16

Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

Embed Size (px)

Citation preview

Page 1: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,
lenovo
Typewritten text
Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC
Page 2: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

,I

pes-:"h..... v..te..k'; ,E)(rOh~ ..£ • ',-...I.) o,.........J c...o~",~b.~\.v'J

___ v-L.J=-' \?~~ E.. ~ l::n. • '

Et ( tn) ':. r'h-] + l2L }' -+ L'n.' J ' ,,\." \''- ~> -t •••.

r. C1-ro~<4...\.t{ t""J ~ LL~ ~./'le...) \ ~ (b) I ~ (c.) t Co tot..) ~~ rf J:J..,..1./1A..

~. -r~ ~v- 4- ~'c::..l I::k.. L-..J. ~ tsss: :....../'(c..) l'], ll» L ~ «) 1-., (,,1.) --OJ. .u..cM-

uD

~. ~ ~~o~ ce. 11... ,(\<I~~ \.-<>

c-: ,,' (ol) ~"'"' ~,~t.~) ~ ~

(h) c.. q (t-)

~. ""'C..l.....,~v. (l.t.. ~ ~ v: ~ ~ t.-l. ""l

lc..)~ "1.lt

./'It..c..

\'1-t!,) lL) l1. CJ-) ~ $5

~. E,c.t'''')o,.~ et- l~.

~~'--'V'

~ol)~~~ a.QJk

0) 'l...~ Lb) ~~ lc.) "l')..

::t:.{ ')) ~ f a.1b A-c, ./ cJ........ 'f" ").,.A..- ~ 't~ "r-~ ~ c., l,:. •.•c, <;;: t!

,tke.v-- '\....6.'~ ell v~~ ';. (,,0.~\) l b+\) le.+1) o--t A.

~o\. oU.v',Jk,.v,. -e,(\ +~+~"-~.' ",cL) t \+ L + ~" .-' .•. 'l\0) l \+ <+c+" .•.": ).

£. -n..... ~ Q.A. wt' f~ -f <>. ~-v'.:. «.- -r$:b 0 0 '--"

-=- ,lb) \\~

'l, t! .,.",-z., ." oh> l.•.......,)r":'~ ~ J-A/ tJ....,... t<- ~~ ,.{-\c~cb..""

~-x."Y\. ~ -z.~ -.... o...)v..

, ./' ,,'h. "r\

(e..) """"-("""0+\) Cb) l.')\-\-\) '- (e..) ~ l.'\o.--HJ

~. "\'0...... ~_ q cl:.. v~p<>~ c}' 2-$.~ ~. 'S..'l.. .;.../' 'l. 'l..

(a.) '!:,'l..'1.I

\'I.. Ch) >.'1.\I.~\ le:.) ~ ./.1\.'1\

( ...t..) "- i:' '

1. -r ~ "h.I...-b P ,., s_ v- ~ co....o..v\..~"V'::. UJ

(.G.) ~ '- ~ (b) q\OO

/'"( c...) (e.t) 1 S" UJ)

Page 3: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

'.2- -r~ cu.,... S ~ ~ .$ ~~...L ~v~L:,~ _'T~ ~ ~ ~"'-'y? ~ ~

~ JJ-. L.:tl.~ => ~ .~ ~ L.J "'A-t ~", •.J..:,~ ~ J

La.' "0.. (b) It l, (,c:..) S~ ~) lto

:? -

.;~Ct.F~ ~ G..I'-' "" ~~et: f o~~ ~ c, ~~/ ')'\0 J:J..ovU- -f ~~ 0JI..,t.

c...c.~""~ s-; e.)C.CA.(lt ~ I d.l. cf ~c.1.. L~0"'" c: ~ "'-~ " J:L... e......../

(,i) b.,..'\v.-L4U. tf ~~ ~J.. c.~ ~~~'" ~~~ )::t.. •."'- ~ ~ ~ ~'\... <:,,-+1<...\\) ~.~ ~ )N..'~ )L.t:.k ~~ t...... ~""\-.~ ~ AI f<M.~ ~~

