mathematics 3 handwritten classes notes (study materials) for ies psus gate

88
I. Lin"qr rAlsebro Irf atrr"ce s Pc.o P-e-"'l-ie-s oF Deterr"inofe : l--Lt *oo \,otoq - ec colr+ms oE a nna-tzix qde jdernlrql , lLen *he d efe,r,minsfe is Zero o lv_ 2-fL*roo-troo-q. oft, -Ao|t+m:qs -[E- A rr+,qh-ix a?re- i-t "U-.og.+ c A. tb *hrrep roto-q Or- eolqmrn-q oF q En-a#X e-lqo@ *L3 L_L n o^"&+.t"g +hq- Sqrrr orr diPFerErrce ob trno e-.terqen1( r $o.o{t- oorn --be __a___: qlerory o_ g2-z r - -QzaL= QAg- q33

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Page 1: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

I.Lin"qr rAlsebro Irf atrr"ce s

Pc.o P-e-"'l-ie-s oF Deterr"inofe :l--Lt *oo \,otoq - ec colr+ms oE a nna-tzix qde jdernlrql , lLen

*he d efe,r,minsfe is Zero

o lv_

2-fL*roo-troo-q. oft, -Ao|t+m:qs -[E- A rr+,qh-ix a?re- i-t "U-.og.+c

A. tb *hrrep roto-q Or- eolqmrn-q oF q En-a#X e-lqo@

*L3L_L

n o^"&+.t"g+hq- Sqrrr orr diPFerErrce ob trno e-.terqen1( r $o.o{t- oorn --be

__a___:

qlerory

o_

g2-z

r - -QzaL=QAg- q33

Page 2: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

'LLo uce.nr -fri q"nlgL.l qr m afrix CLr m) ? --

Ait *l+e- elernents a-hove ,l he b{nci p*.1a

cLio-6on.l ls co.qed \^is (erooo

All *he e-lenneyr*,s he-.loo +hp ffni$sf, -eh-dlqg"r*Lji--r€-rco + is eo.ued u^

h..to+-e : -LF,-q rnc.h,{", -fo_ d+g.r t",o"r ++""tS"tq" f t*pf.t@ -fee d"+erhnb.f, t@gg-P$oCgel olteraon*l qene-{ ,-

L. -L

q2- 5} c2-

R + Rr - Rr -

ecL

: f er-u) Cu-c)C e-q)

l$sos-l?cle---

-> D^ -) o-t- o2-hL o I alho h -c rl-h n1- - fn-k) ( S-c O I hl-r]. Lc(L

Page 3: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

3't:t

ab c

Too cs-l qroros q r. ?d",olfccr,t

bc q bc I

cq b cq1

e(b c- abl

clicu DlAFrY55i6lGMA L.CONI 978423.i98i

:fq=kt. s: ll*46 ; -t*.S

" -5XIJ

is n rt Sx a llq (xI-

- h6-- a hc'

+ L;, tl , -'r-LQrt/u __arrlb 11b-

Rt lfr-f4t-/ c LIc

x/u *LlaLl" L l"

Ir4 g_

CrbC

Qz*Cr , QL*cr -

!t]1.

= c.bc (\* {+Io*t)

rtb tT T+C

Page 4: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

l171

-e){2(t-6)

ek Al+ern'fe t'ne{nnd -to [frrd del-e'rholnoJz l-GnIB f..,r gxa .'nrtdin'-) CShc".+ct,t nno+h"d-)-

o19-

Cr) tc*i{" +ne -eirsLffi +r"e ;dS!t--s!fu.

C2) ri'h"* Md *he Prnduc-l- oLt,e,g dieflo,n.J qmd s.,l"h-"^!,1- -5, 6, + Cqgorno.J ,

A :o{- q+o--t2-*o-l * -L ll

--n

-- c ouc-d ( r+ h ++*+t-) =

-e C8ts

j 2- -2-

-r 3 o

o -L I

Page 5: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

sA-r : Adi a

A

-'. A-\ -- . I - I "to.d _ b c [

_ c-bl*l

rt! El

Aotrc.ho:nglry +hs elsrner,* of tsldjogonel qud chorrgizrl

-8 I

r=tpoj : coll

--ltno.r sE qn e,{..n"'f s1-ft^, p - irr :o -S7u,q:rl rm.fA.t--lhe de*srrdnq4

!E .tbe-s+rq'rp St'h melax lo -ll)-hie-h *he-E ol o(hcl +l^e--co11 oF

- *he 9q-z$sdqf-ejenegf, L,sr -fo- bp de-.le fe-d'

* dllAlkrtr"tc. Clhorfcr'.f ) rnel-hod --1"-

g!4 A-) I -*Z

2gl_t

-Q_-t,c5+g loe tst +€o- -QoL)'o*+',. I<fuu{ s9 de- an "l IfoL -L b&

@tej{s{=&g --'b^illI!tt-ontd - 1.

o+9 +L*\a--c -r = ''L

co_s 19 Sin

:S'il0-

- -&r*!lter- oderoF=lx3 or- sr ots E----

Ad'.r * : frrauslose oF c-o!sc*o]G fd^le$x

@n--9,une,l-&- Dl1 mrror

r 2_ 3rl

51-+

t2_JI

Page 6: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

6

@ -fhen n';+<. +de Rest -l-l- of, dekrrrlh*f" o.t' +oP'

@ fuen Lorle .iLe- -ls+ {oo s:ocos on *he oloor:r a?{e oF

ept_: pLolr 2- o t-

0 -- Ther. -1.".,8e *oo t'oro S ,su ece ssi ve-l B 'crfi d Pfr''d +heleJtrrni

-----.ncro+ Co-b * b c.t in hzr' dirl- et-,d +hen to#te hLrezdcq&rl-

zz

2- o- -3I L -2_ | L

9 -z I 9-z_l z ,-z I L

-t 3 *t 3

A = +3+o-.4tot2=t

I

2- 6l

EJ2-

Page 7: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

I6z-

I

uD

l+ 1 i\\23lrl{2

Sr9:2tfl

I

2131

A L2:+€ I ls-s -+-t2-_ _ =_[_

-,s-!rog

os0

R A M l( AMA L. C H O U D H AR y 5 5 i @ G Nr A r L C! Vr 978. 2 -l b !r 3.l

: $ !a S----Cos€

t i:-i.ell--16lao:soF o:-d,er (rrt) qre (e-lrQ+hq

- *|.,ex-e L--e4t-ecs-L brne -AoL{gn! nr?no?r o@ ; i5 e"isf=

-r+t c*,f.q +t. r"" e--IqcHr-qa-d-i^--4emoJ<-{=b6

&AJ- r

\)_=_ ooo

- @--rf- ll is - -q-orm - u=eva ,nnery1x--;*len+

SGI 7,, J-

-2- I

-Z -3 I

Ll

> Sltr Fr e.:< F --e---Srar9 Cos 0

6nL o o*C\

Oo

Page 8: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

J _ l..lin w Rq",K oF@zb ,

L3) "af i br{-l,e?nx x , {*o?fl

qr'tr r\o! - Zero hq*H"wf+ nnc.*air or lderrl.6

ts t'rn q=F"x oF oc{<r

J (r.) =r)ex l- s(rd :ar S{rs) : B r sC 1:,) r s

8_ _\ .S (aLl_za_5_ -6_

l2 z+ 's6

Alloqef ene norr <e*o *hen8(r:+,)_>2-

-Ck o.j-l <ez-o ,_ *\.e',a, ( nl

lz? o f 1'l 2k+

Oheek *he-f<]- or-c.lenr , ftq Norr -z.eru.'. _-S(A\: Z

l=ffj@c}.e"lq oorcla-s zsr-n, *heo ele@ -

Rg={te* +e"-}-jcel:z

q (A) < 3

tr_tn_n _<ere_-: S (a) =3

Page 9: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

3

( (n)s g

Rs BRrIZrx;

Non Zera- of order 3x3---

to I e-l(v> \r 2 s[ +

I s r i l.r,

L E L--rb l,- z + z ls 2 o -r[ -l a s tLr l:!_5tl-Ll lr 3 + r-l

4 = XE[_Zsrro

I 3 --9--:-1

+--5r4s &-1--5- lL :/

A : Zeryo (edleqs*-+eauaqr

NoLo- ,

cher.rS .frr 2- or"le^- ,

0..lon {ev-o

{ Crrl =-L

I -1 L,

tof

s2-o

4-!: I -o-l,.tx1

2^ --3- 4 -_-:

-l * t-.t o

fo"Jl

Page 10: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

lo-(ts) I r 3 \ 3

I

la s 2 3l =>3(A)<sll3 q ll3x+

llp O Chee-h &?r nr z oyd e.L__-

__ : Non <e A', 3eh

R+- Cr?++ Bz+Rg)t23

Ooo

2L

Rz * Dt't '

I 3 -2

-jrrn ft'Roe g e,.l] is

O+xr',r +hen t +ho.^ it is djPgaeutL *.. eor" Pqte 3o cno lng

-

'rt .t ej+h"r bU- errnJ<.rekr\nees 6r qlg +u9^n3 '

!r be Ze ro

Page 11: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

Rrzllti

, 5(q)<{-

R -- rn r pRz-- 3 R,-, Aq .CArl flrf- RaJ

- ,f.te,LD- ,-^ Lar,a* _+-u--

_lq_ a s= -L ls

' I (n) < g

ChecK_fur_:.+

I -l -2 -+a 3 :L -I

l-- : l- =')-oSo q 3 1o

onr,n, I\.A-.rOlfFAqrq-l l. i --V iTcZ)\6c81

ej.c.', la\qr., ^n

Page 12: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

fr q.J giuen sU.+** oF e1*,0r]c-Fbr)-tC1 z = drO:rgbz: lQ:-Z = dr@rdg

( cseP+icie*t

nqe.l'"tu)

-- hos I so\".

