ffi q"r<E< fr{?. ftro ffi{
"-/\d +^lP;?-
Grsffi Efr aqn fitg E{r {E<R ^
qrt4rq ffiq ffi
e?is-< nfr 5i <re "m< r
C/ >. \A 'Rt
tr
26. {{{ firdrf F {{f{ Csf{{{-$r{ffFd
-d
I tcB'191
q:rll <t qllsKFI u<rt {:n+rt ei-< r wqFr F}<
E[!imzl-{ @iT ' 6{T< qrsA'{ Ftsa q< {iro 6{t :m {$E

I E \r
qnqlq-{ q{r<?r F}-d q< {F wFt slitf T{FI F3<rq4s, T€{{{r{{ oE-<q{FI ob-< qs

es \\\
r

{ffi{delgnnBgriqEqt... (I
 qEl{frGld lEIofSi : olriflT-ot
N\
zt. u€ vt0-* qfr eir+* cst tv6r e ffisrffi {Fr w{Fr q[e q <infi qr
"ttcr ft il [RB'19; DB'l7l
r

EEil oA, CiE *Bu d, E c<(vtm-d Rf'rt-{ s fr6sFtq{{:rFr I

.ld+El= ld-El
+ JAzTBzIZAB;oso = /A4 Bz + rAfisiI8dr0j
+ A2 +82 + 2AB cos 0 = Az +82 - 2AB cos 0 =r ABcos 0
=0
... A = 0, qrFrT, B
= 0, q{<rt 0 = 90.. {E<R cl-connr esF {Fr Ftd qrq qq{ Ek[d
ffir<ff 6st t 90' qrt FE-dSffid ffFFF-{ € frrstutq{ q{F e€TI Tg-{ r

28. qErtcq-{'ttrrsl ffEt[q Fl-d fi-stq6T {F{iF[ <rf{rt F{t{tlp

lsB,rTl
[@ ftx qar utm qlErg &fuEk< T{ qr{H sRr $E€ q<e <r{ qrrcq-r
Elrrt cR-rrr <R, $trd s&frTf <-o nr qt <ffid q-Nfts ga$ (R*) qNftqr
alq6afr6nqF qaqBarB"l]\t (nr) eqqm E{frs+r<r

qrE q+ T€rs Y" t 6{tr.t &r{ 6{ts{l{sm $Fffi {{ I

-ir IBB'l7l
EXEI
6$Ifit T{rtrF p a5q g mkcf br{rE T€r E'rd frFT.ks q-{pfr.r (dt, F* p s.. 6
= ,
qi <sfrq <-gBr+' mtr frrs rlsr lio
{s{tx F, nFt ctr< T€rs {ll-c{ 6rs
rrl€Rl Es {qq qr{r G,l'C*t TFI s:r qrq, coso q{
{H Ifr qk?r I l'r{ F*s
<tVm r qt qft +ero bHk qq-{ rq 6{rk"f DF{f E{ I
30. qFFRq€t't cst-{rfT.[rtr{ cst-d{Et 6{[s eftGt_<rF$I€r
lcB'171
EEE EE F}< rfr fl-:r9ir c{lg{FT qr e<i qrFd ql{ {fi q"mtr< Req"f $, sr< .c-ftq-r*r frere{ *#1
E{E @BLT{ {l<rr 6<r{rqqE<FrE(EEftqirmqqrd{ Ftdr6rREffl F}r<rd {Ts6s{r€ wfs{tq$rsrElt
freq1 ffi.fl,cs wril.I Fb-ds r{t {rs r

a cQ: irsr<l ET{{l"fl{.ffi c* s lTit E-s* >

ol. Efr qftrrd dg.'Is?t <qbid {s'q{ 6fi?
EEE xq sR, o fi"Wo Efr <nei< EtEq c({sF r ftvro1 qq& q<t , frtre p m qE Etr.t et_-=-/R
frsl&sq*t<ttr nc[, \5l{Efrsu"ffr"t s Bst$"fiitrqNfi-o$"frt Fcoso, q<fi-r oA
=
<<f<l rE-gtu"ttrt = Fsin 0, qafro os T{|-{dr<rs-d\5rNfr€Uott$t Fcos
fir+ eFft-Tfrr$tlorquqnu<sqrq, 0 qr{6vss:rE6 r1'Gr Fsing q? {Ft r:r
e <IFlfrf,g ffiffir

E6q qt(
.lA,
Fcos0 €F qlT 6{Ff CC{qft, A\5T FFf ftf Cm Sfi{ BITI Tqq
Esr C$-$1 \o-lp<iFt Fe
qtTfidfrT$qftrr{f{r
:rrtt o'-rfu r+a q.t F r Jr.[E F -{ T{
r-o5T Tjlld Cg}f
6fim
!@ offi eE< qRvn<
ffi
ffit?l rttrr r <t- (i) TFr s (ii) ft-+ r c{Fr F}Gr qtq

{drqrrr{Firr?.trq"r<frr+c?.tFi,(n<fr-o qoqr{€ qF&{sre atrdrvric-s-{rT?.t@?<TjrR

e ft."r ffi
crn ER
GA_{
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03. E6q{$T F rlrsR-F{+€ T{c[g {Irdrl 6{ilr?
p
EEE oa, o o nE {{Fft 6FFt = 0 r efi-{ Efr qr, R = JFmr + 2po cos 0 r +$r 0 = 1a0.,
re?-< e<( qq-<
e<FRT-{E{q.Irqft, Rmin = p-Qr,sq{cmficq, p q<(Q{:Il{qrdIRwErrft-g,
n e<lqwvlqr

rN R q<T,iv{TFrt{IEca qrKT|rIsiq
EFqcTt{6Eac<<nftoil{s w<rs tr<mr
 qqIdfuEI'EIm
04. qTE qr-r cEA-d ffsfc< "n
s{t {tfl
[!fi m 1+ ar os't m].c{r F?<ro sl-?r {Fr Ekt sFf q{rt Ee obcr* ftr+ co-r FB-d,[€rt <l{ r

qF-s Fi-{r{ erfi{ s:r(g cHlirls Bfi qrtr EPr 6-€ (") cEsl <{r mq-{: i a, n fu]1fr Errt e-ffi
celB qFrfd
@A-d eFFr.tfdtqs r qR, I esF 6eB< slr rt{, A * 0 ... = a e< t* e+-o oi-<
f =a

a CQ: frnE <qffi{r$ rst6 {ftsl-< eLTI"t s Ewrr6 qTEIq{s eH e l{-qt Es-{

or. etc6ftaqRre <r<qrftffiR-q o (o,o,o), p(2,4,2) q<( Q (2, -4, -4 ) I IDB'221

({) FQq<cnfi{r+-dr
(q) p € e q:tq<qFr Fidc{ firqtrmemF {t{tElt{-d r 8
--+
I o-"* vrre, o (0, o, o),p (2, 4,2),e e, -4, -4) qrFr, OF = (2 - 0)i + (4 - 0)i + (z - 0)R = zi + +j + zg
6Q = iz - oli+ (-4- 0)i+ (-4 - 0X = zi - +i - +E
qsq<, F0 = o0- bF = zi g +ft zi +j
- - - - -zQ = -aj - oR
Fde<qn, tFdt= JI=Flf-6r = 10

{R,0i=Aq<(6d=B
'i-:rqi<EI?qrq, A. E = 0 {c9er{ r

E = (zi++i+ zR) 1zi - +j- +Q) = 4 - 16 - I = -zo

E
6qrQ, A.B + 0,s1QP € Qq<q{q6 Ftd 1Tg-dq{ 4g I

{s.rqfu
qF.I{Rr
02. 2 krn
g kmh-l IRB'221
qlE

qs"t TIRI I kmh-1 6{r't d-fi Etfrrs lfi-< cq rdI<.d "frd E{ r riF6t {tfu qrq c<['t {ft{ efs rdFrd trEt utqls r

qft{ {q z k- r

('t) Efr{rfqr.tTtRfrs ffit{firs fu DHFqtrfu

F) Efrtmd 6{rr+ ffi aq xlrr ffi {F Er<?'flffB{sk{ {rl'{rt ir< | 8

E CkEd 6tt, v = 3kmh-1

d'sF ctt, u = 8 kmh-1
qR, qk6{t = *
GIE <|{d Eflrft frcf ffi, v cos 0. + u cos a = w cos 90o
v+ucosd=0
cosa = -l
.. o =.o.-' (-l) = -.-, (-;) = LLZ.oz.
lrE
v=3 kmh-r
 qntFiu[{.Hqa
@ w+.ttRr<mm,
Eqsr<< {kq8{R ffirq,
dkr<ut,1l/ = \1;'=7
=\1tE=g
= .644
= r/ss tmtr-1
sk ffict{:K, d = wta =} ta = * = # = 0.269 hour = 16.18 min
cstxtfu<ffirq,
d
TT{.t'= usino '
= asin90o cr = 90o
=?n
8
= 15 min
.1 tb < ta q{t\ <-<F6t a1fu6 aq {q{ EFIriI 1

03. ffi @trrtFt{qt-qrc d = zi + zi - ir, B = oi - gi + zi(

qqf = iory + z3)i - (3x2 - z)i + (txzz - y)k lRB',22l
11 d s E eb-<rr-a-*zwft'ro eq{ Ntrdffi qa r

m Eaqro 6 Nf?Fqq{Tftn ft-{trtulQo{r 8

E n"dq<ftfrr+oe-{,
ti i
lz z
kl
Axs==
le -rl=i(4- 3)-i(4+6)+[(-o- Lz)=i- 10i-18[
-a zl
Fi{.
- -'' fi = jri i-roi-reR
= ...__!]9i:-1!E-
- = i-
g 4
,142s ,l4zs- ,l4zs
€<s-<r
lAxBl Jr2+(-10)2+(-r8), '@S
f es-q<arftTs€GFF -<,
f =@-#-#
E c- sa< v14+ftt re<t-rr 9;rI curt e = 0 {cir r