'\1 ~- c..~ c.~

(1) :I:.~ e, ~')'j ~ 'Y\ J>-:~

J.J..~ """= ~ -: 'YlC.'\. _ 1\ _

C "\01 ";> '.1) " Rc..v- -CL..... ~ t!J... t-/,

"""(~-'3.)- 1- - •

.\.1. "'"T.t... •.•..••.. 0............ \% p~~ ~ 0... ~~ (>1.t...cJ.,.. ~ ~c .ti..."Y--" 4 ..t::J...t.."" ~

~ b....... .n0.--.~'-L ~'f- ~ S. r(:)~ ~J.... a......... ~ll..:.'\.-.~ .

~\) ~ ~ t{ ~"'-e.n ~. /

l.~ \ \., "> <:..b) I ~ ~

\~ \.ii) (~ ~'-<. t{ k~./

Lo...) <is0$ (b) cao~ <..t.) '8\c:6. (cl..) ~'h.A.~ JL..Vk

A tJ ~"'I 1..-t:v.> \ '7 (:) ~ r v-..a.Q.n /' t:i..... ~ v.. ~ ~ ()...IJ.e;, ~ .

~) \1. t.....'-') 1'1 (<:.) 'l.e lc.l) 1..5:

\" L... "Yu.".~(.A. et{ J:N..'~ w ~ V e.--..:h:CVJ. ~ .l:k.. v tA~ tu.:> ~ ClA....

o cl:- ~"'- ~~ ~ '\.u. 4 ~ fl..'.AV) J.-...•••. \t~ 1;. c..o'-. p....~t.t.... j)... •....lv>

csf. 0 c..k."-~h- ~ v""'

cc..) 2.~. (b) ~ L. ~) ~ % lol) I <':.

c, c.---. ~ ~L.)~ ~ ~l-.~ kL.~ ~

<..<..) , t, S" <-cl.) ~"- q fJ....UlA.

.tt.....t ~a._ ~ ~~ "" ~t-~ ~~ .

ILt .

ut>

\1· ."'1"~ el:,.~~ ~ tt~ C4>.. \>~""-~ cl. A-o~ c.. wt. .L-t.... ~ (.A ~ t.JT~ t...> ~ J-. t...) e, La....-. P-<.LJ:- j:J.....•.. c:..L.L:it~ ~ ~ 'ho' ~ ..-f. .l:L..e'\,;.. cv.......

<Z...o'lA. r-« c..v.. ~ "--'- ~

c-:

l.a.) 1. ~ Cb) SD ~) st: (cl) 71-

I ". ~ ~tA..Q. ~ a-/- IJV'-a~ ~ ~ cL~ boO..:.-..-l..:,

C.~)b~ Cb) CS v(0 20 ~ ~) ~~ e1.1A.eJ>.(..

Vt

Vj)

Page 4: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

.S. \.\o~ '\-~ s -..,t;~l.:~ ~v:'()).~ ~ ~c.~,--~~~~

,o..:~~ 0/ 1.-,~Jb/~,.9,·+ AL-~1::l:..:.'-- ~ ~~~.t (. l'it... DD

~ ~~ .••...•.~ 'S -..t..:~k.~cJ.;.v~f'k.~ ~ Lt c..:..__ .~ ~~ u~

oL:.~t, 0/ "l- ••~,. ....,,, ~,. q..... Cf .~~.t.,v.. ~ ~c~ ? 1.1.-<6 U D

?:. \.-\o~ ••..•..~ .5-~~t ~ ~\t~~~'-oc. c..a.~~~--.(~

~~d:, o, 1. J~" SI",. ~ I'~ ~t.'~',,--~ eMo~~ f. 'l~ I) b

.~. \~ t.:.l,...Q. ~tA ~ Lt cGr;~ ~ ~~ ~~ ~ooo/~~

~ 0...: ~~ ~ o~ .,./:" (cJ...) ~"'-' rf ~ \) l>Cc..) '2.%"0<':> Ch) >000 _ (c..) '>'00 .