- tn O'nsisie,.'f SUAttro -AetL nn S o\*,

(h p're-G--9tr-3{ae-}j {CA ) : N-lo' oF UnKno".r.,-re*ffi@__ce ((+e) :((Ar- No. bF lqt]oollrs

re;sc+l-IUor^ +1"^ gts+.^', ,

t^

OMq hqr rno 3o\*'a*14 +o b e Nr\ cons\ ste-"{

--- rlnen +t^o (dslerrL-a--g*fo ia--h-M

-

A,"nd tao"rjro J'rr,F-n.Hz- rno " sP Ssf---$:r- l'noro {'uqrn -gqll

Page 13: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

I3

I, Tne h o. oF Sol*.s +o1r

|Iff + L =3*he S 6<+enn oF eq".

x +21 +3z-: {-xt tSttl_= 6

@o u@-L .O2. @e-sottl +F - eil4rrr-q--fts*orr hq"e rr&+re sd* 6 tloL:nilr Le

ex E !uCed ']eQqdl,@eiakteol*,]Nse:G +e lA8l L hn"teqsle- nq..f,i2l .thtiulqs$ Col* mqs+llehelr*de{

(-Dp*rt tryc'F.e e.lennacf, 'rrero ln-lo,a+ col* ) -*---- -@ --tp,-'-Ag' is s q , q* -oJ#x , *ievr lq c* -Lr th , nn q st' Je-

Tn cllLqJ-asL.

- @ m++e lqs{- col* i. --erc-l qd e d,'u-a-freseu *fug =.td.trl:Lhsrcm(- -@--IF-: lu AR Ro{rrr l< ( .

-=s!-f

e - A - '--- ( Rs.lorn €le )

.'. _grcluole \qr.* colY3r q

q

Rn * Rr , Rq- R,

AJ_?_3: ?

'' u lr.rs .t F S o tljs---fl qi' qr{ e

_ ConqicBrf---2----2t * 2-J + 37 *- 2

Lx-*z : 3

sol* )

: I I *'2 3

e o -3 1 l-- 3tc{srn5le

_L=l_ol <e ro

R+ + zls

Lrnlgue s *,\ *

ll

{o --g 3

\l

Page 14: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

lLr6Tt3)+Lz---)r* )-J = 1

loy+32z_ -Lax -3y*Z = S-

A r14SoJ '' J- Ag = S,+qqlrln "fix

lr-ri &nd o d"d )Q- Lp- -9L-3

_o_

g__z

lo q -2-'l *t -3

Rr* 2Rq , Re * ERq

32--1 2 o'1

--- {+e e S- rri*h h-,, lns r--i$- freselol--,*. Frfil : \

O,lo 3 -)'L -g -r f

R',-3Rr Rq - LR'

Page 15: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

t-1€-L. Chec,.l< *1ne Co6s

' efeney 06 *tre

4lc *LJ {-6 z: Sx 1- J: 3Z = -llrx *'?Jtgr: z*l

S3s*c,rrn ok e?"

-l 3

Rr-+-Rr , Rs*-RrI -8 I

S o ogb o o( 3X

@ jqc--e -Ld-e -qLff .rrl

X4 :!

lA\ s, e :o

[1_e-$o_o4_$o:r_ s o I v ir'.rg _gy s+ep _ ok € ?" Jno,.tr'S ons_ -S _o_L: [: __

O )c-:-A *r B@ <rornrr)e-r's fu..le .

)c : A_L_rjy__: Az x = A.zA A A

OXlo

ta(3t' tr Y')qt-'

--llJ-32: -lrlq: 3

-J r:lI+-5-32. = -r-

(fnrc-e hre- h,q"e- dtsc-rlgsn4 fr,e cc'r") s isfrc.ncff o h o, U Reo{S;s{e,nrr o b ?4,^

. o. S o\* o-l ocoy Re-*l h o -J

3ol' : feel ; I

CHOUDH,ARYS51@GMA i COV 97842Jb:lti

bg1_ar]<-_rr o , oF -s o_ll.

Page 16: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

Let Z =A, fohere k e R, l{

) - tx : -L

1= l<

t:-Iftr ol l$ vo.lres -uE lri tJe' lL hqra -di![-. -so l* errnql- tLe.ssSolt qrp im Eni*er -- - --

-2, Ch rck lfe-,fo-nsis*enqrJ

3r-\ *z = A

-t +J - z = 9

l-l__q_ qoo

_r_2

_L_6 ,

,]

Clr,

/Al u*, d n{f +n) : 4

3(AB) / (tel--

E+ 5 + a_;_6I + zJtAz = lo)c{-13 l.Az =v

lem oF@

-l

Ru*I

J- -l -l Io I o lgo 2- o lso 3 | lq

Page 17: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

l'rCql Noq-

sol* Cb)

Insl:un?e1ue Sol* Cc) Iflore *t",ort ona So\*

I qs* Crol* li. ekc lt^"{t"fSx,t '

l. 1

)1. I

I

3

,\

I1o

1'r.L2-lt 2-

-Rr-lle_3

B-e =

__t

I

,\ -3

Cqqp ff-P X*

Ssge= 1-- ---,Lp )-3 =o ',=-.L:lp_ :_o

i{Ag)-;-J.[aL 2- <s,t'\orre fhql cne Rol* , iA-)-:,:*-:r-lq

OocalTl- ? -Ip .\^q *o

J (+B) = g (A) .:Uojque (o l- lE ) :L.3-

4,'TI + ?r ? )_z : B

2:c* 3.t * ),2 : lrPol- ?,- A8--: L? i

es qnd- Li \reolorq f --

1-L*f,'q''Le--.pne g+lre, m@ urrn cr! l'b.

-- I to sol* -iF

^:g- - xf to

2-g

Page 18: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

& l\rq nneler ' ' lB

qlr^re tl^oofs o F the ta er ro Cs.fu eC drqra c leu f, -. lics 1oof if, - L**.sd roots or H5;ertr re,-{ tre on Pnof<rr

Yo] ti t' I

fre" y@lael-oF ..- = I XrI&

hlhfch .e+={d"rr s+er+. ss,.U-sEas *he'rnot-a{- err ,

[n - f fJr --eo.ma-spmdftr:r

e+nVe c-to^r of *tne nn otat x

.l.tat1 l:iS elttal {o

.\oA{s

r{ T

O rixe--pro.o

+o I ts --ol e-+enrroi"nqrr+:

---€tr,= G b/--_'-"@re @,Lr3

AT: A

.\ : l,r

e--S-- Dg-en--\rc'Ju: oS skero -Stsioxae]^'ic -lo*,tlX Cq"ll -r pt t^relJ irnafi inqr3--OS_-!e-lpS:

(ar:-A)

Page 19: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

l5A: o -r

1r ot A'=forl, l-t o I

A7: l; j/4T:-4

---@ o're--E-f-er @ lo,aer A- es tr*Pgr {-*t*HL

{i %+sl eJ@4- E'e.ol rqlqqsrz sl

*a-L * f5.n-- YaJ t -es ,

_ A_:- l* 3 , 6 &€ern' hq.lqes

- * T" pio{-g+* -ure,Ltres . N-errea Cso w o*f }he Lgs4q oP: tqsHx

- -l-r.en *Le Vdller grl tct-ot',g'

r3z\

A-A:It,

): l,e-{. F$.d \hsrdsen -trelre oE sa=sLs---_-

--- l..to aee4 jq I"k" *I-rkng&g Fod,,* [-I

3e- I lg 2- I

I I-'l--23

II

Page 20: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

2oz"slg{c-l{:-*- s,s

a lt, s'

'-;- 4n q Shot^l d lhe" ec^te oP Qoo,

: L- c---u-I

4xr + Arg; o--*--ru" mgj+ t""-+ {oo trnkrrnu;rt

-'. r r {- r a : o ---

i.l- cen, he sol,reol ort L,qi[@T r ? - I" '&nlS hh€n ellrof, +o Zero-Ar-h-ile-

Xr

srrl-rl

l-l , *rJ T

)c:f-f l 0rLl ,-,]tLI,l

-*-=' r 2=L I UXer,-Vq.f,qes

Ve"lors oF *l,p-_ nnsL{x_,

crgonoL@

>-6) :-o

Page 21: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

Ttr A = 6

\-, :, I li',]

LI

(tr)

-)cr {- 9:cz :oXt =4xz-Tr :xzt{l-l!

[.+,-rf]

-0

o . --Ti-e--ehqrqo{qrisffr eolt olli-c--mohjx :

- @ g, r, B -G_t, ", t+ L@ _adrlrQCl r , 4r j_

2_

A =__l_q8 .l_qd_t_4J_:_LB +28 * tq3

"-''- A : o, ' . Avrs Prnd tret eh ould h e ze-a

Q0 1, Lz, 4 (Il_O*2,4- (cr -r ,1,4 (_dl_,2+2r2

4. rnc,-hi

-(q-r- 9, 3 +si , d,=-[ --- ---- (h ) *(Lsi_g-rL_a. L

--cc)--3ti.-l.i*s4i-- L{d) s* -t + ei , *t -:itn ++fs fl^e,

the "ti*$*r"l

tf vo-lt,-agle.rnentc .