({a,curte=ixd [i=*3i+rgi+ranl
li i Rt
.v,d=l ue -! *- I

lo*y + ,' -"3x2 z 3ru,2 yl

= i {*a (s*,, - i - }e - sx,ti - {i I f s-, - r) - *g (oxr +,,)} + n {09 i, - :x,) - $ (oxy + z3)}
=it(-l) -7) -iGz2 -322)+[{(-6x) -6x}= -2i-o - 12xt= -zi-Lzxk+o
mq, V x d + o, vtE d csffi 1ftftrr
04. s00 m drqa elF lftro 6 kmh-1 6{r't qls eRlfu Ecq ret rftD {t{ e ffi effiq-d Brtr.B {vR
6iF6 "lR WTfd frhlg frrqt | {tt{ 10kmh-1 6{rt @lrsd clr{ d 6{lc6t q1( ffR 9 kmh-l 6{r{ qEg.it ctFt
qf\rlm fu-d rtFrg tsF TirE I
[Ctg.B'221
({) c c+trlR{FrsoqrqTtQ-d cilqdqnft-qq'rd "ttry cffq(<? I
m Eft"lsq-flrd mGvnz ffiB+fulEFrqrEFrsnN3 8

I
I

E \\\ 'tffiffi EvirrRg",iqffi... (I

 qql{frg14 bEI : o[rtil{-ol
'tE[

E qqfq, ql-F< c<ti, u = 6kmh-1

qtfo<< mut, v = 10 kmh-1
qls\e flfrr{d 6{crldwF66sFt = a
qkcstct,o=90'
q6-<[ffi,tang =-J!I3- 3 sneo. =; = #i#k
o = cos-'(#) + a = 126.89"
= 6 * 10cosa = o +
TE-dt\ @t(Eir cftq 126.89' 6ftc"l sEld sli?E $eK fiMl1&
q*r qlcv c{aftp
"lRr< I

,rt'E6v tflq, u=
E 126.89"
qttr+< c<ut, vm= 10 kmh-1 = # = 2.78ms-1
)
1ft3Eq,4=5ssm
qft< Eg r<l{{ nltd 6c'l<t Sott('t
.'. vm sin fl = 2.7I sin L26.89" = 2.2234
qM qlfr , d = v- sin c x t. ; t. = = ir*9l :, \n = 224.g2s = 3.75 min
$fo<< srrTq+{ c'{T, tm = 3.75 min
"** "--3-

qffii, Fkd ctf, vn = 9 kmh-1 = 2.5 ms-1

FR-{{f ir<q(g ffi{ q:Etr, tn =
* = H = 200 s = 3.33 min r. tm > t. c{ls, frfi &sr{ I

05. eTfr-{ qFF qsrqrt Et$rlEt € <lEIH3t 6tl 'tt\eTl6tfrql+tl4r{, ICtg.B'221

Q= 2xy2z3 -+xye V = (y2cosx+ z3)i + (2ysinx- 4)i + (lxzz +2)k.
('t) (1, - 1, 2) ft-{rg A qs.Td qqcEf{ cqffi +* ffi r \9

$ &fu & q$ffid <tvt{ 6i'tGT 1{l QrqT ft-+ g1ltflfu+ frcavt< TIITrl {s qls I 8

q*rq< q'Net,
fB* )
E cqsTl qES, & Q = 2xy2z3 - 4xy
qrdqtfr,n=*i+*i+*a
affi, Vq = (*9i +
$i +.9n) .(zry2z3 - 4xy)

= l{z*v'r'- 4xy)i +
* (2ry'z z3 +xfli + !Qxy2z3
- - 4xy)k

= (zy2z3 - +y)i + (4xyz3 - +x)j + (oxy2z2 - o)k

q?F, (1, -1, 2) frTrs
.. iQ(r, -r, z) = t2(-1)z' 23 - 4(-1)]i + {4 1(-1) '23 - 4' L}i + t6' 1(-7)2 ' 22}i
= 2oi - 36; + 24Q (Ans:)
E m\e{t qt(q, & fr{ a q!6rEr <181rffi 6<'l
f= (yz cos x * z3)i + (zysin x - a)j + lzxzz + zlk
q{'drqrR.' V=:i + +i +:fr
ox oy' nz
<]Etrq c+fq I{q frqft-{t dl curl qa B'R fi6< +m r

I 6i i
3
[t
n
... curt v-=Vrv=l I

lr'.oil*,' zvrii'*- + z*,1'+zl

rs*r' + z) - (zvsinx - +l} - iffi raxr' + z) - (vz cos x + z3)}
= i {$ } o1

+R{} (zv sin x - +) - $ {r' .or * + r'l} = ifo - 0) - i(32'z - 322) + Rlzvcos x - 2v cos x) = 0

fm Rre Curl V = o, rc:t & fi'+ <lstcq mtr+ tfi Re ;II

80
r

\\\ qffi-r*qs,rTBeF{qEEt...
e
 qqI{EGEI }EI 9E[
06. c+IiRt q+fi-q ro
-s-1 6trt {tvlstrrlB'tgfrr r o {{'* .{+wr<rB z0 ms-l csrtffiUfuTrfl&T{{rtSs,zzl
({) 'll&ElTs1BTc{rtFE"tfrErfteT{r<t{? e
(T) rfr rfl&r 'rG{Rqftufi'm zs ms-1 r<rof <E{qtqurrvm & lrfu Etm@1E 6trs rur qRfi"r,{e
{ffisr-{r<Tft? "flff&-$frrs{rrdffi,1Trshg-b"|-qt'r{Edr 8

E fEu m't, li.l = 1o ms-1

T,fur'll& 6{'t, l%l : zoms-1
rnBotr+@<mrt"tffio<i<m+,i". = i. -i"
= i. + (_i.)
1;.. 1 = rtli. lTliJ =ll@Tffi = 10r/E ms-1 = 22.36 ms-l

t6 ]# ='*= , - 0 = tan-1(2) + 0 = 63.43"

o=

,. rtlEEf{f
IE{ F ll"F 2 2.36 ms-1 mcet v{fiK rrtret 63.43. 6$.lccl lgrs fi-{rr I

E fP<<"t, li.l = 1o ms-1

<ikflB.<c<'t, l%l = 20ms-1
aq eFIC, li,l
= 25 ms-1
"ttG-<
frqfts aut e <E etr6q{ Efr, n"" = d + (-i.)
= (25 + 20) ms-1 = 45 ms-1
... +iqIr{ il&-< ure-$ 1E< m "rffio $irtrr, i,(ac) = i. + i".

.'. tan o, = Ji<! =

ff = +.s = = 77.47"
=#04{-isr-
= 46.1 ms-1

''l' {rs {t, "e'( 'fi& Dtsrff{ IE &.<t "tvI{ 6Ft frE E-{rffi cfret 63.43. "f<-<frp <"R'flG<'tGT RrAs r
m
fr6E 25 -r-r Kctl <l{eF{R trE \s-r6 @ cico 1W c<rt q* E-qrfd {kel ZZ.+2" r sdn
Et mq fE Ciro "iVf<
qRcHqst{c<+r
07. p(x,y,z1 = zxya -xzz qrE 6t,atl <lFf q<( f = (zx+y)i+ (3y + zz)i + (-sz+x)ir qsB 6gt{ ElFl
,{a1f = iOxy+23)i+(3x2 -22)i+ ltxzz - y)ir V"m,rcE e}<<tFtt
lsB,22l
(rt) (2, -1, -2 ) frTrs p e< celffifitfr++ r
I
$ i
Bfrqra rfls e B re?<qr$* crq, c+FrFcfufivrq e{( 6{rrqB q{4T+E Et'fiffFr fr6drr.|r r.rnfi
qqQc<r

E p(*,v,2) = 2xya - xzz

p q<cd&'Grt, vp = (*ai+fi+
fik){z*v'-*,,)
= 2yai - zxzi + Bry3j - xzt
'1 (2, -1, -2) R-TCE Grad p = 2 x (-7)4i - 2x (-z)i + I x Z x (-t)3j
x 2,
- z2i
= zi+ ai - 16i- 4R = roi- roi- +[
I E
 qEr{frsrd lElrfr

S d e< oiu",g"o"" = f I
.
= (fi i + rli + in) . {{z* + y)i + (3y + zz)i +(-sz + x)R} = 2 + 3 - 5 = 0

E€<Dire.gen".=V.B = (fii+$i+.lt) '{(oxy+23)i+ (3x2 -22)i+ (3x2'z-y)R} =6y+6xz

.. Ioffiqfr+rutqr
ti i R I

Iq<curr=V,A=l *-e- * * I
--i(-zz) -i(1) +R(-1) = -2;-i-t
lz*+y tyir' -sr+*l
liik
lrad
B€?Curl =VxB=lldxAyAz
-
laxy + z3 3xz - zz 3xz2 ,l
: i1-r + zz) - j(3:Z2 - szz) +R(6x- 6x) = (zz- rli.. I s E,a< cflI{lFE qqftft{T{ r

il 0,=60"
F,:4N

o.=45' IBB'221

Fr:4N

C@frfficrsftr rei-* o RtrsftF{tftq<Grqr

(et) i, s f, ei-< 1F e+F *'*vRcc< E/E <RFm4t it-{F qEBGt+trd cs-q'qe ffi e.t r

g1 F1, f, e F3 cstrd F,rF-d fifru ve cor+ ftT + fust +<r<r {fifu-+ R'eF.tt{s {sti rlt r 8

fE <-l
E
Fr = Fr cos 6o'i + F1 sin 6o'i {R, WE + x-qs
= 4 cos 60"i + 4 sin 60'i SN -+ y-q$
= zi+z^li fi\e$ ctcs, lFrl = 4N
lFzl= 4N
i2 = F2 cos 120"i 4 F2 sin 120'j x-9.Gffit ltal Fr q< CSFt = 60'
= 4 cos 120'i + 4sin 120'i = -zi + z,/$ x-q({Fit qkef Fz q< 6$lct = (30 + 90)' = 120'
rl li i
q?F.FrxFz=12 2,11 ol= (4{3 +4V3) k = 8V3k
l-z zv= ol
." lF, x F, | = 8J3 sq. unit .1 {ErsRr€{ cElF{ = eV5 <'{ .cs-$ (.ans)
I
Il 'q'cco "rR, Fr = 2i + 2tl3t
Fz=-zi+z.l1
F, offi
x-vrm {{lqs frc < {k"l 180" + 45' = 225" 6sI.i Eq'l"{ Frd I

:.i, = 4 aas225"i + +sinZZS'i = -Z^ln - 2,18

F, + F, + F, = -z"l^ + (+'t3 - z,/Z)i
I
qfrq Es"rn r+t"t = 180"
... 1-g[s1 qf6ef
- tan-1
l^fi;ial = n+.a"
I- \\\ rfurl.dEstmfror+tu'rt.. e
 erql{rfrsurd oNqa
09. fi-€Ttqlrq, qTE Cstrf (E- f = (6xy+23)i+ (Zx2 -z)j+(exz, _y)k IBB'221
('t) (2, -1)fi-TF d er cetGmtffi F-{r
1,

$ tttfiEr fr[Hst td {Hr<, 64-ett€ a f csffi qfr -fi vln ffi q1{s.trtE q6z 8
-)
f-E"r
E A = (6xy + z3)i + (3x2 -z)j+(3x22 -y)8.
FA-d<tFt<cdffifi{rqc-s-{r
E [= (oxy+23)i+ (3x2 -z)j+(3xzz -y)k
aivl = (fii +*41+.3R) '{tov + 13) i + (3x2 - z) i+ (3x2, - y) R}
=6y+0+6xz=6y+6xz
..' I csffi {fr{TstE.BI r

li d i
d
nl
d
cr.t f=l laxdydzl I

l6xy + z3 3xz -z 3xz2 - yl

= (-1 + 1)i- (3zz - szz)j +(6x_ 6x)[ = oi+ oi+ oR =d

.'. [6sffiaiqEq-ftq1 1a.9
Y
2)

10. x' X frcFqF'rr-d q<rftrE{ q{ar{REs{Et\s : IJB'221

-1)

('t) 0d csirtr y qrT-r {tc{t ss 6rt t Eqtn +-*r<r

Eftqus-d

$ Bft"imd 6f q<( 08 cst-dT{,R."Kq? ft-{t- dfi&<5sk<: uttra

(Ft 8
r

E q{r{, A(1,2),0(0,0) q<(B(2, -1) A(1,2)

or=(r-o)i+(z-o)i L/
=i+zi .Y
tft. Y w6up4 4;6q o 66Ft s-GI r "l\
-tqfq,trno=j I \
B(2'-l)
.'.c=tan-l
O=zo.sz"
tr 'rt'<u,0f=i+2i
6E = 1z - o)i+ (-1 - o)i = zi-i
0f oE = (i+ zi). (zi- j) = 2 - z = o
q?Fr, .

mreg of s 6i e< dot e"l{ 0, sli skt qt

I "t<-t< r

E
 qn{frcld iEI clilt : ulrflI{_ot
ICB',22l

ffitr' v --5 kmh-r

^::AB:{Ar efq:2.2 tun

t20'
@I(sit 6< f , u:3kmh-l u:3 kmh-1

FE: >> G-E: t

2.2 km qr€,{sFlftcs qt[sd 6{ot s trmtr-l rdvf*qFetffsm ffi ffi ffisT|<r qrFt 4e]T {vfr
s kmh-1 6<r"rt tr-q-) q.{ttcr,sr ftfrs ;frEle q+Q crr G-q-r c-{fi[T dsF qrrs F-fi r

(t eqT{sts-dErR c<{fi{rv< r I
61 B-s {vr< effism$4Tq {lFfursfrrssotF{ r 8

E Gl6O-?t C{rl, u = 3kmh-1

{-e-ls< 6{tf, v = 5kmh-1
SR m{, R = fiz + vZ + zuv cos a = V5, +-5, +
... Z x- ilT x cos 120" = 4.36 kmh-1
qts-ls{ sfr m{, R = 4.36 kmh-1
E a?FqtsIs(nft qRqF{:r'{,
d q{tF,d=2.2km
22 v = Skmh-1
= 5 sin 120o = 0.508h
a = 120'
rr*\9liF-d4ft "lREFwr{,
-d
La -----:------- d= 2.2 km
-

t"' 2z v = 5kmh-1
:. = q.in q0' = 0.44h a' = 90'
... 1, 4 1, qefq, {-s1-q sffilgl$ rr futr Frqft << I

lcB'221

(rt) TE Etrd Erfr x-qcqd qlc{tr G 6s'Ic.t fr'stftq vt fi{T Fd r

(q) <EEtrdqR-{wqF-+S"ttct s ErtB"ttrrtqT[cr 6sHF fiFi? FERrslIg 'tfi&'$Ifucqq.t€ I 8

-l
t-trEr
E qR, p, € F2 q<Ek F1 qrclc{ e mlc6i ffier
tane =
Fr"ino +0=tan-1 f--&ltry!-)+o=tan-1 ( 8tin60'
) +0=23.41"
I Fr+F2 cosq \Fr+F2 cos d./ \12+8cos60',,r
x \T6Fd {rcal E\'f{ coFt, e,. = 90" - 0 = 90" -23.4L':66.59"
I- \\\ 1ffi{dEs,mR6a'Ft'qt.. e
 etSl : allll{-ot
'tqI{R6H lEI
E fi, ek x-qcsit{Icct 66.590 csnBsffi{r{r<{c[TdeR<{H,
"'f'EAo
R = l.Fj-E + 2. F1 . F, .cosq

= 182 + t22 +- . 1Z .8. cos 60e = Jm4 N

R e< E{-NF-f g"lR1, Rx = R cos 0x = Vs0a x cos(66.S9.) N = 6.93N
R q<BffiB"iI$i, R, = Rsino,
= V304sin(66.59') N = 16N
'.' Ry > Rx q{F,, Efr.{ Bq{ BaRt cFt r

P(l,3,2)

13.
IDin.B'221
l, 3)
x
Z
6mr p e a ft-{<qqr csAd<llt+zq F s 0.
('t) AoPQ qT @€Efi{TT-{r \9
g F+QeF - Q ot<m +v qrs.d {tc{t {q|{ 6rFt Es'N Trd ft-dz,tfi&r ftGm,I{e c\rt{e Elts r 8

E {qfrTit{tto(o,o,o)
qfi, F = (1 - 0)i+ (3 - 0)i+ (2 - 0X = i+ sj+ z[

8 = fz - 0)i+ (1 - 0)i+ (3 - olR = zi+i+ st

t
li i [l
a= =n - 2)i - (3 - 4)i+ (r - 6)k = 7i+ i -
" l; i 3l
sR

..lF"Ql= JOPi@i]-sy =tB

... AopQ qi cENr{
= t x (p x Ql) = i "lli = +.:s <u{ qqs
"
E *$*r.e,F=i+ai+zR
Q=zi+j+aR
F+Q = 1r + 2)i+ (3 + t)j + e+ slR = ai+ +i + sr
F-Q = fr - 2)i+ (3 - L)i + _i+ zi_ [ e_ slR =
F+doffi+v-qm<rkarsq"Fc{IccF{r{ 0r {64, 0r =.or-r (ffi) = rt.tr"
qT{<, F d ffi +y-wrq ag.1 g** corc"t:r
- fi{ 02 {6{, 02 = .or-, (ffi) = 3s.264"
F+d ig F - 0 eb-<Er rall-€m +y-q6ffi qkat s5.5s. q<( 3s.264" c+tct EvF sL< {kFd {Fr q:rFr dr r

sc{te,, csA-<E{ +y-wv< ftql qrt qRrFf 6+,l.t ts"rr

or< ;rt r
 OI-GI : E[{tflII-Ot
''qMRgld lEI

14. [Din.B'22]

A qr".ra r<'r 2 kmh '

S@ @Irc-d {fts qqw{ 64-{r q:F 'flg qip q'R "rftg qI€l|F E-{I 4 kmh-l r<crl AC T{k-?r ffiot EFIIfiI iliF
iFGI I C{ qelrt {try o fracu cdrq r BD = 0.5 km, qft{ Eq = AB I
('t) ift{cqABfi{rr-dr e
(s) AC {itfi tr{t Dffi eT. s{ffi qeR {tca c"tqm e w+g EIr't q{q Kcft AB {dFrd fu Effi e-T T-iffi
q"ll{ c,tRrs qir 666q TT ffi ffi Tfi eFlr{? ffi &-s Rrnnqtdo EEr{{ fim $il:re E I€ r
"llca
(-tr* 1
tr
TA-{ el"{ T<t<i qFqts Ery = AB cftof< c<tt, v = 4 kmh-1
qft<<r+<qGosEiy= BD qkE<f C<tt, u = 2kmh-1
qft-< erS {dl+d d-fi< c<cer< O{(n = 4 sin 1100 + 2 sin 0o qfi-<4(ald-sk 6crg 6slct, s = 110.
= 3.75877
qft ({l{< 6il-sR c<ruk E it(.l = 4 cos 110' + 2 cos 0' BD
.J;
= 0.63192 ms-1
fiffi t q:nr Rrslm frr+ strots trV & frr+ r-c't< S"ttrrt< c{E&' r

AB 3.75A77
v =4 km/h
BD O.63a92

= AB = 5.948L74 x BD = 5.94AL74 x 0.5 km [BD = 0.5 km]

:. AB = 2.974km
A u=2 km/h
! unrm q'K qlr atrn, tr =
ac <<r<< 'fr6g 6rco -fo
E?F[CS@, v = 4kmh-1
a = 110'
d = 2.974km
EI{(aI.
2e'n 0.79 h
t, = 4 sin I1O' =
qFflT, AB <-<k< ffisf EFfr{ qls € fr+t< x<i<ff c+H a = 99. I qrFffi qK qcs{:t{ t, Erd,
t, = -4- =4 ',"1..=
Srn 9t)-
0.7435 h

{g{I( AC T<FK {3r qN AB "fceEr (5rs c<Pl il{{ qFlrzr r sdF,,

{ft ffi
as "Icet <srf F{r{t @Es{ {TrT
q{:T'flQCfrSIF9ll!Tr{I
l s. frTtfrr t{trs rrqr EF frA-{ qHts T{ttq[:r p(1, z, 1 ) € e(2, 1, 1) r R-t EE* w,r ft w+qn csi-d {erEd{
6F e 6d I srqr{ Ftrdq{r$ {ERs <R ffi qtrtqR'r q!fi <r-{rq R frA-d qFrtiE R( 1, 1, z) qr IMB'221 r

Y Q(2, 1, l)
R
(1, 1,2)