Coro e o.Ns o "f qrnq nn c.l-eh e-! l,c i++'

L *q 3

-6 T -q -6

9_ L S

Page 22: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

LZqld *r'e E€", re c*o 'r ?qe8s Fi"nd

1T" Xr."n I,ta -+ e-

I

lo -11__ 2_: r +_:_4- 4_____

- jorlL-9-o-lI: - *"j eillUqgoosfgqmcd-

Lr I

5- 6xg =O --

(x r + aAz {- j)c" :o

S{r : * LXz*![t : fCr

-"12+gxg=o

2- --s-

Page 23: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

23

:l l1\ ="

?,2_rl

0

frt):r -elolo

:6 rr+--

*lLXL**2JQ--:-O

U rC, = 2d3 R A M K A Nr A L. a rl O U D n A RY5 5 1 @ C M Ar L.CO M 97 8 4 2 3t 9E,-)

.;u _:_ :(s _2- tl

- -6tr -Lz(z) -t- 3 ( rr) : 6

6

Ia : 3r Iq 3r2_ 0r lz-

ll 6(

Por ):-)x

Srcr L2x*=*2_x3:oK x, * Lrz-

_ 2ct. : ler,1 q-

Page 24: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

L\J, fionn the hn.,trfx '1"' =

I I L

to o

,\ : l,t-13

One ol *he E.v , tchi ch on e k- ec"ogr der leo si l:vl -

*re Ev

Col {{, r, tJ 1 u-(tr) [ r, r-+Lkf+;-r , ,-J' rdr f s_+tI

_-tr*1!C3:Q __ Da-}tr +XL=oTz- =- 2-X-:

oL3

1[r = )ca

nr Y_ t l-I:--a^T r - 'rL2'

-

lu--2_t

-- Lx, , r, , {.] . I r,-r, rJ&. The re-o-l rn- n rns+;x ,_

t /..1

i_:J

Qr3-_- :

= - (.j!2-

Qa t Gar- Oe: - ().rvr

et

Qn, Qy''z- Q'n,r ; Grrn

11 03nn

Page 25: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

2"s-l_ennq H^

)L-B\ - 2-S -9--

ffiil:l =*-Rrr + Qrc., : o

Ir:XzX -g 2-

D(t - !C2-

I -2_

)C

- 1)CrFUIc :O2)c' = 9Y,)Ct : )q

lo-z} *s-^+-A2-- q :gl)*-lA+-6:o

S, Tt,e

sg'*t-

f-o-rlz'rI

fn**rfx safrE e-l-jt oon €Lurat*zr$lte{

Page 26: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

2L\a,e+ \qce > ,oi+r'''

t" ='lq + 6a:0checr '

, as ,sl - \ 3s p-Q 1 * \ e o I =\, 8l lt lql lo'J-L.:_H :S_ = R, H.S

Or , -6-L-;, lA - AL6: a-r ::1nn---!a1t-l64-l : -lf *A

T:rLl-l-u- r -l-he *he rn*l = ?(gt o,r Ct t r,rl Ccl - ro Gt - m- _

R.ople"ce ' r' hJ *tn s t_ r,a0_ A-re --r1- : -nn3rrn-\:-nhl Ia- = A-t

pq-l * _ rnl1t LrJ 'lo' tp a e x-? -,,ho c,f Jx to i*n e hor e ?* ' -

>"+ 2'+al*-l :o --'{-hetr P-l = ?

L. lf *lee Follo$i nS rre-Pyeqelh +Lq ery- oP a{ l-fme +k'n +he /hL--

(o) o,o (b) 3, d Cc) +,3 Cdt +,d-

Cot f}+P+r (h) PL..t_pfur \k{--(-PtfP+2-1) (aJ )rLP>fP+lJ

8r t-rJ :- 3 2-

P*+ tt". o?tiorr 1, +h. zVnCh"ck LHs: RHS

6x +o + 4J -g:-- r) =-o

Page 27: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

.o 5-;-tS a.trt Tfl Coa"rsr\+<l,rf sys+<.ro

Rq.rrn oF'ff'l+iH 3en Ve cfq6 oP *h e roc4?-tx.

Rn M KAM A L.CTT OU D HAR\ 55 1 G,GM A t CON4 978,i23a:!i:l l

Ar :gposs ihle

Gi\ et'n

oto\t\rL-

z.h'kOF ecy

(qr 13. tnhich

q 3x q tnotJx cnd I\- *he", *he hlgh" s1- P,

GrL (c)goP *he Pott oo\n3 1\

1s o q\o s D

looe-

I cb) \ol Gr ]I lEi tL -1r\ Il--- 1

'n i- -- --l

furrs ?- A P\T : -

e I

od) I

o L-1

t_l

- e " Tt^,e T.orw:x-e qL G nt]o+t{- ,

Ar,s:.-

oL

Page 28: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

2Q+k- Rmntr op

J' Tbc Lcnn

of Non -Zer-o e,lerne,,.f, fn *rre dic.3o,-,J

.-hnix

0 lrola

r t*) {_3

--lnl zxr*o

** A+n ;--?

0r-rh : 2-,1 a zl : 'I6(oLs

(s)la (brez (_e)eq .-fu)/rE

dektr- rn jrre.,\t *hr* r-u-si-

Pouoe'r 'n' rn.{st be t,c\-i }k$ ,'-

lsL. _lgfl- 23 I*\

--__________-___-- _

- e [ro:\

3ot2L:t_3 3

L -1L

-9oJ- I =-

l'lL

Page 29: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

-ir [ +dj (nqia; | =

=(

Adinln-l

l*l n-t ;

rt-l

2-J

lnl(''-l]' I

I,-ktlal be- q gxs rno#x eLrsd lB' E j-r,,n ad+ 'ql qqrd 16:&e

- lBL:6q +herr-$e lnlrj__91_L*L- B= Adj_a

le\ : I al ^n -lr .2-

6.1 = lA\-', Lol_= Jeq

- n.F^rr3hd-lretqe oflalJPe.'E .au.h-uqJUe, LoTU$-epetLrqo-L"gj[utt-,.rr

Otr-g?* l* --5 qn,rdz hqvo o -S!l5--?2x'{r3! :-(, - -'r+-\+L=qx+Lj*Z zQ

-S$--J(+s) = ((n) ';

-- cqse-t. r F (o +s ) +J--

lal sP21!;--l(o+):j, ((a):L-gsiffo-is- lb r .ns1d3!s+

- -Urn-Pqss"r b,le --' '

"'-'lhL-i. q Ccln trl"de]ioo-@

For *hE nq"+o{x

.3tr q/ r1t

-rt rl I

Rc * R'

3"

frl :

1

tt3o

o[v .lq

.{

3_2_

a_

1

Page 30: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

L++ L-f*i+H FCx) = Lt

T-+A h

, Ri3rrr . U r"n it ,

L+ Ftr) = Lt trCatnJ-+, )c-)q h+o

RAMKA\4AL C!C!DFARyt5l 6.GMATL.COM 97E42l69Si

3r

F ( q-l-.')

-)o

L+e

)'f Rx),'l-fE

--Tng-Uril+ eE M-j{- Lol{ ffrL teft ar'1d yf€bJ Lr"il-

G{'e €xl-s*eul-cg Cnd ere eqrr_*L

lr:-t(rl *Lt 'tr(x) -,--__ _Chsrhj -

- - ,f-:)- Q d: Q

It l-arrr _IL4X. + O Xf z*7<z-

2_l

_eD _

rg ) Lt L-qas-+:r llfasx - or . Lt [ * coJrX-+O rsinri \-l-CosX t*+o )c'Lt -b tt -t"'+ sia:;

:c-)O tsl.n X( r*(oS>c) x+ o 2-x:-l

. -t--

.2-Lt 'sltrclx ='l : t

----x + o lTc.:I:( l+l 2-

--e-- l[nSrc.:=L-cusi )(, -:--U:3qstel(_r+ crslJ -

:* l € hoL*te \tq!n guq& -qud--ds,moomeec @o-n olas+nic

Fnnc* , ther c-l o.'* lse L - H "s ? f +*^.1 tta-l e-'

-I

Page 31: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

{Eorc - s'in:<.3z

3. LIr+o Sins rLt- giq-q!S"-r. - \Fx)f+o

__LbI+O I

Sio-r]=e4I4

Cosx

sbr

( t-Cosx) (t{-cosx)

= -L = l_tC rtl) 2-

--L?!- r- -[?c t*l

.' l.t- - Ttz q'i,n t TElel ---.- -q{-(cesCcx) ) : -QsoqrcL+-9- o- -l--- ,. ol r.r-:c) : -1-\rr

r I,'f1

f->o )C

a. tt '-Jl:xa - J tr'x+o )c-

J, t+ loq (ttrl;;o x.

B, Lt Lvs-zx-+ g le- I

s" Lr s _[L;rc * s.Il;L :_S_r-+ o r

-'. L+ tt+x)Lg B (t-x)13

-r+o L-- Lf ffi,1 =1.-l'"-ritlrr+0

: Vqrfr +VqxtI

o<+.r

T

Page 32: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

336'-LL@r-to yzIL x_ -[:-a:c) --]-- x zrr+o z fi=rz- LJtTx-

_Zlc

=- | | s

-lI6aL rL-

l-LLlp€ LLr-xI --, !L-l o

E_.__kIi o x-B It-o

8- lt_-----xVt-z ^ o-';.-

-

r_ig r.-B , 0-6

k__r+8

(:irl )9 , J L-l' r.*if't "-

)c, o

I+ e h1-

lz" lt- 1'+r{rq t'. l.nz\+q 13

x_-) o.1t-- l+a;;17-H= ""to)!{o

lo. lulx-l-F--x+ *\)c-t)

rr \rc(r- Vr

- =-:e=r- - ;z-T--\*'g-Lo

T-rP

f,. f x*s-lII

t -- 2 \'&ALJ

Page 33: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

3Yll. LT

h**l1- 2 + 3+ '*n

q1L

Lt m (n+t)

yL

--h.=e---lnL L

lr. lt tL+tL Fs2-+----+x-)Q-- ,7'9 'xn 3

It -(nr-D- Crrr l- t )tt- n 6r-

_4-2_alrga:-1- 13n-e Gnu (

':' I

-t+' .t-t n:Lr@ .--lt!jb* O I tL+,tL L' L ,nL '7 s -

2 r.3h +oo lyu 8xY IQ!) _

.r-) o JCC-

ls. L*- l- C-o-sz-1xjo l-Los3:c

r -lF

Lt h

2- l ^L- z- lj

h4P ffipr)13L6J

Jt _ rfff rn*rrJ- X-21-g-+ree f..J r

= Degrrerh)P , 3

,: lx 2-la

Itle x lxlrR

lfsnlU' - D-

Page 34: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

- )4. It- .+ r +^ +-J- + + I - sry\* oe t'l- r'3 3 V lrCrr+f )

* []h€.rr *te so/r,ir3 cF Prcblenn & diFftculf orro] ]re d or,'i kmoD

Q.J sf,, frrnu,) q *hen Ql e ' :

! .-9n :. I T-, '

Pe4r#d fuee$oiD en rl+p4,

-(-Arg J<r-ao - snn q4 l<rno )o J -- -l- - t nnvrili"iil'1"" *':l'11':l

n ntl h(\ft )

@ SLa-rc-qr-id -Co.sr qsLqliirlou,olrs Rmr* fx ,

- cx orr-o Cortinup-qg--Eraf -hx ,,(tx cp Pf c-o-c I : o =)--X -:{elr.f,fl-g

- - ,@ r,ot:c orrq Cs-ce c r qgp f-"rpr,,;", f-",*" - f-z-€xe efJ- --Sr_ra,*-L, x fu_