P(l,2, 1)
I (rt) c c+lc"l1:r6 ffi F{ r

(g ffia16p-s apqn qc(.Ttft fr'gq {rr r(r ftir {fi&{r <nr8t F{ 8

- r
 q5FTfr$IcI JEI ,E

(-EC-l
E 0?= (r - o)i+ (z - o)i+ (1 -0)R= i+zi+[
oQ = fz - 0)i+ (1 - o)i+ (r - o)R = zi+i+ R
6?. 60 = (i+ zj+ R). (zi+i+ k) = z + z + | = 5
l6il = \T +V+T = ITTTTI =,,16
p|l = ,/Flel=
= tT+T +a = JE
qtqfiffi,6F 60 = l6Fll6Qlcosc
- o :.o.-'(ffi) = *'-'(#r) =.o.-' (i) = 33.56'
Calrl{ {l{ 33.56" I
.'. a

E of, = (1 - 0)i+ (1 - 0)i+ 1z - o;R =i+ j+ zR

pa- : 60
-op- = (2 - 1) i+ (r - z) i + (r - r) R = i -i
Qf, = df, -00 = (1 - 2)i+ (1 - 1)i+ (2 - 1) fr = -i+R
FF= bR- oF = 1r - rli+ (1 - 2)i+ (2 - 1) R= -i+R
lFdl = vt1, +GTF = Vz
lQRl=vGlrllTP=vz
IFEI = (-t)z 11112 = 12
q{f{, PQ = QR = PR
qr<Tq fr'-gsT qlrolft EEq't5{ Fr< lt I

{s<}i APQR cqciFlft ftW'tbq s(< + t

lDB',21l

10km efqRFlE e+F qftcs qEEd 6ut sL-5-r I qlcfir {IRt 10kmh-1 6{r{ @t[n-d ctBl a rsk"t q<( {TTIRI
10kmh-1 c{crt @kE-< ckel qrslGr lft qf* <o $m r-+e r

(ef) a csl-rckr {lT su qcE ){{tR cqlqtdF'ift-d q,rd qmg c{Etr<?e

$ mt{ {lRr {A-d q"ii{ "ffq qlc'i c"tqfcE {|{r<? ffi&-{ ffi{q rEFs n€ r 8

tr
sfr (E) qtu (d) q< t]-cet 90" cstccf Tfq srd I

tan90' = j. u + vcosd = 0 V: 10 krnh 1

{s{f( *ffi
+ cosd = (-) = -* = -!,.,=.,'-'(-i) = rzo'
q1< -5 kmh-t
.. s q< {Fr 120" qa4 >l {tRr cfqq& "flIg c{ql(< r

S xffi<ffi-<eiqT{f€c{cq<Stt = vsinq = 10kmh-1 x sin 120. = 8.66 kmh-1

...){q1ftk 'i1gSrsT:61, tr =;ffi
= r.rs+z tr
qf<f<, qt qlRk Tfr< eR <iflTd c<rsk E'|I$t 10 kmh-1 x sin 90. * kmh-1 x sin 0.
= 5
= (10 + 0) kmh-1 = 10 kmh-1
.1 r{rrftE s[rl6ft{{5Br, t2 =;Hh = th ... tz ( tr
l{$Rqr{cfrqrcs"fl<r{r
IN
.'.
 qqI{FGIf, bEIet-JI : olr8lS-ot
17. i = zi + +j + zk,i= ei+ si+ai(q<(i = i+ zi + kr IDB'2ll
SR
l--------.-.--1
all
lt
B/
tnt I

('D E€ f Ffdqif i6sq{sRr$-dffi-qrqfi{r<xr

$ Eft"tt-sr Ftrd etG :tqsAr ffi ffifrr+str< fie|.+t s-d r 8

(-Ft
E E s i {<r {68 a1qtsft6a< (ErrFEr = lE x Il <{ev+

ExA= 13
ti 3i 4lkt =i(6- 16) - i(6 - a) + [(rz - o) =
-roi+zi+o[
lz+zl
. lE x Il = ,{C16PTVi@= zrl5E... qmffir$< (EFF{ = zy':s <{,aor
F}-{ffiqng,ftrE[q,{rEdsFil {0s$r{€<ql{E{Wcr{ I ,'l,q(s "I8, E x d = -toi + zi + oR

(Exd).d=-LO+4+6=o
.'. ogB-< ffi ql{t E6E $Fr€< ql{s;r
W I ... qfi T{sft{ |

18. d = ri+ zi-

k; E = i - zi- i<; d = +i- zi +i< IRB'2rl
('t) i Tdr<-d E,4< E? qGaryr fi4q s-{ r

m ln-dl > lE- il , li- il;l6-+Fifroq{-tEtt{-dr

E d<-<t-+<Ee<q?qBrs"r, lEl coso =ff=f; fan".l

E I-E= (+i+ 2i-R) - (i- zi-t<) =:i+ ai... lI-Bl = y'J2.,,-,p= 5 qcs
E-d = (i- zi- R) - (+i- zi+ fr) = -si- zR
.. lE - dl = tCiP+erF = Vrs e<( I - d = (ai + zi - [) - (+i - zi + R) = +i - 28.

lB - dl = ",lF :,22 = 2r3

6q-{Trtr, ld - El , lI - dl , lE - dl ... <qqrtrctsffir
19. IRB'211

Glt'4ms-l 6li54ms-l

Fwfrm v = d-Ef< c<rt = 6ms-1; R = qft c<rll

('t)GE.T, 6ey5Effi 64 pffia-61
(s) crlT G-ql1ltc( qft qcv irr r{s eFtc<?'flfltr-{:Elm frrSq"t +-< I
I
"rl< r

M 8' \\\
,tAo<auqr*m+q"otK
 ,tqHfr'slEl bNotS[ : wtfl{-ot
(-trEr-l
E trq'+' coek GtEd muf, I q<{tcet 90. cslc"tfuiF'-{(q,
.'. tan 90o = *a# + u + v cos c = 0 = cos d = - ; = - : ... a = 131.81.
€?li, Ekr<'I, R = vuzJ;zT zuv cos a = 4.47 ms- |
ft-fi{'
qfrr<t, n ='F _n, = ",1@=4 = 4.47 ms-t [v = 6ms-1; u =4ms-1]
E
trq'e'coqfr< efq <-<FtiI ffi('l{ U"lI$, vsin c * usin(0') = R = 4.47 ms-1
... ll<qcs{iur, t, =
ffi}
= sas.szs
q?n-<, Fq-({) o, qft{ 4q ffiK{ c.qcrf< Bfi(-t
= vsin(90') * u sin(0.) = 6 ms-1
-'fl3qreqT{, ,, = #+ = 250s ...t2 ( tr
,. {ft 't1-g q6,.9 s:I 4qS EIrfF
Fq-'"t' w{ry63 I

20. i=zi+tl4-.,11i<c<(E=rfri+si-zi<qsBfr'wq-{EtrcfrRs<rqffit+r*r Fs(csi-<a*v"r<

qrtrfrErw{Et{RRp<Rfift4t+'r*,6{rtrrF=r;*qQ=}Br [cts.B,2t]
9z\ d/\
/\ / \
RF
1:t1 isEeavr<ffc+Ftfi{r<<r e
$Bft"noBBFtsfrqqsrn-dfi$ ffiffiqfroqmltrnl-et-rr<?'flfffur<fus-dr 8

a- n'= lAllelcosg + cos0 = ,''.#;#,J,9i:;i'0, ='*'12*'*

E = 0.e31 ,. 0 = 2.1.424.

E EeFrfrWsi< ffi"s-q = 1;I x El <'ien-+

li i kl
Ax E', = | z A -"Bl = i(za + uB)- i(-4 + 3) + R(o - V6) = (s./5 - zJ-z)i + i + (o - V6)r<
lv5 z -zl
... m,m = i ln x El <ufcqs = ; x 4.383 <'{qso = z.1e <'iqFF
>r ftgre<
i = z(zi + tlT -,,1s[) = oi + a./Zj - sr/5R
d = ](.ta+ 3i - 2r) = f i + ;i- k

u,u= li,t
lf i
tl=i(-arz+f)-i(-o+j)*u(,-T)
-, 1

, lF 0l = 6.sz4r6
"
.. llftyce< cre\T = 1, lF, Ql <{e<rs = 3.2874<'fqss
.'. ch:ll {Er, lrft-vce< ffiaq{ lqfr'gq<(\9- c<Ftr

.'.'rr ftEq qfio ql-n:tT qrl4 sr{ |

 qn{frs6rEr JE'rcl

21. rfis crF rrqrsr sfqr sk fi'm rtm q'16 cFF qft eRtfur {er frffi qq<r {Irg El3r
<t&-d "tt't
p6qq| qsfr{ {stcE @I Etf{ tF EsTk 5 fiFE tr( @rc< 6{rt{ {tc{t 120" Ftlc6l 12 kmh-1 r<ot esF
ffioF +rErq-* Erqr.r {s{t q[Err r lCtg.B'211
(tDlftcoq;rrc{6{ttffi+qr
$ .rs fr I{ffir{ Ftr{ E"|fu cF {Fr{r ffi$'6qr< frcN.tr<r 8
-l
f-E*
tr q?F, @IN< ctqt 90.
v sin 120'
tq ft+ mtqu& q'Er qIre c{q-r{ I ql(g<r F f d q< qtcalqk F, so. cfft"r a[s(< r

u+vcos 120o
lkm
+ u = -vcos12o. = -rz r.-rr-, (-i)
... 6qf(sit 6{al = 6 kmh-1, st ft{ <-<l-<K eFIlfu r

d
qkmt, n=@kmh-1 = 6V3kmh-1 12 hrh-r
.'. Erqreftiqlr = ## , = 0.0e6zzSh = 5.7735 min
.'. s frRF< mft wq-{ qli-r<, .lls {all{Tcr ElF qr tkm TI r

...\../,r!- 4km efq estr ffi-d "lT[V-d 'A' frT qco rZ kmh-1 6{c't
d:4km Q : 12kmh-'
..\./.!
qffi ffifi nft{ q'l{ 'lt@ {t\3Tk wtr {lEt tsrF TirE I qosiq
6q1o' d-st Eqt{ eiF Ego qqir "ffry .8, ft-AIg 20 fiFtb
A P 6kmh, qrefFf srd Erq crldT r qftrE qtrEd c<ut 5 kmh- 1
r ISB'211
tr-q
('l) fu{Efrc<rldqFrffi+.{r
F) Eft"rrs qc,ts{F m+tr< clr{ (fi-st{ {tsREr{fr-{t? ffi&-+vtr<<tutQr-<r
(6) 8

E
qk mcuf< ql{ n= P2 +Q2 + ZPQcosa
=l@iE +2x6xlkosgo'
= 6Gkmh-1
P:6krnh-'
r
I rq"K
"[rs {f€Tfd 4a{fqft{ crT, = #; = ,*;
.'. qc"r+ql+ cql$F{ ql.[q ffiER qtslq Er<
= thou. = 20 minutes
l

23. m = (10 kg) sGGr eTE {gd Eq" qi:t qcrT Df<F {E fr-{t +<rq, sl ) {( frrE cq-{tr{t qrqf- lsB,2ll
a : lms-2

2N
fq-r Fs-q
I ('t) ) {( F@ T€trd S"r< ffiq
ffi <rmcF issz
(T) ) ii6(aKqtr{ItrF c R fr@-d :nnt T-< nfu
lrmt 8
r

Co
 qqMfrstd tEI rtilt : slrfiIT-ot.

E
1t*
<------+------->

L* ^IZr*
frq-:

E 't{Ce {f6, R = 10N

.'. ma = 10 + 10 xa = 10 +a= 1 ms-2
Efl?t6, tano =
i *, = tan-, (|)
:. 0
= 36.87" _k:..
... ) {(FcE<lgrqfco rl(Fqq6or

24. frrE'E;{F Ni-dms?Iqtcq: IBB'211

d:zi+ai++ir
E=si-i+oi<
i=+i+zi-si<
1r1 i o<rd e<wrqr+Ft+s?
1x; csi-< f e<r E "t<-t<q1t ftT ftTtft{ 8

tr I = zi+ si+ +R;d = +i + zi- sft

A'C=8+6-L2=2
i.v
cos0 = = _a+ 0 = 86.045. (Ans.)
I I= zi+3i++R; E= si-j+0ft

exe=
tilz 3i Rl
-i1z - iez - *
ls -r
4l = i(18 + 4)
el
zo) + Ls) = 22i ui - rrU

.. lI x El =
"[D + s, +-CW = sr@a ...
lI x El + o
,.dis El<-l<estfrroffiqmr
2 s. frrD-d Bft"FFE es', T-{ cfi tr{ tNlfr-$ ql{ts rl{qfr A (3, 3, 1 ) q<( B (4, - s, r) qF fi n r
IBB'2ll

A(3,3,1)
x
B(4, -5,1)
(t EB e<Tfiqdlt,{$E qs-{r cs!-dfi{T{dr
({) o,A\e Bfr1c'Tqd{s{k'ti6sfr'EqBs+Efirorftfrqqfr-iL'nRB-rRar.qfuonmwmnsr e
 rril{fuEMgfE

tr A(3,3, 1), B(4, -s, 1); AE= (4-3)i+(-s-3)j+(1 - 1)fr= i- 8i

lAEl=v/itTcry=J6E
... aE e< wrs${ ,{+-o ea< : fr (i - ei)
E
oI = 3i + 3i + [; oA = J?TTiT = JTd
6r- = +i - si+ k; eiP + L oB = JO+ =.@ A(3,3,1)
aE = i - si; an = urfzal:fi! = 135
, _2 _ _2
0Az + 0B'= (Vle) + l"'la2) = 61
B(4, -s,1)
AB'z = (r/6s)- = 65
.. oA2 + oB2 + ABz
... 0, A, B frTry{R< qRCT]r'i qfts ftEqfi mrorft fuils qx r

<k{rq-d <tGr qr6 qft qtseRft emRs r <t&T ccffi {fi-tr 6+ q"r< "llrv sk +r-qq
{mq I 000m {ts r

qsfr-{ {<rlrq 6{ FFr E*. qsrF fr+ ro fiFrb olr( cTrc? c{crR qtcet 120' cstc.t 10 kmh-1 6c'Et qsF
d-otsor{rwdBrqr.f TsTtfrrqlq:Rcclqtq,,f{"[qFtr{<'cqrq'Ct{imr fqa-det1g-qFFr{rinEiy
csFtrFlr{E{rq{coqr{ rl uB,2ll
('D tfr"rsq$Erlftm qtrsr 6{ul+Ez
({) rfr-{ fr lqficclT pn-t Bqfu ccs qtfi(c'ilR6'r Rrsl.t"I(F lgtqs Et€ r 8

E
mqt q"r< {rq c{er6{lT q-{,,
, usind IV
tah 9(1" =
v+ucoscl =.120"
+v+ucosfl=0
3v= -
-ucosd = -10 cos 120' = Skmh-1 frot< m"t, u
= 1o kmh-1
.'. @ICE-:[ 6d = 5 kmh-1 qkEEI C<tl = v
a = lZ0"
E
<R6-< q+< "mv cfrqrcs ir+l,
d
L -_-_:-
- uslnd
1km
d=1000m=1km
... r- = (10 sin 120' ) kmh-1 = 0.11547h = 6.93 min
..<kqq6+qxmc{qrrs,ffir<r
-:j:::-- I

27. SEEEEfr{P € Q q<qFEFffi€rtqlcqr oF e 0dmmrqft-Wr<qqqlrcsA-dffirF<(ar IJB',21l

P(2, 2.-l )
t
Q(6,-3,2)

otr<sa-{T{r<66rFt0 qr:rFrfr(rs-dr
ttt) oF e od
tg oF,F0 e 00 cst-{ffi qsiTqurEq{qFr{r<fr-*t 'tfiF+-v-tr<stutt+{r
\\\ qfurlrqearftsa'Frr{r...
8

e
 em'I{frslf,!trqa
l,_\
L_s-s<J
E op-= zi + 2i - k: op = JzriV +Cry = 3
od = oi - si+ zR; oe = ,[O + =7 eyTV
0F. 60 = 10?1168l cos e + (zi + zi - [) . (oi - sj + 2t) = 3 y 7 x cos 0
-12-6-2=2tcos} +e = cos-1i = 79.079. :.0F,od ebr<<q{l{66sFf 79.019.(Ans.)
E 6F = zi + zj-t; od = oi- si+ zR
re = 00 - oF = (6 - z)i+ (-3 - 2)i+ (2 + 1)k = 4i - si + 3k
t2 2 -1r
lo -s zl=zes+ 10)-2(18-B)- 1(-30+ Lz)=2-zo+t8=0
l+ -s el
, 6F, 0d c<( F0 Fi-dffi c<'AfiEraq<Qg r

28. lcB'2r I

frcoFie Fr<aoroftaruffirr
('t) il x Fr' cq< +< r
\,
tg E+ Ee E-E"r-rq<mfr+srdt{dr 8

t-Ft
(-zi + 3i + R)N ;F, = (si + zi + zt)rrr ; F, x F, = l-t, j
ls z zlil
F, =
E
= i(6 - 2) -if-+ - s) + R(-4 - 1s) = ai rL ei - ret... F, x F, = +i + si - rsR
E F, +F, = ai+ sj+ ak;F, -F, = -zi+i- R
(F, +Fr). (F, -Fr) = -21 +5 - 3 = -1e

" (i, + Fr) e<( (i, - Fr) sB

q< qcfcq 0 T{ r qE e$ qr +*
"r<9n r

29. FEF qr+-< cr(frru-d q,*erek t-sa Et€: IDin.B'21]

A(2,- 3,4) B(1,2,- s)
('t) zBAc GsFtfi{TFd I \9
AE, Bi e Ei et<-qr q+8 qnsrt ermmftqt?'nfffur<ReIt"t+-r
(-Ft
1n1 I 8

E EE = -i + si - e[;aB = J.rf +g +e* = $fr

AE Ad= |AEIlAtl coszBAC+ -1 +30 +sq = zBAC ) LBAI = 2o.os4"
^1107 ^lncos
E AE = -i+ sj- o[,nC = zi+ j+ sQ,ef = -i- oj+ o[
l- 1. 5
qqr | 2 1 -9r 3 l= -1(6+ 18) - s(LZ + 3) - s(-rl + t) = -24-75+ee=0
l-r -o el

rEl .. Ad, Bf, aA Fbflqr q6Q aqsrEq<fu 1

 qqr{ftBtil oEIeEI : olcflEl-ot

30. x, lDin.B'211

ftaldl=snslBl=au
i
Nt-dItrdqR'ffic+r
('t)
m (d - B) cEiqF x qrr* qftat $s 6sFt Eq'tn Fr<? qc{Mfu Fq''{c irt"Bt F<
(-Ft
r

E ffi,cst'{ Etr{ qk E, st E ,a< .firQl o a{.t ?uR Frd r

R = JA, + Bz + zABcos cr = r/5z + B, + 2 x 5 x I cos 60' = VfD u d10 ,/R' -.

e= +o=22.4L"
trn-'"*r.o,*@ H('
t'0" 'x
E E= ai;i= scos3o.i+ 5sin3o.i l,
r=Ti+;,'.r-E=Ti-;i=d
lI-El=Jae=z
srE
X-qrrR CIc?t Bq'E 6$Ft, 0, = cos-1f + gx= 57.787"

31. lMB',2l I

x
('t)Bft"tmd qfrEtm zAoB fi{T v< r

(q) aAoB crFtft fryq ih +-*rs fr-{t -'tfi&-s frr*sqtft IEEE qr€ r 8

E 6f=I=si+zf+ft
oE=E=ai+zi-:Q
Bl=^!P+VlT =,1T4
lBl= $izt a.sP = tD
f d=e+a-3=10
A.B = ABcose + cose = 1-
.'.
10
AB= ,h4xn
0 = 55.264" :. zAOB = 55.264"
-
E 6I=:i+zi+fr
0E = si+ zi-:R
AE = -+R
OA=J14
I oB:,'122
I
AB=4
i