- t- cAn$ lt"-n eeu*arr.rlb !pl4e &',c-:-o

--Xo l*: - Lr,-, P (rc ) :- -L-"" -.,p{ q D = U.*,, I

h+o h+o Lit-Vo-h

z L*- -l '_*- =-l- ?J-l-1h-:o L*z-a[h lH I

-: - l- :I:QI l-ar, A

LI+S _,; RHJT *u r a Drk- Qoq"",#n--tfu

Page 35: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

3oL, Ftr) =rc

IeC""- t )Ir --, -)

r*), Cr fo) F(oJ:o

gqf r- L."- F(x) ::c-rd

L*^ F to-h)h-lo

: L.r^ = h, L"u." -tJh+o --Le:Y^,tiL -: -h (o-[) =h-qor l.n -a. -!l'' = {to:€-* =O . nln =Jo =e-&=6.J.Lr

UqRt) - = Lu-S{ olL,-) =-l.* r, I /h-rlffro h -ro h-+ o (ekn+L)

_t* h. J^LtJelJLJ..l-o ej'h[lltl-^)

(tt#

C l+o)

Herc

A Fnnca F(x) iq 4e$.ed bJ , '

tlhi ch o F *h e lot torri.J i. fuq o --(q)-+ n 0oo.H:nrroqs-,e4 r:-o

(-b) r' ,r b?.r'co,a^t-.irnor. c c,-*,-L :-.U --CCI ,r ,r C-ontrno.fS oJ--v*:-l i' "

@,+rneSolw: * L,,r-- trCr) = L"^^ F(x) i {:Co)

rc-l O

o ) X(oIJO

X) o )( =-C

o

z*._^_-l-_(' 2__

Page 36: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

37-l- *xL

:-l *g:\LT

L.hl Urn f Q[ h+ L.:^

QIJ { <TL r)l1

X.:l

tlL - t/y:o

{lz:--lz- -r+t l; -4 &SH L;a-l - Ijrcoqlr:no.+r

+- A f;inci E{rLt -gon 5:r,-_ECreI_: o; x(_o

X = c,r,z--gf r Qlr- = to- ---- ,-i ) o D( :- o

: *q

-lri)=x <L t)-r- I )c:_l____

- Sz -Y qx}' 3:-r -- Qx L* 3:C

--!1xr . gxi- = !q r *Sx I

__:-l_

- Ctrrr y<g____ \>L )C:-|

- !x2:3x 3:c Fy 3rfg

f{s- t-j!t D--! -L; Etq $ :ECql_ tlef+S'q'rd _ h-+o (-bl

qe}rvJa.eJ

olr,d R+f,r: = L^^ Flstlllg) gyists qncl e:rehlo h g..elf.rFrcq)

(-E LH o :- RH D = F'Cq.)

- -er-:-g*-o-1 : (J-9x1=-3f-*-I S!<L

: D"r-qslitfuNss

A R^c lli fu--se..id-!o hp .l't+<ueo'{elhle- gt }: er -ip Ura

Page 37: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

L-ist o F' f-orrou.lq 3 -Ctl d (xx) = ii\,)c h-l

Cz) dCJ?):-1-2ft

Qq) d sin-rrc =#"Qfl C(srndrf +J-

.[ r-r:1r-

Q.rl d fu{+ild= l-JF-r

3e

G) d,r*L)= il-\ 'I-r/ zrc*JTt

(s}-!lf-l= -i= Qe t @"gd

csl*# tgrqr"tl { CC'l = o'c losq (?-ot 4CUft} = d$ t {L

d" "lr dtrr-: -dr

-d:t {cc

ApLd C+eCIr):--ser1(trt d (-<,a").\:see,..+qnr dL\ tr/ _-9f, 9t-dc\ \r

*tI l-rLl*x>

ltzr dCsiohrc\: co.th^r \ts1-

CU rF -U-: s i",-rl--zr \ {t {r -

{' ol*(l.,-'' /

- r -l.- t_efn ,

*ren -d:=- * q-{:u I t:tL

YqrI I. -Lrs x)\u )

))L

t

ts+'-LJd Crl+

_r_

"l:!'

Page 38: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

"1I-qa

e-)c -J

- )t Eg.-\

v@) XJ:

)c( lt Lps>c)

=)d: ?':ilrt

- - tgC; U"g,e1?

5 Lo3* : :(3 .to4-9y lo3:e: x-J

_J LoSx tJL = L R A lvl ( A \4 A L. C r O,J D H A R Y', !, I ..r G i,.1 A I L. C O M 9 7 84 2 3 6 9 8 1

y CLoqx+l):X

_4L -= -(-lt t o-g:r.r :.,rr)-:{& ) .

Qrc _ (t+tosytL

- .li

- Q.r L-oJ rrJL

Sol* i 14*I 5* = f oS 'q t*'

-- trg-xl+-Lp3+s= rln+n (t Sql

J "lr.

R dr - *nnl4r :(

nr

Sot*-i =- L"s -(xl:b) = Lo:

-o .:t- !s:L lqr:- c( +b

( x+: Llrba -LoJx t-b-[o S ] = el]b--Isg Clet=D

b_ tl-JdXcl r

=,-r- ct tb olY

ectJ { rc)Ltf

\: =?c-v t*lo3r,

iq li2

.l

:' -91 ] :_D\

-

Page 39: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

f b q+b-I dl o{ +b.

LT x+)J dx x+i4o

Ulr+hr qJ:.b d) = ql+-bx:!4!L*clJ{r x ( x+91) CE+y)

gt =J-qxxlr IJ--n.t 3h,= er.ty)mr1

a clJ =Ldx )c

5. -IP 5l;--

vL -

_ qU : _c_als xdx 4) - r

dr : F'cr)

drc X(r;-t

L:-t6

({-ILMTL}-

l-t')'(rr-xgJ -=ry(1*x3) !r )cJ

+ =Gx:qr)

y-! +sfn>c +J l

dx 4\ -l

c. TPJ-:- -SfuL - ( r*x2) )4::cf[ \-xil- -

Page 40: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

7- 4t-lc T.F

Yr

y= er cos Ct-rS x ) + b cln ( Log:c) *l^en

:- -esin CtSt.rt lc"sCLaS:r))C

riy.*{J, ^

r), :- -asl?o (t-"gt) +bCos(Log:r))r rJ-trclr_; _q l

xoJr f-?9yr = :{ q coJ Log* + hsir 4ng:c )

-- r?y2- l2eyr-; a)

/-arrtlil51,

-*=e*ge--

=J

bL\z t:)?C

l+-rlo l+ Cil

i'1 : J-f xJ---_: q2_

Crr 2-

)c= q)c: 3

):- 39C:Q

7: Q

f o. J-F O)IF G)

aojrc; y- j fti,llgCosG -Cos30

si,t & - sin3 &L c +- sin8jL t - Los p-l

Gl

: ,tgr.-I Ji

J : gjgrn-t Ja

a-l

- = - +<te\ :-*

]_ siorc

Page 41: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

L1z-8' ePJ(+) i= qfosfleB*c.rsg] ): q fsirD.ocoseJ

dv-',' :- 2dr ,

& dr-igg--@ oq; 9*/qe oxgc*tsx-siroJjL =:te(ole -3cino0-qqs S9t sC-srril-s cj}ru({ioe)

Eine -*+en0- ccs8

- : 3e or9 ( l-slnb) ;=csrq€--- S s in 9{- l :-s qrz&) s, a]-9-

@ + : @{{otstq+== zs,m €laLqrql+ @jL- vj 4gaE€l1i!o}o=ors+?_,f t q ( r*c"ser

-__-_-- -y-r.,.-s.-glL - ?i qZarp *'lno r - si.osJt

a -ct".,.q f"+<rvq ,x e&,lJ :2 e---I Sb ,e,ren lr"l-err.a{xe(q,b)

-i+- e (,r ( b ---

2- -toll]s fqpqr \.e, lqe +her-a-xn, ee ql = FCb) , Lq-f FCr.) A

E+-,r' dedned sq.A *Aof ,rj-', Fg-rl ri e"-#z,oqs,--io-L-e,bl

-ein-Ee:)-ln!ta ia Ca,b) -ArrJ -Fe q)

= , -J utetsr

-----one \roJc{e (€(q,51 Sc'ch:taed F (ct:-o

rfcu)F 'C c', = s loPe of +ae?vi- . oJ* )L ) (

---:sl.Pe o& trE= H,--l t

)Cu * )C,

= F( b):E{q'

o, F Cot) FrqO

: r3*fx fn Eu,z-J

F'Clc) = $)tt-YF'(.) : Q

rtr) = (2-)3 d*d z o

=0b*q

Page 42: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

3>c- -co-

=qv3

FRo8t-erq s- 9- c:c )

4sz_ *zot .,7 t7

L1I

c = tvJ3p

J lo:c,gr

C

in f o,

+Lo;rsfur*o :j : ogol

l?G) lsiatr- o :-. oa1i q-T

RAMKAMAT.CHOUDHAqy5sl@GMAtT COM 9784235931

p( a)' (er)'

:- lqsl--liur qCal-g-=-slEg

xCq+b)_e, \ o / b> o

F("tJ : q'+-abA( at rb)

b(qru) _

F'tr) :- l f >c + aa J1 -5.)C_a-ro1

D'Ir"):0

--- ez -qb =o_-- (-a = c,.b-

JF llhlllrl Cst t']eb)- , R"tle{ &ma,,l-a-& xo+ zf4-e*Ap

,I

c = /a-a _g_Lq b)J

- lo .!r& c.q-!L, 1=4rr.4<"

Page 43: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

I

1eg :q"03!__a3!2 Ve,Jtte. 't hzo^derr)\,1ot-

CIe@

: slope oF A.E

: Yr:,Yl-Xz -)cl

F'lcr": @b-q

Fcr; E eo"r#r,uo./; in L",al€rr) rs ob{gerenlls$Le- in Cc*.-hl*FCq) + p(a) {1.e61 (fq.aj{_o) Lr*+lpn^s* oae vQt^ece(q,b)

. p-c.2

:c :b Z=c

/ -tp(o) ,

QQ F(-e) = l-eTe = I

c e-lt:C e-l

Pr.h/erq:- Fr:") :.r3- e r-+rtac +a ion f o',v

F (.r) : x3 - 6-1-+-ux-6

P(v):6V-96*vy:-C_le

Sca- l:c +ll : (*(-6)

3"'- lrc *-0 : o

c*.: lvtJ-tqu-qtr( : lz-+ t--:_ .l gg

C=y'!nb-k t ks

2_

=r3c> 2-t

Lb)

::::

q

F(z) -- |

Page 44: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

L.+ F Cr) w4 g Q) bzeot 4 qLe- Cstr H?roqs

2 ,4i *1,r, Qefived7, *he close4 in*.r'v4

Yr

,3qcl. +l. atr cl; F

Iq,bJ

Lauc hy ! l1a-qn Ialue Th.orer\

O f at{ g er-Q- 4rkLntsn*s*Ur- fn Aq.b)'@ 5'l(r) #o f relc-bl +-heryE e+1e-a51-- e1re vdq4 ce

Cq,D- Sqelr *t^efi:l.