L.
I
rE OAz +ABz = 14+L6 = 30 + OB2
" A,rfOn q{rstftfugsi
'#qF{r{atr
\\\ qffin<esir*fts{"F{DE[... (I
 \\\ qErdfrsrd SEIofJ[ : cFtfl{-ol
32. FtEr 40 km 6fiit "I( ftr u-olt+ qi5tr rfiG-d Elar+ Esd frr+ Et,Irt g,k 6<rld fre"t 6{rl qsF bfo rqro
ch-fi I tt'ft-s r-{fqr x-qr e Es-{ fr-<r qrtqr y-qs frr{E-{t ffit scql rl IDB'1el
('t) SFFEeqs'r(scshfrrfF{@ e
$ bloett&fi-<er1vc<"tmvrqq<fust{B-frtfrr+q<rtrcsb-dfifup<trsafr"+rffitr-o-r-tr-<Rras.t
<rrR cF{f\g I

t-Ft
E xcqsR,ffiEs<frr+'<qrere coft.tuqEer
elTsr<{, vc = 40 kmh-1 ; qk9lfus C<rt,
"flB-d "fiB< cft"frs EIrs-{
vrc = 80 kmh-1
irc + ic = ir . vi = vic + vl + v, =,14[z q-6p = 40vB kmh-r
... tan0 = = _19 =;... e = tan-11 = 26.56s"
*
sdn, ffi +oVE kmh - 1 c<c{ E-s< frrq{ q1-cat 2 6. s 6 s " 6st"[ct T,ffi ce uq&q r

E vrc = 80 kmh-1 ; vc = 40 kmh-1

".ffaq,
0 = tan-1 (#)
= o.-' = 63.43s" +
* ir
* ic ... i1 = Va[2.,r- sbz = aolE
= Vrc
er{-sr<irs EflIir"t eroft <r< qt, ic = 40 cos 0. i+ a0 sin o"
"-1A"."
i = +oi+ oi
ir = 40VS cos 63.435" + 4016sin 63.435" = 40i + sOi !t

u.gtnr+ea-<N<., r=r. rr.. = l+t j [l = ;<rt-i(0)+k(40x80) = 3200fr

l+o ao ol
a?frffi esfr cA-d fr4t+'-<l {s< €<r qfr szooQ
c{fe,.o-rq-{
33. 30. GFrrGt qFlto csF "I|{rw Erq 61rs 72 kmh-l aqr{r4qsE<f{s{<Elirqrqqq{{si{}ts@6ms-1
rTr{c't
"ll9l
frrF 1vcs tsT F{c'Et I 1E rl{l em ct< s-<r+ qxEt-€sttrt <l{eRR spErqt IRB'l9l r

(tt) \eF[s <ltlEFr$ FEi cffi 1E eiv.(g cRqr< fr4r ra r

$ TEe<trq< q-fi{ <FIEFFF

"llVT
ftrE-{ frrs 1E 'ivr\r cE{r{ <R S?rtr{{ eEs {R € Ao tfff&-+str-q
Rrss.t€r

tr

vbu,
v."in -- 6ms-1,v6r" = 20ms-1

...W=Jv:"in+Vfrus+2vrain.vbuscosfi=W=2r/T5ms_1(Ans.)
1216 = --fraliIg- jl!l469'., :.e=LZ.Z3.
vbus+vraincosc= 20+6cos60.
q{-yfrrTi {kel = (30" + 12.73") 6rf6t s-r< 42.73. (Ans.)

rM =
 qqt{fru,lil bEt et-J! : olrfiEl-ol

E arcu .'t i.,18{ {fl<ul i., <EeFrrr{{ c<'i n., <Rq-<lr{d E{Qfus 'fiG< ft"ir$ @ c<ut i..
vc = 72kmh-1 = 20 ms-1, vr = 6 ms-1
i.. = (i. +i") -i. = i. +ia + (-%)
i.. q<qt{{l{< Fa[q, efRl6E-d5lq{<R<) E'ltrnfrcs {i,
v, cos 30o - v" - v. sin 30' = 0 = va cos 30o = 20 * 6 sin 30"
.'. = 26.56 ms-1 <l{ eFIr< ft-+ wryfi-evtc< <tcq-{ frro
va r

34. fficE d rtFt+{Errq f = 4i+ 3i+ si<, B=zi+i+zi<e<(i = xzyi + y2zj + z2xkt lCts.B'l9l
('0 Bft"fns-d I € E ren-<sffidstfizo csF qqr cstd qd ffi r

1t1 Eftlzo< d oBm-< +tr4< vttstrcsq t+i c6{ ftr ffi&+str< frrtrs.t F{ r 8

IxB= la a
ti i kt
sl =i(o-s)-i(8-10)+t(4- 6) =i+2i-zk
E lzrzl
.. I s- dr<Er.lr< q? frro qaF qs-<r FB-<, i trFz
.g =
ffi =

E
ri i kr
vxe=le ldx
i
dy
el
dzl
lx2y y2, zzxl
= i {u! sz x1 - i rv, o} - i {ft @, i - * c,i,)} + [ {fi (v, i - * $, il}
= -yzi - z2i - xzk .. V. (V x d) = ftf-v,l + fi{-,,) - fr{-",) = 0 * 0+0 = 0

.. d eAc-<< +trfir ulnpkruq 1lr

35. SqFq+rq<' lsB'191
Ct--------------- B

dl g--" I
l.." .l
o F 'A
F=+i+si; d=zi+sk
(t Eft"rmdqtintme-q<{Frfi(T$-dr e
({) aoAB € aoBc -q< cwFrcar< TqE cltlgfr{ oaBc-qi crc{Erf{ qfi{ ft--+tz rtlfffrTst[< frff<.|e< ra

t-EEr-l
E Uft"rceP
" Q o<r<rff cot"t a r

F = +i +3i.. lPl = v?z+ 32 = s

Q = zi+ sR.: lQl = ,ITTV =.{TE
u)
ccl{-cosa=+S= sY\tze o=.n.-, l'
lPllQl -!:... \svzel =77.t2"

0.{<{Ffi 4(r,t"no="ra.*o...]]8=H##=0.84;.0=4o.zs"

I IE
t\i qfur+*ss,rtfts{nerFal.. (E
 om{fruwoworg
I-
It R=P+Q=ai +sj +sk
j{,rd, px R= li 3
i [l
14 ol = rsi_zoi+aR
14 5 5l
... AoAB (r ($EFFar = i lF El = 13.124 sq. unit r qFrK, aocB << m+q = i ld f,l
" "
- exR=lo z
ti i Rr
sl= -15i+zof -aQ .rAocBq<(scFFar=;l0rEl = 13.124sq.unit
l+ s sl

qmruko oeac,a< ffi'sq = IF, dl ... F x { = [

I0 2
I 5l = 1si - 20i + 8R
5l
= lp, dl = 26.248 sq.unit ... A0AB + A0CB = 26.248 sq.unit
.'. 6$-4rFaI
= JrlqlgR-r Oagc q< csEr{F{
... AOAB re AOBC qR cErffid qqB finigR-f oaBC q< ffiTrdld {{F I

36. qfrs d= zi+i-i< e<rE=i-zi-si<@ oi-<frmsEh-dsb € qrT e6Frfi{Ts-dREr fr frqrqtGr,

csts-<ecr< rrq c+n"i-< Tl+ qFF fiffi {G{Ft qful +-<r-q slrq-{ irE s 6q qorr+{ {F }FIF[ iN r [BB,r9l
({ f s E ot*r cretFt qt{lgRr{'-d qRfu <rq m E-e nwkns-* ffi-sm ffi F-d r

(s) qFrs<,ft-{TcctT {AFrftrdTq{q{qqsEls I

8

E
ti r; [l (-e
nxa=lz [(-4- -si+ si-
lr -rl = - D -ie6+
1) + 1) = sc
-z -sl
4qfsfrr$< cETq = (-5)z a 5z + (-5), = sr6 <rf qq6
.. r

E 6 s 4-q sqr;rd{l-{ JljIFr qr{, AB sin 0 = AB cos 0 - tan 0 = 1 ... 0 = 45.

qfi, A . B- = AB cos 0s + cos 0o = '#* :. Oo = ZO.g"
:. 70.9" - 45" <t ZS.S" flR:r6 s-{r{ d s g Ft-dc[s{ EE s 4{ e{cl-1nfi qqfiq6qr
37. ffi'{qr{$'riq{TfiffiftTrqTtsratt@c{ p(t,2,-t),Qe2,1,1) ,4<( R(3,1,-2), milr{, F,d e<(il
ew{istEqt{{{k{-<qlrrsfuffi{q+qlqF?-{ft6tFrdr lrB'tel
(fl F-cr€"K0oAmdesqfr666o1-s16fi{rq<r e
F) p, Q q<( R frTEm-< EF:r {(r{M{ E|{t Bq"F F -derqt
sFil {SE 6-q qFE T{rstft fr-W 't}q Trd fr-{t
stdlffBr RGFrq-d :rt{m <II"BI s-d I
8

E q{lrl, F = i+ zi- R;Q = -zi+i+[; Q q< 4t

F.6
qsrst = --i -'- '= -- I
lPl= ./5 ,C
E R=ai+i-z[
F0 = 0- = (-2 - 1)i+ (1, - 2)j +(1 + 1)R = -si-l+ z[
F
0F = R-0 = (3 + z)i+ (1 - 1)i+ ez - Dt = si- sR
RF = F- F = (r - 3)i+ (2
- 1)i+ (-1 +z;R= -zi+i+ R P(r.2.-t)
.' 1Pq1 = r/-r+ 1qP1 =
.'. PQ2 + RP2 < QRz
.1
"/-:+ lFFl = ./6.

APQRftIl-slftftYEi;ITII
l_\
QR
(-2, r, r) (3, 1,-2)
 qal{frgB-tEl 5Ert3[

ICB',19l

oB {rFrd, TRcc{ 6{5t = I ms-l

OA {dFtd 3Rc{f 6{5t = 7.5 ms-l
('t) TRT FE c{c{qfi "fl.rg cfu{? ffi F? r

g) <Rr {[rri co qftrt q'R rfiffF{t fuHrqTq r\rFrE

e rfrra-< 'ttry c{qlr<? E'l\3 | 8

f EE<-l
E 6SIN< C<rf, u= 1ms-1 ;+Rm< C<ui, v = 8 ms-1 ; a=120"
gKEk 6<fl, w = 1i!z + uz + zuvcos o = 7.55 ms-' (Ans.)
RF'6[< (Fce {T{. t. :3 = Y = 66.02 s [c{{rr{, d = 500 m]
E 7.5

ofrrr< mrq mt, k = ,*"" = r,*r*_-!9L = 72.169 s .'. <Rr qlrtt v"r< {-ry c{qfr< r

39. Fq q$Ird oABc eFFqEvrsErq<oA q<(

fi'cE-d oB <lqE|*tl$ cst{{elt@r{ F = i - zi -ir
q<(d = zi - si + zi(GrfFlgqffar lDB',17l

0z

('D Sfr"rs \n1ftT aoAB qrr ffiErrqffiq< r

F) Bft"fo qtrtca 01 \e 02 ,4< {c{, 6FIF <V Et ttlffFr fu{c.ld rll{r]r 6{iI iFd I

I aoan e<cw"+c = il0I" oEl

i kl
=iti -z
erl;r,oExoE =i(-4-3) -iQ+2) +fr(-3+4) =-zi-si+R
lz -s -11
zl
.' loi x oBl =,tCD +Cry +P=-
... aoAB q< cFE$Er = j Joo <'fo+o ,

E oA oE= lffilloBlcos02 = (i-2i-R).(zi-:i+zt) = ldlllGlcose,

+2+6-2= 72 + (_2)2 + (_1)2. "lZ2 + (_3)' + 22. cos9,
+ 6 =.,16. .,ffr . cos}r+ cos0z =
#n .02 = 53.55"
.'. 0r = 90'- 0z = 36.45'
{E<I(, ez >

e
01

qfur{csgnsFffi"lED-{r...
\\\
 qEI{frsB-H !Err6
40. c+n;n eF 1tr{ ftT {fr{f qffir '[u"f qiMT Cr{Qq !E E-s?sk< 6 kmh-16firt {GE @ r {fuct sF
s-?Fr, <rsls qs-q{ 6{lrp 4 kmh-1 c{rrt qFrq q<( qeFrq{ 8 kmh-l 6-{a aR-66,6q
{tA r qrrr E-vrnr qlvl
frn&srftqd-flElr< ffir IRB'171
('t)Eft"trsc{1tEsl6ff$trd{k"irFqgslBdqkc{'tsE? e
(s) 6t[t5-s-s 6{ITE< q<(llQmcqr-6rs qlEEdqlvt qrti-{r{qr<<itormqq-{RctT
frrds"tqd r
'fffi 'rfiE-s-skq
8
(-tr.r -l
I cnsaf vne, @< 6<rf, v
I
= 6 kmh-1, 6Efr$-?1 6fft,u=4kmh -1
u= 4krah

.. ek mq, w = v[z fiz = 2/i3kmh-1

tr
ch€{I \flrq, 1E< mrt, v = 6 kmh-1
(if 5-as ffiC.T{ 6srq, u = 4kmh-1

(I) = tan-r (i) = 33.6e.

I
... E{ra< ficat 6fl"t, 0 = tan-1 v = 6kmh

qltrsrqE{s 6fi5p{ 6a6q, u = Skrnh-l

Ch€TI qf(q, 1E{ Ctt, v = 6 kmh 1,
{troffi< c{rl, u = 8 kmh-1
... E-{6fl< q1ret 651"r, e
= tan-1 (s) = t"n-, (i) = 53.13" e
v = 6kmh-l
. (6 raro ml.o s l.lerorE E-crs ffir$< s|g[ q+is$Tgtr< <lom.a:t r

sqr{{tft q{6 rft

31km eFfsr EE ieq EIEE 6{tB qv|Icl& ,lR qs{Rr
q-{I A qrs q&{ C{cut {t ET F!Fr, TtrE{ ,qTE AB
41. Tirlfir q'r{6 Ac
<dl{{r saF{Fqv1Erl&,tkrErs c R{rs cdqtr{€ o ft-tcs cdqmrfrffi
[Gt6rT 6{rt gkmh-1]
A ICtg.B'171
('D Bft"m cF fu{
q&{ F"t Rff{ <rd r

(t) ri-o @ o+8 mwffi-s qr*

"iw c{emft-dt {tffe-s fuHrEF{q rsm\5 q\B r 8

s cns* vrrq, @Ire< 6rt, u = 9 kmh-1

:rr{ +R, dot< r<t = v
qq[I ci-ffi v Cr{ qftq< qlcat 120. crk6t q-qf \eS
€<rE Ek F{ GIrET clral
90.6fltEq"lfrFr{t
... rane0" =
#*ffi; :e u +vcos1200 = 0 + u + (-i)" = o
+v=2u=18kmh-1
 qql{frgB-ra!Nqg

E qlf<C{tf,u=9kmh-1
frot< c<ol, v = 18 kmh-1 [(5t) {rs]
lr ffiot< mre, ek 64qft-6 qqagaql
.. efu 6<fl, R = fi, + V2 + zuvcos 12oe

= Jra, + s2 + zxs x 18 x (- j) = 15.5e kmh-l

3r hr 1.989hr
...qft {Kq[scr{
' = *e
ar6 c<'1= 15.59 =
It ffior< c+cq-, {ft{ efq {<Fr< frsK 6<tt = 18 kmh-1
...qft{r<5.Csr{Ts=
ffi=*n.= l.7zzhrt qc<r<Wffi ws.i "ttec41qmr
EEtr
t
'jfsq +--'l'----+'F
J
42. lsB'r 7l
Cqfr"r

6ql{{1fir,asF "IlFl q$o-{ yFi qqfq{r{ rqlsm E-Vrq r {B,5u 6-a ay{t TVs <lsts'tRqt t sw r

('D frrE-d oc {irFt-<r a&fu{f<rq-<qf{FEa

({) Ao :rdf{{ 'fi"Ik {lrfi{ ffiRt"t frE"t ErE {Fltr crFfrrf tp[-{?
'ilfiFs le< {tvc{ $lrRI <r{ I 8

E
ek, n = v5z+ 52+ 2. 5. 5. cos60' N = 8.66 N

$ ao <<l-+< $6l< flftTIq ftEq arE <E e6{ = (2 x 5)N = 10N

N

1
w +--t---+ E

J
s
<dlrlclit {:rrcff crFt, a = 60.
.'. sN Trd< {Irel cstct o q(4, tan e =
*i#*r . 0 = 40.8e"
, "fiFIF t( frr+< xtr< = (40.89. _ 30.) = 10.89" 6$trq EEr{ |

43. firs{GrEEtrfrT,t e n El{q+ cq'{tqtcq: IBB'171

B(4,2,-3)

('t) AB {(wt'l{'tft csir<-< TFr Ffs <t{ r

$ Efr"tc {frwqrmrft fryq {f{ s<r< ft ? firdnlqi+rsFrs Rts r

I-
 qErdfrsdrd uNrt3l

E 6f=zi+gi+R
or-=+i+zi-tR
... AE= 0E-0d = (+i+ zi-a[)
- (zi+ :i+A) = zi-i - +i
,.1fr1= Jlzy 1Sga 1-ay =.1v
E dI = zi + sj + R ., l6fl = oa --.tTt? +T =.lTa
oB----=4i+2i-3R
,. l6El = on = r@iTIP+;* = rDD e<rlEBl = J7T ('r)q(sl
1C.3-{ <fq, OB = tB qacfift frvq ng I
; ft-q 08'? + oA2 + AB2 ... 0AB
++. qsFd-+lFeE{tft z.s kmEr€ ,4$ErftF A q{qFr qcsE{-{ firs AD $rr.Krrt6ql uB't7l

A --------+ 6lrq{ l:+,t

Q{ "ItfrFo d-+m r<rt = (si + ai)ms-l q:R Gtrcjr 6{rt = 2i ms-l ci[ qTF crrq ffirs AB <irrirsr
e<tE&NEr{ffiEsr
('t) rft< {Isffi{ q? <-*f+< eF-r csi-dffiT-d r
e
tq Eft"rsqflra r+f+cemffiqrcutq"r<€a c{qrr<vt
'fiffEsfueTq1fsEs<ns r
(6t 8

tr qFt< rrrscE
EiE Ga<, f
= si + si
E = zi ... tft<c{\9-r{{qt{{K{ cA{
= f xE
ti 3i olkt
exB=l: =
lz o ol -6k
... qftnry"_rdmqr$rr< qs-s@E-d = I ffi = tf tft
=
E Eel:r cs.c, ffiq-q 6q, 7 = (si + ej) ms-1, otcu< 6{rt, = 2i ms-1 i
.'. ek sq, f + i
= si + ei .'. +ft< etq <{l{< eft rfi"F B"It$ = 3 ms-1
.'. nft6 oPaqsryqa
= I3l9 . = 833.33 sec
fr6-s c$[q, g.fu = vSzJJz ms-1 = 3r/7 ms-1
Grq, AB {{I{< d-fiE {@, ffic{g AB {{f{{ Fftd
EFrt-r{I Eq <irl<ro c<crB S"ft$ = 3r/Z ms-1
... qft {f{
{{{ scgi = Egsec = 589.26 sec
..fr6xcffiqffiq-c{cfuc<r
45. qtrfr'{rffi-$qnts<KqrqFitssr<qqn A(1,0, -1) qr( B(1,1, o) r
lcB:171
('D AE r'rfl-sitt[6r qTE ,rF.s Ftrd fr{T s-d I
Firtrd
$ EF R'T A € s qr q{ql{ cEB-dsffid x qrrq B"r< qt qBr+"r ea yr+{{-r Rr${.t

I
+.< r 8

t$l \\\ ,tffi-{{esn"ft6TaqEil...

 qEI{Rgld DE ct3t : olrrt{-ot

E 0i=i-[
6B = i+i,AE = 6B-0f : (i+i) - (i-fi) = i+t
... nE q< wrs{r{ qo+ et< = : # = } 0 + n)
ffi
E afr'Wq<qr F {, oI = i - Rs nfr'Wqwqrt Ftrd, 0E = i + i
oi;li)
.1 x
- \5rcffi B'K 6f q< q? qes"t = =$ =r
qrq?q6rs{
x - qc+<E1< os- = ffi = + = t
.'. x qrs-d bem A s n fiTem-< ft q&rs'f T$T r

a cQ: fi6tflruMermHeBw-s*nq-rcqqaq$si{ilBs{ >

01. qq4 s v6aqe1ff csfr cp{+tB'Fa{q1 qaal qsF{{{Ersfrrq{frqFqfifuqtq[vqqt6{ = zi + j - zk, i
B = i + zi - ai< e<( d = 3i - ai + si< ftnF csi{ Wtl qdraTl,s<( vtdt Eq{ !T {Trsffid q'{s{ ffi
q<rsdtrqtr
(rD f 65ffiqmqfirqqcsttBq'f{Trdqfi{rFdr e
(s) EFr-d eGfi Ftrdftlqtrdclj{rslstqv'{ffiF{co eraftrqtF+t- qfie+qr<qlflQr<r 8
/....--\
\gsr J t
[ <R, an-e oob< zi + j - zk; & y € z \5nF ,rlBl +rqar c, g s y 6Ft E\'frt FK r

i.(zi+i-20 2 z ." c cos -'(i)