I

fu,E_Ld_-:*ECaEEcq)5-q

- #-5'o't<ttgg@

lq-&ed

--_-q9- = eb- aqf c e-b- e-q

_ _a_)(_*.*(e3_qU

zb Zq

Psorleq-€- Pe,.) =.Ji '{-eid :*- IE-{tu4

Fl{r.L:- /- , 5 'Cx)-l^ .z'ii zrua

l16 ,* JJ{- .--- -FI

Itr'l--;l-

- T& Jir \'r c(

-c: -(E-:-J3la-g-.Dl

J-2,.0

C : J c-b I ( ct, s)

?x6'Cr)-:--:-es

Page 45: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

P1-o!I"!?r J:Cx) =Sinx, 0€G) =co{rc lo,LF'cx;: corr

^,o tu.il: -s'i,,:cCosc . = Sirnlf = Sino

_r7t_)

96

\. +o- -S irrrC

2-

Co-rJI * Co-ro

-C-o*e = lz- o-l-o

Ctt* e

' e: -L e ( o,7r)tl L'/

Le-+ F- ( x) h,e o 44.4 Al3<re,t [.r

- C,^,qtini r,qS ije L q- CLa-hJ-

(q ,r I

G* r-t *h : )c

Q=r)l nt n/

Fcx)

#"en I l*eee -aLe--Volqe O +e o,t) Sqc-h +hq+-

h = r*clF( r.r r FCq )* (r.sL_t_C cq) + ( x*tJ_LCq)+ (y-qf F't ( rr)

+ r-rtt'"*=-:-Ar

e{^"4 jtoAljq; rfre-t"lq )

t (x) . F(-o)* xF' (o)' *xL F" ( o) +,- x! P't' Lp) r' -2-l

o 3/

F(t):Qx= eo=lF'C r) r ?x_= I

fl" Crc l = -eX: I

€r>l.t-r(r) +xz 0) |2)

-{ Lr) +3/o

reo*c =- *f

F (c'thl = F C ql t:h ?tcq) +-* F" Lq)+ - - -

Page 46: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

e)c: / -t*x +lS:+- jf+ - &Lt slC qsn &t T o*" o fo pven

F(x) -* ST",:c =ouFllz-)--; c oJ )-c :-=l--

11

bblgsr-

F-l' Cx-) ; - sirr )e = o

F lll ( r; = - c-oJr -;-J,91t,-:c . trr!) 4-4f Cq ) +-x' p1csl,vf:S""P*'TK):+1'ry':'l

a{t!- sL-L- -o-Jx tt) +s',er)-.Ld : C-l) | - - = c.r

Bl

rl 1t

- cal rc : J L -rc-r -l- :!-v - :q( -J . F

L____ ___:J q{ (LLJ-IIP -Pg-oble -Z-- PCx):-tn" x=oi---_

/+-r ) : O

3J

I '-l- x

F"(x)--= -l : -LC rr:c)_

F_lll_C)c) __7'_ 2-

rl 3to

I *ec

Err Clc): -2-Ct 1-r )Z

_I 1-..

C ,*x f

Page 47: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

F t,, cr]

xl+r

(r) + - r -,

Ltg-6-3o + rtt) +

: t-Dct+r3*:cY+ 'xT#

=- r{l-:c+xl x3+ --':)= ;c -r1f Y3-.rY+

2-l€

r--'-Efo"l- .tae co,@ Coe-FF = :l

l]e--gxP-t^, of Fcr'\

2- t 2_1ac

= <I ahnuf xz:--gqLI----F<rl = tr-q)l()q=sl-F'c a) + Ct=q5-EC"re

z-l

P=Ueerl-.ff o+-:cl-,:n laz @'oP FCx).-:,e.lr eAo"f r,3u-

- E{reJ =--e,-' ,L C - ) o.-l^'osq5 -} : q

: Co-cr CoS

TL l-)L:

- >cl 12 - : -)c--L

- , *Cr-qItd'(o)'+-----art\l

- -'. 0rr+!j.tnr,,* ot,2L=t=-rS-= +kl -gt--+l ft

2" Ei-rrd+Ue eyP*opFcx): orL-I-=-l-Sol*:* q-r l+x =.*L b\ l,+r.L gl 11 4) l*xtf=xlf

,2-B:-

*=ft ]! or PcFro q lgeh'1ec- p-{.1..c * oY--h,rq + * the nnos t Lc n-

ecrr) €x pr-€ss :ct els >&,

_ (rl )l

x1T

3r

Page 48: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

3. Ffrnol *t",e exP'of- F(:,)'sfnX-o-bo,rf x:ftt'o

s ol*t ^ Flcr):coslq: --rF" (r): -sin)( = o

P Jre ) :- - cos r:-tg- ,

o. sln r:--o__-*-lx: tr) (-r)11gfl-aq } (!q=1,]3 cl) tLt.,

91

'3 I

Sir>c; - ( x.rt) + (r -rrr3 - (x =rrfl + ...

:-E-Cx L --'-qq{rol elqr ,

Ir -- [] Cx+g I

LL: P-cx-+J*t) I

-&seest-'

too .],rs:Lon

ts lq-+-i J de"i,.olr re J

' tZ - A:z I

D)c bJ b3 a:t

-- l-{omo€eooqS fshoq$iqj :-__

A -koc* F e ,,1: ) rs Sc.^.q to-l"e ho'nncJn'n-qsql---ol ol€ne-e \n' tn: o,nd \ . ip --- I

F _( k x*__Ky)_tel E_L<+) .

furn-Pls l'

---Crr Eft*ll--:#

r ---- L(lsrq, K:) :# : rx.q arly---=--klAf-rg:=*3 r

h crn 03enneoqs OF c-\eg,reeo'. T+o\S

:.rvqf rc.l 6-!

----r,- 7=: r?--)cJ +J2-

OT by

Page 49: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

N$€,1. so

r-;-rih. h-od.{"t oP *too honnose-oeoqs fu.no' is ct6z.rn q

h o rn o g ernol>L{13 ftrn c*,u --, t- r--'L

Exi- FCx) =(x3+ys) (x'-3')R:B+2qf __

^ o TP er&-fu"L:Etr,il h-rrm* 1p X"ft "-,t"",+o"-de.m orq\d.-l-or oTp h orno eenqoqJ --&od-_henr e 1+"e 2.a++61ncq

o

E":- F 3-z-:l

€t^.1. r\s - tv.eo-e,"n I - "Corq no@of@- eOd-s@__

--te 'i is o \omogeanoqs ..,4 fq

r -crrnd J , *+"onn ,

drc

&-i-{Lr Ip z . c."t+J'J b

t, * u),.1

Tt = 1 - l__:_JqSzoGuo TF - ; 4o >ctr +- q -xh t* er rcTfgl+----. .- .[_cr,nn yL--

-' t*our _x i4__:tr ]

ErQrl C?z

, r9J--p rbz =?)AL &JEr,lef* nof PossiU\e'

-(0 tF K= #z31|Y:J-= !'z i' hqm de8o-ee: Lx.'+y

Page 50: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

NPTg t-#(.)

TFZ( Al*crrn"*tve)li n of oJgebz'c'fc

n(") ti *l3elr€^l'c arnq

Y ' *her

qtrdt.1OT) lrt

Lqf, S)c qrdhonno sneoqs oP de-saT€e

t()! Re nn orre Log- &ronn Qqes*r1oo .

S Cz) = e3e *]+51 Rlr*o'no''fl"'o*"'t1o'T'""'l':"

- S_t-e) tsirr r:.x:tf: - -Yt E 2=l :--L2qJ_ _

- r . rc zx *- 5 ZJ -" -J':Sfin< - 4ero3 */7,-:Cos z

xI<_+-- trla:JAJC a3

[tora 'rt

x-y6

bx -ql_ - -- sqc3z -----eesz

APP@€r-Le n-$eqresq- t -'

qrn{-}-, tLeq-L9-c)<,L-J10! =lDzt ---Eror aJ:

'rr -l

b r/r aZ^ z_

b3

-QrD - lp

-\) )CZx -t_\zv : ?____-.--'

Page 51: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

No+t=:4'^ J -fr' rs fit-nr..lq {'s c.PPd Cq.ble c.,lizeClJ i F 'Z' is q

5Z

Alsebre*Jc

&pc-1bqs.nofle:^

- *aem *r4-4 ftcal)z- L5=)+ z- ?