"'
coso- = 7i7ffi = ;6 = - =
(zi+i-20 1
.. cosF^ =
i
ffi=5=;
1
.9 =.ot -'(i)
[.(2i+i-2R) -z -:-
-2 i.Y= -'(+)
' = -=#
.'. COSY =-= cos
!/1.\12.+L2+2t V9= 3

6i = zi+i- zQ; 0E = i1 2i- eR; 0e = 3i-4i+ sR ; oADc

16<crcwn= lodxdil
qfr, olxoe =Mr
sfl€trdq'is-{ = oaoc @<cre'+n.B6Er (oH-)
= lMl l6El= lMl.loBlcose [., on = lOElcose] XA
li i Rl -- ')
=M.08=(6I"oO.d= l,ls t -zl.(i+zi-s[)=
-+ sl
11
lz
ls
1
-4
I3l= -'*"*'
.'. ql-+etrd qrssq z qq cq<r r q{\ vkt smsq fift FTro c'rmfrq r

Emf = si + zi + sir e<rE = ri - i + zi<

(rt) i
e1( BBEq 6strtr{B'Rm,{frt raqfr ,qT{ cs}dffi TrI I
(s) c q<( F q{Frqrqftz ffiE+ftr*ct{q$flnsqt€ I

irmr
8

w
 qqMfrsr{!Hqg

E
ti i kl
rxs=ls z
b-tzl {=zi-i-rrft'lfxEl
72 + (-r)2 + (-71)2 = ",Eq'+ l:aT,zI = 3,,18

... E :-A Oif<-< Ei _Z=#

f< q? q<ro @i-{ =

I I+r =ai+i+sfr; f-E=zi+ei+R

. ^^^^ _ I.E (s.3)+{2(-1)}+(3.2) 7s-z+6 1e
m-ffiffi=EE=;ffi
(A+E).(A-E) 16+3+s 24 24
ffilu, = -+ = i=
4
cosp =
li;Elli:Ei = ffiffi = .1 a+ I
03. 36m D€g| qsF {ftrs 12 ms-l 6{ttqFF d-fi Emril <oq r ffi qft {T qm fr,fiv q6g C fiTcu
cdqc I iftcs @lrc{ Grl 3ms-1 r

ffi
('t) {fttrd
"t|tv-d BC
qr
Ergrs? I
m rfi-d ffi {Wr B ftToo ffi-s fts dRrq $fiE$ ft <i-rqr firo 6<z rtfiB-+str{ frrS+t T.t r 8

^
tan u
12 sin 4oo
= 3+12 cos40"
:. 0 = 32.32'
.'. BC = AB tan(90.
- 0)
= 36tan(90'- 32.32') = S6.9 14or., 3ms-r

E ffi e cef$ wtlcR a R'Wu cfrre qc'E fu

e egs-< mcfi qfr q<( Ct1g;d 6{Cttr T!.Fff 6{rt,t 0 = 90o
(re{r<r ffiq s q1ruq 6{n-?tr{r<ff cflt aqreqF6t'it,
tan90o = ;*i* cos(l- vr + vbcosfl =
vr+vb Q =e 6e5q = -:rvb
.'. a = .os-, (-*) = lo+.*s" [...vu = 12ms-1]
c{R, clrsr firs< rr[et dot urqt(g qr{
104.480 corc+ r

0a. i= ri-zi+i<, B = i- ri+ si<, d= zi+i-+ir

('t) i € Esrfi qosqt{Efrrqd cEvEffi qr r
I
m i, B e i s'sDrqc,Ttft fryq {r{Trr{ fr? E-€-n-d wfrn Ifu q,t\s r
I

E IeE1EcRRE<rq<q,

r,.E= lllr f-s Ilsl =i(-10+3)-i(ls-1)+R(-e+z) = -7i-LE-7k

rM
 ,MI{fr{EIET
'EI9ET
I-
E I = si - zi + R ;a- =i - ai + sR ;d = zi + i - +k
.. ec+ ., ldl = Vr+ ee.{
'
lrl =
^ple*ll
tEt= JWe*lO,4s<'=V3sqr:F
.'lel = 1EIT+C$ qos=VZTqs-{'
,-.2 ,-,2 .-,2
q"rm, lBl = lAl + lcl
q-eq<, eq-s Fi< ffi
cnIstft frW "flq s{c< r

0s. Etq-{ <rfu EFGI ctro RqIr{ Eca {qr4r{ {q,flt € }q{q'IT 'ttr{ F{E 1 51a1-a qfr'lrF dlq,fia e hsc-'IF{
'
EFrtr mrcn (-40, s0, s) q{( (-60, 100, 10) ttrwr ftfi. eFFt t F-dt wr rl
('D <ffi
E(g fuT e{.t Frd k{hTd cdqt(g as 'F[ q&-dF:[ Fir6s e Ec<?
(q) EFFtqro(lq.tlfr'E-fftft<,futrrc"It-trdfi("+u[<qenmqr{R6Tffi<rdz 8

E q,cr rfr, ur+r< ft{rs <lq"trtd w{qF Fbd,

= -40i + soi + sR 6f
q<q WoEr Tlr"rcshq-{T<-d q{qf{ FA-d, oE = -60i + 100i + 10R
-. <f6.rr& <s?;mq,{t-tr{ E<g,
AB= oB-0f = -60i+ 100i+ 10R+40i- soi- st= -zoi+ soi+ sfr
i. lAEl = v(-20), + (sox + (sX = s4.08 ft. fr. (Ans.)
E os qrqr orFI corrr ?T{q'Irs.d E<y q<( oe qrqf ElFt clrr nls.fltd ERV r

og e<( oA q<x<l<ff 6 o "t r

6I.6E= 16IlloB ; cose'. = l6fl cose + loil coso

ffi I
_ (-+oi+soi+sR).(-soi+rooi+roQ) 2400+5000+50
fr3.65 km o
J(-60)r+(1oo)2+(10)2 177.O47
+

... <Mrftt'rT{$Ift <rfu ?qtrR"F<T ftro ea.es km qelc-{ qrnRE

Efsf catrs r

06. d = zi + ai - k o<rE = mi - si + i< otr<or 6rtc;rt {qsm ffftq I

('D m=rqrq,f q(E"rdBqdqtqtrrF{ffiFd1 e
(t) "m {drlc;r?Eerd(gfdsr{d,isr-'firru<ITflsrFrqatfrffR-{r"EfuE*qv,qBft,laq-$trd{mqdr I
(6t
E m=1{C4,8=i-3j +k
(zi+:i-[) (i-:i+[)
I s< BoR B- €T dr' qfusq = 4E -
lAI Jz2+32+l-1)2 = 'z-=' = ],lv
'114
E qffiIqlfr, f .E
= o cffi @Ad ilFrqF "6-6as g* **,
{fr i.E= 0q{,giECE, 2m- 9 -L = 0 + 2m = 10.1 m = 5
qmqfr T{E<|E<sssERr{.
i = i = l*
= -2 ef,r{r ... m
fi'rfi {lrq c{, m q< frBfr {I({d w{' eB-<cs TFil q{ Sq'fi T<(R r

olt c+ sm, m qn $rq;r BeR cB-<scm l.rrq?.ffi qt {:rrs<Fr qFFI frfftr{ EfuE rgl r

o7. qsF E&ft6f s.tR 6sR-dT q{ {qr6d q<qlT Ff{ i : i cos rot + i sin ort
('l) T{rtrdq{qf{ F 'd s Elqrfrr 6{['fdq'{ e$r qr{Ftfr4Tq{I I
({) <o = 2nrads-r'q6q, 1= 15s6 {{rTF.tlRd 6$t,S<(ydCqir{Ft{\'EFI? 8

J E \\\ ffil6rl-tss,rlftsr{qu.rt.. e
 qqMfrCIEI DEI'EI

I i =icosot+ jsinot

.n=$=*1 (icosr,rt+isinot) = -<oisinot+ r,ricosrot = r.o(icostot - isinort)

ixi= 5l = t,, -, -i(0 + 0) + R(tocos2 rot + tosin2 ot)

l-:"i:l ;ri*,
= oR ['.' sin2 ot + cosz tot = 1]

$'"t' cqro++tE< 6E:f, i = <o(-isin ot + j cos ot)... ... ..' (i)

... a=
f; = f 1-i, sin r,rt * io cos rot) = -ir, .os r,rt - io2 sin tot

= -or2(icos tot + isin tot) ... ... ... (ii)

(D{({ftsi{q<s,
.,t= 2n[-isin(2".]) + icos (2". )] = -iVs"-i. [<,r = 2nrads-1, , = 1r".]
.1 ltl = n./(-./3)'z + (-l)'z = 2r .o<nF r

iiil r( {fi-s-d"t Ers, :. t = - (2r)z [i."' (2" . i) + i'r, (2" . ])l = iznz * jz,lin,
r--- "
ldl = fiz122 + (-2,15)' = T2,l4ia2 = 4Tr2 (<5-or

08. i = y2z3i + zxz3i + rry2z3i< e+E ef< m r

(t (1, -1, 1) fttre d qrqisr*rqqfift T{ r

m Efttmd sq'-s cE -{ cmF $-{+6T ft{t- flne-<r dbT{lsFE qts I I

(E.rl
E i I = (*li * ;; + frt). 1v,,,i + 2xz3i + 3xyzzs R)
= ft{y'r') + fi{zxz3) + ${ant'r,) = 3xy2 . 322 = 9xyzz2
-1, 1) w{qtn, V. I = s . 1(-1)r(1), = e
(1,

E mirll Fbr-Td <q{.Fr qcn v$+ftn qt ffi r

li i k I

i'r= | * * * l=rtfi<'*,,,1-firz*,,t\-i
ly2z3 2xz3 3azrzl

l\rz*v',' I - fr {v' r'l]i + r, lt px*1 - fi (y, * )}

- 6xz2) - i(zyzz3 - 3yrrr) +R(zz3 -
= i(oxyz' zyz3l
.1 (ffiT4{qqr '

IM