---i )c:-- alqqJ a5L

S o l- tr : t(+-rlxz-Lt:l x Z-

oJ

x+! D xL__ *_ b

---&l.-lt ( t=-Dz:0

_ *hen,xla< +-z:c''nz.J & =-r@___ bxl D)ca) ay'

+--cpe$o+ion-oF Et*lerb - i*#yb e-cq qse +e.n Fqmc

*-*rrern- Nel

, '- A \terrn#ve_ hne*hoC_ -:j ___xe+ ds--= ctT, e z

Dx DJ

_-- X E24, +ag . | * yA.1 : 2*Co-t2tlZbre' ,b x i: q:_ a)C

--*-4-J-L.1 -A-<- = tlerszz - l) :. a_a) 5' 6rro3 83

I-bi .+lE:!.!-a = LClss-La * I) x oa,

-llqL - -!4al Dr

.'. e1* CLI , ,

3€

- -&qdlEt*r+"1"

Page 52: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

a (: cos tz -t) [= jg+: UlL )r a:Jf3t. ' [z

LCoS 2a Sizn ZZ\=$in4L Jr (zCosez *t)x sio2Z

slz', *z * Sln za4 - I€ z; Ez si", ! tU - y,-+q,-' (+)l+$een-tg *here is lr,,,unrx'e- furc* *hesJeSrel o+ orogte L<u*

pgmc- erc-irepl--Sxisfs'

x h< .+ 5 dZ -;-zzDr

--5-. T-P z : Lx3+f)dlb f,Here cd-ss cI -cx-Lqr:,en+ie,{x zx t#-Lt?iL fuqeaf. <eo-c, :

ich, \n:g-@ 4r,,e- Forrrqor d++ler" y : F (>c) or r : S (Yl efc rc r"Ur4-9tra-

I. v:qd-_'l-t'rlcdr : er'r {- b

Iqf

aJ

dry : 2adxt

9*3# fh p..o i" tn o df gereulfu-1p L!rh! dedrrehve-og €xrquf

wt-<rmc ,

--- fo-J Rr"oc* uthi-ch-li -rn of €xl]_.srt js sqr,4 +o te !{{ltmrrlci*-n+"nrc:l

,l-o- G >e 1+ ztuelr + bg;L+-O

- $* t..rh L"r b.,rer Frrrcl -&- {loflrcrl $nc-A:{g-en-@

Fsrn{l

Page 53: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

sv+ o'o s ho1* cl*t frirmuJ e +o Ffnd I 3+ olttd rc^f-irre*

b-Edq bxE-r

b_)

.t, - b FDX

bF ( lhrc tahlr]- -bu

L. z = FC:c,J) tl.'eo_ffoslP : Q4 - ct,=D<

Brc E: 3vl

C=,rt iouo"

-4'dxe a,t t

fujrqre shorler$ furnn*,{q *o l}joC Mo P ol.rt 1vr Pl ie-it F.ro.* i

( Pq'6-h"c!jqr R^raortrq)-

- l F c4x' f- zhcc5 + b3|- : I C:{'} '--,

d5= hLclb-qra Lhre*rrr& er.nn pl et .

-

l. L:cz+ yr5 J- g g-e_t *CI:ar,h:3 .Zhz_!j_,

;: - ;ll^3tL

--Q:l , b:L-, Jr =.lL

Cr-x t_aj

*-1/ u k 8L-:-_r---;-r

55 ftrHe--Coyr -+ :f, oSc"e@LC"-t--f-r ave -' ?1r* 1rs [..-l otr - ze,co *-he.q tg eC"rn rnahe i+ L otr '\ aPFt3 *+re Fo-rn q q , b.,t Lf *t"r e

Corrc* f's Ze'r-o , *t,,'erl N"1-o'+PUc.'t(e'

C:ct- +l

, .' h :z !'l

Page 54: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

S s--l+ thor*cq frtrot,^,{q (s*eodqr.d fo"rr)"1o.{ ) +o Mrd zn"l{ea.h"+,rr":

tF'qxz*zhr3 *bJt+ 2-g{ + 2-B +c:-o*hen,

doH a qbc *Z-

,.j

dx'? Ch-:c + bg+fJ3-Exsorrel-

ot = 2 - h : 3, C t l,h =L :--3-:-l JttyPE'f :l ""*''1'r ioN'r e724/"'''r?'1

Er_Uemg E1

- l. y'-*-eLJ qleo .: E

---qsLS*ffn dqr+ .Srrqq^lq-

{ r.t _c_ryf

A,JTF 'z' fq q &nnc--. irr .rc'qr""-l 'y) qr ltqte +rye

7: FI

-lz

- fo.{-o-] olq,a-rve.l-1're oF o -eoP,'lPcsfu ApC-----

- tP <: f {r-y] Ai *ct) , ) z h(t)--.J{en:tte lpt,{

dervo+,ve oL Z rr$ '+- (s dennl.d' bg qli, cx,d l*

d< : i:--qtdt

eL lJ-g"c.,"q Jt b e tcl-.st "-{s ri'tqhlre- q-E :lri:e----.=--

in d"Per+{e,..ct- r,qdo-Lle 'C em{!-:-- - -- -i\/+ - lP--q- r- P -(T-, J ,*)Y'--'*rn eq d!- * l*q

c{t brc. .ty +-'h !l---d3- -.{t .ba qt

*y

Page 55: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

s6PggHs-nu: -l. Lt eI sfln: {z ^ o

o{t

: ex slnybatI3:,. = e*{gs1 . qlre =-l*, -dy :2tal qt ,t {t

__ gi = _er sho} T-co-s;,{r*L

='( -f s i,ny + z t'c,osyJ -

:r2 a6L -1

I = e,t ,y = C,tcos ?f ,z : eztsim 3t - -= -- d.g ?3"{t

s!f-l.- q{ e"t)i.L )y+ Coss 3 inBat

, Q: teQ-', dJl = Be-4t'

dt

-

' . gz+-ecr'.?+-sivrttJ . Lelajl:lfofe;--e.*tai-U = gzteot,_*'

*'. ( d,.{ I : 2- pzt' cos LJ--L-qL*( *t c in a-t* )

\ {t/t:o -

L

=_2tQ_X epSs _ l

z 2x lxfn')

qt

L-

Page 56: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

s+-s'V::c3-J.e'

T =f,Y=t-,2= la5t

:/I

id-q\\ "lr )t=t

- ---gs]:-h u: t3- t, El^St L l.'-eLo!* :L{ a gc

t

;_6x32 i]j z-

-6, CF F--=Jx

--J1r4.o }3i =z c..l r->2 r U :JErc_ay _ '

--

-f rn e *''od j -(c\* S- D-Lr:e+Tl e dcrll = n.xH ]

b5

Aq_: (J-) )c'

e2F __- utbxay - +

I rq:-lNq-l- qkqL= q] t"g q\J

bx

q"F = e_y_trlS*_rq3 +_Dx'a3 r J l: e-x lx o 4-.t_

AI

--g' {e--q : eryl -J:!rian----r}r1-- b xd 5bz.,

..X'J L(,

/ a)ct--\\J

@Lqil.4

e'l'-J '

y=,[: x'-.e""+

( d,,r\ ;61 r\ Atlt>t _ 6xzr

Sol * xYz bq

Page 57: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

seb'q z aTYz ( >.y *+z)

o*5L fn?. ! Lr!*z+!] +xyelY'[tv *E)

&. IF-r : ycos tr, ! = r sirn \iIten;Cn a:e +e + ?d:c? D 5t

Dt ay

aLlL

-Ci €g* = 1

x o\J

Sol*B:@__ ?-Cos S {

r\ t9= ler6-/ V ' ,

-

)C

G) -'.3s - I x :L0x t+yu )c-)&

1 0s : -YDr x'*--?-

_ ?r.- Lx_r+,r-\z-

- llt, aLg : :a:cJ

D")C

a'F :lJ(rz*y) - yx:-yDrqy Lx_'+v-r1r z-.,

ar'"t1--*-lr, cos)9 -a.Lrble J ,. cr<lg - si.nb

: aos rg

DrcajDzi€*r f tt qlva.*lq'f'z-J

aL 2*x_

Page 58: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

*r2- c-os 4-By\

= - C-o-s z0

sy

tV2.-

RA N4 (A\4AL.Cf r O U Drl ARYS 5 1 O CM Al t.CCI\4 97842 3693 1

--o-4i:niHm-l:___ t^rrqvm \.oJqr_l!J_

0E ct QorrH:ootg &{c* E(Xl {r+eonques +o t} Cvo{"ua- Va\r-t e qn - +,jlnet lreL[,tg- [. c*l-{ e-d Nt Otr

_- -

[P o___C d eerael Es_Jq a C e3-+{, Fn

-- ---vs=Lqe -enloLj+1.,cq jrD-sreqses

-:fh"+: rc<tlqe t -qq4|ed- *

-aG-+1^.F\ t/gf-utlc'-

N-alt-i"-

Lo fhe mqi[L_chod -xoraran q Ssc-grs o l*er"no{-tve{'_Euln<* Corr, he\r€ -(pvrrb-l hnx i In-urn clnC Seve ro,{ rn,n-

vel,Ue t '

cfue- h,"lrt Vo-lctL

---!---,{Le Jegs* f,"r'irn 5,uerlue. ls- -Qg.Uqd {r" n G )o ho,,l fn'r-l^r+mr-q-ol

- $r qaruersul hr# a'.'d thq hbhqrL rnazrw vc.lue--k- -

-@ho{- hqx* o-r--q!rrrer.*] hnar* -fu+ ^2-IoeYintq 0:r trn'wrirnq :- ' -

F 'c1):0

-lae fu"nd-

Page 59: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

6o

Is TP :c ? .o\,P"'.("() ) o ', F- is rnit q*lc

= 4

oo. lq in vo) ue = F (d.' )

.'. l..,qqj Vol qe = F (P)3:--CPre:1.

F' ti ,s .'i d -Jo_ he- slgAonqq ,

e-Ip- p"] ( Y ) + o , +t".n -'[-L1s sqb to be the ln ete .+f o^n

polmt,

I' FCx) -f {

--SsIf - o d(d')=rr (t+logx.'tF'(v) =rx( I+log.,c] _F" (x) = r"u :L +(t+IoSx) (rt(t+tojx.t ___ .

-

Xffi_-=-ff i

F" (:rj : r II-{-+r. K ( t +l os:ct'No,r,

l='e.):ra(t+losxJ =o-l t+losx .-n

' ,l osar :-t

r is--:Hrq -+:tlrsr *-e q'I-s i. +ve ,

ooo ) oi

- " F ls 'rnim o l_- >c =.t.ehnlr kqrqe =+t+L{+}f:w

F (x) = 1A. hqs it< hiqx af: : e-

Page 60: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

rqirVo"l

Fn clx\f c.l.qe

of

-le

qe

/e-

xLx gle .

6l

% +C>c) &s+

Sot*+ tr:x+CI_g:L

P'(x) : J x"t +-lbsrr r / * )x rC u \ y")

-f (r) = j.o [r- t.s d-Fl-tu-l-=--r3- ( 1- I o trl S

TE 12- \- rc )

-Iuug*

r ,, ( t-log>c): oT"

l- loSr : oII oS e- r- = -l- ----Y : EJ

no, x- : e.

- NJotP,

-gtr(e) = o--{ (.e--r-- dl-.t^O," F ls tnad o*l(:el4q4r-rlc.f ue :-F Le): logs--+-

ee

$-l -i-:- "'. ) : LoJr-rrIL -

! trlogrr;-Lo5fl

e

?C:q

Page 61: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

61

I .e{ +-{\ =)JtC

'-*, ::be?e:.-ltaea zrce-T +2 P-.r

-', -F" (*l -: e-x- fr gE r4:)-!|,rt,], _

_o -c(( :r::c1 =o_

-'. DC :._ S 12- _

A+ rL = o

- P" (oJ =e -o (z*-qlq):2>o

- Lis h',?^n tT se =oAtr: )

Q:

Fr]. nnqrecfr -)g FCic,r::r3 e-t ha" iA finq- otl_= -

6. Fcxt : r3ff_

-'- la" Cx) = 6:c+ 6

d3krp_La_,',

Frcr, =r' -g-2-- -t3 :o '

. 5cz

19* / - o

yL= I

ll-2o

Page 62: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

63qs oP' l" nnia alx =:L

A+ )c= -lF t'(-t)

)

6 (*l) + aGt )"

-l z-< o

'. tr it -lrt-qx of x : .-fF-e4f Lqeqs :c- 16-rlrl r L c-o,Nujn)tq-l_re * C -/ qt+-b- RA M

oo, o t cf l o u D H r R! -1 5 1 ai G M A I t.ao f'',r 9 7 84 2 i59 8 i

- -E)qx rJ = c-+:J@srY" Lescf yofrreo&-,3rorx +- qs,:al+ 2:[.?

i +\-* ? t z*!a?+s'

- 9 - <--- ^ - ?

___- : gcos's +.*_c..sp - 3,? sL g_ta_

!_.lS_ CoqD :J-&iL e+2-_2- 2_

--- rnox.\rc4qe: " -J +*f- Fi'.,d *t"e h+ 'oE.1-1." eo_l-e_cf }rqx_val'-.t+tr @

1r, q sp hp re oP rc4ai-r cf- t -qn1a -

-- hL+* : tL -

V:__l=rr LLh5

=-I Ct-n)crrh)--V =T ( r+r-.-

^--h*)3\

\r

Page 63: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

t"=q (-z-Gh) 6g(o

I b'.LU-l =o(an-r) (h+t) : o

3h=1.h :J

a.J

He_!r+ : Lfi

=4lt-3"

in q sPhe.re -o-e- J.qdl&

.ou V: trr:-n

---ry tel *lh

tku-: sf1;P

- ht ='-Lql-3

I H . l'orn,{o' fot' ql hlorln a SP h€rL '

p

/'njo.r'6o$

Page 64: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

6s:l+ z=FCr,:)P=e-,- ,97=Dz. tT-_-a'< iS:& 4. =a?-

fort' Q nnqxirno.{' or' rslininnq r k1' t P' t O q^d ?-o:-f +n{ 79rt &rnev ln :rqo{)- { so uoe hq"e:{roo

}rqry''d ) rsplrre {qre ag-s ,:qrnd S"J re]- -tetr-{Xe.n^gle:=

r !) __r4 !:f -Y.:l r !c =L - - :

- F.-om?leJ-

-+ Pq +aL':efa ejfioe-J"r_l:o oy 9_:o +tlaen 4Le- eqa:h',&',n.4 +o c*u."]@; lt-r -.-lr"*he- e?-

e?t i,rqo4J

f :--lccr-2e S z-

)aY"i], Yre xo, 5o)-+tx+JJ iQb-l?)- -'- s\e+onqg P-oioE on citesl

- '4oiaIl'r,? Oote c Poirt {x--Js) ,H',a *ue VrJq€J oL.I, S qrr!- ]6--

eus_e_f i *

- Se-o 4l+e &ros6. oL+aiy,-q irs ntiri.d*l+qt-tarnf^'" fqtrrn -lrdLre : Le&rfd

IFtt-s':-Qqo4 r<O4ht M Erne.. ajg"irr" f t= msx l- qt- 1l4.i- golxt

Cqstflt- - -lF a-t : sa(-o ,_t!rr

Se -e!I* -,- ltqs no ' ex.lermrp ' Vnr tqe -NdJ&q5-tni11

'-

- hcf -xroy*-,''

-Ert --q1:-othq++q &r,se-f* d-cstrlklll qn"l n!*C Eu,nrhtr--

o .o\:owr*r$q$q--

--+ t0heu -iTrn-- Arns -og+iercr' lms |Qi.-/.qqx '_-tr 1q-.foo=o31i olttnen \ul *+.a. Vo-lttEs 'rn f =o rffsr{ } = O #hon Chocse -he

C" rr""f aons ,

Page 65: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

*fh-oU\enns t -_f " -fi,.-R,",c*Ftx,J). d+st l-s:c ;t:*hos

66

l

\_t@ rn'rn * a.f (-g,o) (bl r,n.rt* c+ eSp)Ca) !to !!:lare rrduat$3*o) CC, lro,are

r\;r5_e| q*

P:3! = art:$ s n =OE .: 2]Br aJ'

^6:- aFO:cL ar0J

NLq $,

-.- 2o Tr,e Func* FG,y) :t-rL-J- L"..r? . -_fA) hnir,*o:1" (rc,o) (.b) nnqddf Lqd

(e) No €xtQremqrn o*Co,oJ edr_trtpn_e_

O:< ,AJ Dra y

_ S : o _r ^r L_-L

B- !F F (:<,r)-;-rB l*33 -€r j ho* '- :-- fQ) rcrinw t-f C t , tJ G) For' * c-T (I,I )

Cc J rn i n* qf, Cr*r,t Ca I rqc,\.t *c{ (r, *t )

--&S:= 3r-

S=-Q F :: --3"1.

blp = 4DebYa: dr'

aJ]_*', r* *-CL

= q6--g =t7- >o

-'.'Y:6> o

Page 66: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

\, F(.>c, l) -x-2+5,+r5 *-x-_qytE*Cq ) r",ia qf L2,-j) Cb) nr,qx tt Lt,-s)

u-@ rnln at C-z t 3) (dr rnot"' of (*r-r s)

e7

Y'- 3:- a2bx- E :CaRi fV

KAMAL.cH ou DHARYS s 1 @ G M Al L.coM 9784236e81

f : a'c -Lbb'

S: -It e Rt"c* FCx ) = 4r+ffiJ 2-8r --f418 hqs t T(o r nr-,,i^,.,8 (5 r rna. -€ tCa) frn?n I o (4 r fuqx lo

3o/w3- F=3e --8"c-,* , ? =5=C = la-y-*&x cJ

)'=&ap, =R S=alF',:o ,t==azc z-lrBx-L D>cr 1 dJ'

}-t *,(L.= ?6-o )Ot

33

[22>o

C. F = sio x +-sinJ +sin (.€) Aq c 2

(qJ tn?,,, .& (b ) lnq. - E (c) rniin-@!,.,G (d) rha":, 3.13-z- 2- *T 2*

: 4--;.- = lo t,

-, <-

Page 67: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

sol*9- P= * =cosr-t-eos(--rt) 6a

A ArYV - Dtr :e".ry+cosCr+3]D9

Y = OtE' - -e?r,r - (Shf. rgr)arP

t )c zY

Ftr f"rg - *t^e t.c. lc/e o F ryr frr 'p) "e4 *-

s hovtcc"r,C-ns2-:f : Crrrl '- t-.*-'*JL=,?:--a'rr O

Cos 2x : Cqr CTr*:c\ 7

3r:7f "-,rJ(o

J:7-/s c eee (l8o-&l =--c-osroIt

t = Dltr-: -sfn) -sit, ( v+y l

Page 68: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

D e Fi nite qod trn pro Per r"l+e J"..{ t -

e-v"n

' ra ,iP F('q-rt '*F(,.,t

r-rJ-z) ( n

67,

k =-r , iP 'rr g oddk : I , iP 'rr' t't ev'e.nn ,

2-

Prn,n rn-:) - - - (rr -l) Cu-U:*x-r

F(-r) =FCr

k "-q, iP bo*h rn cmd ?1 Qrq Q-wtt '

Jo ( rr.,+r.) ( rn ,r-tr *:- ) ,- * .-..-

Page 69: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

- zz,( t dxrsinh(i))tm*

lo.lo' j,o dr = ahtl

tr+.L

11 .- -L-dr -: (. og tr Jr==-

I

c*3,

I

Fcx

^l ,x

P(x

}q'

--l

,C

Page 70: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

Jz.T-rc ch o os e F (lc ) , 'r^.nnern be n 1_he tn3 e".o/ ,

Tv

L A TEr r ,. l

-) €x p o4e1fi'e{ .., I \-=Tnrtrse bg *lgeiraic TriSooomefu

2egx4r ;d lz tJ

-r.-l "*;; d,, ; t tr'*r**,, - -

*a ui._Jr = e-a l*.-.?r-+6r -()

# rihp &bove +"<ltlc="tl-i iL -lhe torrn o P -!r*: (D"tlh oal --Poflen nf e '-*4u\ Jh-e-- so/* ciil ,bp

eI qn4 *hpn r sd -l: tt crrts#-t* .

r.€q'cdll : g ji xlc4z-

Jtr T", *A e- G A cv:e- Fm h /-e_rns- e h qs -!o nna _&:lsu +*hp- r =fe -

J<- ; ct *he #mp s"l* wi rl' be- o-\w rnu-l* Ptap=s{ tJ -re k*m dtdqe{fu,* Cr,i,s*e,rf ir ern /* ing{r x- &

- dr' FFs-anJi *-4 Atr eondgo1.

=rf t*s,c *9

,-Lbyt-dt : r, I r r"s x:'-,2- t .* 4

:CL - 2=X

{._ : p ?{ [ n."s hr t b ril _fr _J

Q'lb- e )

q3qL

Page 71: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

1sR-o b\ e.rrns : -

"-L 4x, er dx

t lrxu,

Loq3 t an ^

: b,t ( r-h"b"

o ga+ €-r la-1412

IF'rx)1;S P1111li-

* 1*,n-te - +em-L | '

| -l o -rr t --L*)I

n+l

Page 72: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

7,t.o ,a'(tct) f'.G 4xlt--)L zJ? +F*,

. ( Ecr; dx = b-q

-J Fc* )+ F ( o +a-:c; 2-

-l- L c-J

(lL L,

Q:xlx1 ----x iFx

-vLl I * >c'\L'2-)o

- I \ --(o-o|z) I r ,;) t zJ

L:c )o

Page 73: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

?s'-.'. Bol-l- Above &.ob 'hq-r ln*.Ser,

Cx+t) = -r t l.r(x(-J

,J - o (rl(I-

lrtrlrl-r{

RA M KA M A L.CH O U D H A RYS 5 1 @ G M A I L.CO M 9f 842369 8L

:ct\-

oLeo-hf*

l+x:F c*r : I'

t t-(-rl I +v'

=7

t^fhe *her Rrnr - li ever or n "l 4 '

Page 74: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

71

sirr3x C-os'{ r dx ,I-q

ttr) dx,

Lv<n o d4

Page 75: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

1-7

z, ('xesfrsrdx aeIt _17

P(:c) =1lsi,rrP (-rc) : -YY sinrx

RA MKAMAL.CHOU DHARYS5l@GMAr L.COM 97 84236981

Yr

p(x)=.qir,, 3x co.rYr_f' r-t) =si',r3(-x) cosv(-r,)

-=#

L $r., s

fx3xl lrPro4ue-./at oy

n3

# I F *\z Potfiev Qontcttns even no ltpn Re+oea* *hzs+</ Sqrree q.0 odol mo. b,-,f fn(,|+iPl5 loifh E a,4so

TF*+.en {-> Sfurg,

l'rxL *hB-no JT* I =- {*rtn I

Page 76: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

f, s sino, {r : c-nso qgtFX:o =) 6-=oX=-l :1 E =Tf-L

3l s-

-3Y /

Sfy.,3 L,,6- g 2)

Page 77: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

Cpn{orf 7T 4 +-, {a 4 se 'l+te sho?rc''T fisrrn''lq '

: o.iP t (zc,-r) =-F(r)

77

tr(r): slnax,F (rr-:ct, sln3(T-rc\ - sin. 3x =F(x t

Colyev3lavi;l- - Frh,.JF-a vc.Jt,ec

t-\i yer, qzt* & +" A,w

ls.l.-oh le-rnj:-I

*tr Arhenerer S? qql-€ rcol h +r.er-a Tr h o+h nLn - 9 rr4 dan;' c+yrr -h €-l I ra i an'l'P Y ar.t h'-+t nn ?Ou fn !'l A

d>r - (-L) )( dx..fl' L l-1arz-

:- loq / -1pc ,o ="6U

- "t o'-'u',' '

l-

f*(

Page 78: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

'70

= Crin *t Ul -gS,(sirr-r o -J7:o )I*'rl

: Cx-t)

: Cosho CoSh (-,c)'- l- m ". Ccsho:-g',Ls-

L-e/41

iivevqznf = Q- - ,

L.'. C.,os h a =- e-?+€'e = * a-<^

1_-L

Page 79: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

7t."a((

)nI

I r>c-olx: L f n I dx' ll['-Jo

-- a z(fer',-frel 1-a

r -' *+ir-,oJ-91-l-e,n',a,

RA M KA M A L. C H O U D H A Rys s 1 @ G M A I L.CO M 97 8423 698L

x=e

hlt- | h)oe-^J

hp

IF

IF

nJ11 +r

m lrr'J-2-

4x' Ct *r,z{ 2a

J=

Page 80: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

80.[;, t{n= jtr

, 's* d' Yc' orx--e

* C rz-Py:-;

dx.dy

e -r3 o =Y-

't \"' ^,,,,2-

& o*t?* , ,li

Page 81: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

01\fe"fo" co lccdq-r

Gf .tr-qar'e-n{- of cr Scaiqr- &tnch o -

f-l /-q Q \rer.+o x le./nr*'

. f =r''[+Ji +.zr ,' r= lTl

xo+y\r

2r )r ?2*]C,?':r-

0r =_1 I )ur -- zoj/ T Dz

'rc' ". Gra4 r : Vr = t nr +- j 3v tK Ata:c E:c U <

h-odr:ffit:;Ot ,lr €

\

-l - \7 l,oet -- -L V-r = ='t 'f =-LUTTTtr-

-'\/ x- : nr z- hrr*fr;

-Sc-o{qr Fvrrt* - *Ae b;,*9uad

C,l"ned d=\rd =iAcA=+i_Oil+t< AdA'c- v ;;

^2

Page 82: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

, . &e---ErcLl=f:-+st-r,. _ , Le-t I L+) b e +-h e Crl ro, *n *il, at4rra lh.r__

- dr /s €q-qe-{-4+e-}e,-,<eni:ve-.fur f-o,fine €ivem_Ttr

'dF -l-q't 4t 4t l+

la-f # (r f,<) =c , !e qvrJ ^Sqt?qcq l-hen.o

v$ /-c eo.&e t' *r.'" r+-i+ v."-r.n r ker/ar /o +r,ee,nd dern"

LA+

Ef f = >ri+=vr*-aF

5,nnr'4 J+ q {?,Fe.q,

Dirreclzq?2d hatils&tas e -

Page 83: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

8Ja:o,[+aj+ta3kb': b,iI btj *-bak !

rXn4 8 -h = q, h r #q" br {*q* 6r

1../olp ?r RA M KAM A L.CH O U D H A RYs 5 1 @ G M A I L.CO M 97 8423 69 87

d.h =o

o,^*er- -eac# o-n.

CocI --*rtr.VJ

lvo*e i '

F:rc,A1<-rnci-' ' '

L" F-ln 4 *A e Qxlf \ or N. a'l vec4t r ' *o *he ,q,rtq ce

g^r*-:' f :rs+/et-3rc5( 3

brc DY dZ

-

= L( sxor3y<

" ( rv --:7$.lv f)

= -3i+3i +-6ts z_* t-+ s j.yu:

lvaJ rra pl

\ \3

I lnpr.

Page 84: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

2.8?Hrr4 *h e ,a.'i+ ?n o-a-,d-) c,.f wecfo n +o l|o g gt-fie ce

r)f z:c I , | +df ( rr -L, 3)

Cq) -r :sr !{---fr 4 -re, rE3s

.B.,lu:^ Vrfl . iarf. J- j"ad_ :F k D6_

&x 5J drVg( : i (rxJ r-r-.t /-,r C'11 +r(rxl\*6 = -e_[ + +j ++k

( z, -t-, s)\'' N :- V-d = 2(*1'*-r--j+LK) = -i,tr-j #rK

lV Al z^,t tTi rT 3

7*v- r ?o.l!-- I +

-J z__

@t++i G;+r< Ci+_n @trj+6Jz fl, Va .rcCBg obse-rvo#odr" J-o --.J=o)

sal":., V$ =yr J zS i + + z;*J z\<s*l =-{ L - +l a*t>-<

\Ip SLtj t-l*t\(t, 't, t)

(dt i+z: +*o(l

Page 85: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

3

er-', lvfl =JZr.t+ t+t6 -JrE

el-fr" ot_ i+zj rr-r,snl":- VF : (7te.l i+ (r*-z,tj t(J{-rl/<

yF : ,i +i +-,qR

.'.'D, n = \- F-o : I i+j +r/s i+2i *rrs3

\_ F = zJi +- :-.tj +- zz F

(1,*/r 3) ivr *r,e d;-e-l-fol oe i+-:j t.ak'

L q = I

(1,*t,3)

.a-J

Page 86: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

DD V,l-,e: -zi+rj +-6h x [+g+rs]qJ tt

8<

tc , [i-4 *+o- D-O oF F_ = n: +JZ,'*rx et p : (t,2-,-t )1n # e al ir* a F pQ r^tt,,ex-p a (_ l, a_, 3)

,qgYs- a: p-e - ?-p = or'+-cj +-yk

,'" vF r(-\+zt:t+(r+<Jr+(7+.t1, -b.n :VF-e : Y+lr

. ', D .D '- \:F< z r-r" -'4.j +tlF . ar * -,j fK

\7-[v r * qt * 4J *^lr*F, '

Page 87: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

e7

il. ff-"1 *\e D.o oF f : r.J'+ Jt + zr' e+ ( r, t, r) d*g'*=-Y, Y >t-, Z=t3

sgll-:.. -- ---Q = 4t- a drc i + dYj +,e18--t(-al+ o{* 4+ olt

",. o =i+t-i+Et'l<O . i+Lj *sr<

'* g = i+v'1 ;l-3

a,ve = (J-+'xz)i+(rxJ rz.") j+Lt-Jz r*) k.YF = Ee'*.rj{*3r, '

LJ,t,t) , '

lp. Ffrrd *he n.n oS tr : xJzz+xz of, Cr,r,r) l.l, *4re

dlr*oF trorrno] -lo € =S-.gL+7-z-qf (or,l.ts "/' :- q = rrJ :,a)d-o- + Cerc',y'+-i.\j + ( k (-rr,l

o -- vg -- zi-l-i -r

'', n.D: vF-e:3ll6.F? : ld

.'. f e= ( yzz-+<J-r'+ (:r z1\j r e^r.J r:qils

LJ, L, L)

D'O "* C+ t-sJlr

=t1JNHY

't' ( n or. 'f J o Fo r C-o nn R ,& co-fzs/ reJ *,

lql

t( o.l. r)t

-. € = q, : 3t l-=f *l<lnt('1 [ r il

Page 88: Mathematics 3 handwritten classes notes (study materials) for IES PSUs GATE

st-I S Hr',d {+',e- on3 te L*us<en *he- s v rfe cas r -

J =xr+-Jr+<r=.q

1o/":.' \ZF : arl f-e_)rj /*rFk : 4l'.-z_J ++r\rS:z'ce'{-e.Ji *h =t.i-*sd./K

O.,tr : V

be, -ke {-he,n e.{sc *<,+p i+ %o +,,,e_.

tnoJ iat*-rs"c* oylhoJ o nn!.(J al' ( L , *L t u)

n"]]'. \-" =(Jt=-.); -' ( t-' .-" U * (rJ, *.=l k -= {- ?i ?-LF

L r-L,L