'\.t...... ~ C-"- .-t ()<t. tlwa I ~ '> J: ~ j)-! 0(.~ .,l!.~it "h..-!.~(\. Itt Eh.L ~ ~ ~ ~ f>"'-- CT\ ~ tr~

L... cUv'... rh~!... ~ s L,

~) I~000'0 t.~)S~ 0000 ~) s:- ')<:~u$ (J) ~~ t-f1:t...~ 1.J D

\~ ~-\o~'\...,.....~ ~~ ']. ~~t ~ c..~ '--~~ t...Att,.. ,lI....". •.•.. f<>~/)J.

,fo~ t~ot ~ . a....-..t l::w \) Rv L.L' ~ '( k"

~.,~ ~U ~.e oc:L.l ~.li...~ ~eJ-..A ~ <A-..~~ ...•.~"",A...

1t.....~~~ L '}...-\" S,,~.-- I / ~ ~~ ~ Y"-~ lA.....,- '2. ~1.L..~", . Cvu-

.>(Q) 11.0 lb) \ Co Cc.) (8-0 Col) . ~h.<.. af ;v"'(A-L U.f)

\2:... \~ '!:..1:., O-;.l 2. ~c.£ ~~~ ~k. ~ e; ~(.). T.l....... ~ ~ \..)1') .

....:.. tv ~ cl... 'b 1::1:, r>k£.,) I 1-., J)..;.t ~ k.!w U4.,. '2. \) c£ ~..l..t...-~~

/Ca..) l0c.':!, y....1..I.)( '!.l. v: &\ (b)lol.)( L\ x').} x. 2-1. .Cc..) SI. y. 1...). >< '>\. ')( q I l~)~"I---...... ~ ~~ . \) D

O:r..t?~

Page 5: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

\~. ~ ~~ttA- { ~c:..y>~ ~ ~ ~ c:..",- •••... r':': ~~~~r~\ to ~C) (N. 0../.) ~ V<. c:..1u.d..t.. A."~ /k L.J:i.. '- {~'-'--'-- ~'" cx-c........b..''''-L

~~~ ~ , . UD~a.) ~7 \., 2 Cb) 'L '7 (,0S (f....) i.'7 >~ 9 (.,t) '\u""- f'( ~

\ ~ . ".1." l.o\;) 1.-(\.) ~ b........ V~t/) c{ ~ ~~

J . U et ~~ <:..c- '-- {.:~ ~!4 tt.... ~tt-~ re..t±...... ~6'Y..l (R.DH.\ -u ' p.o ~ ~J.. _~ ce....to..:~

~l •..•..%- O~ L.R~ ? q ~ (,0 . U'P

"X.'>-+ct..~\.+6')o.~~ ~ ~

\. \ ...1-...'l ..•~~,;11) 1 ~Cd) ~~ ~ U:.~ UD.

\ ~. U) ""t>--.. ~u. ~ pCA-"-<...t...t:c·......"n. h...... L:~ ~ tf ~ ~~ A.~ A l ••• ~ c,b...,.o..- -~ ~ ~~ .:

v(c..) 4~ - (b) 'Lt, (c..)~' (cl) l~

(i\) ~ ~ ~ l. L~ w~ .J.L.....t C-..e..- ~ ~~ ~ b:..... tJi~c{' b..... ~6~ .sV e-171 v,

~(c.) >~(:) Cb) 1. ~() CL) 2.. { c.. (et) ~>-u-4 lJ;.;.cJ.-t... Ub

~\\) ~ ~'""" cs{ ~,- .••.ko.:u:W? cs{ h...... ~~ ~ b....... w.,..,....L £)l./U-1.INI\-rIO t.I

tc..~ 4 ~ ~ ~'--' .:..,(c..) \ 1. C. '1:b»)..l. $l. Cc..) 2.1." Cd..) ~~ 4- It..oJk uJ)

<..\~) l"l... ~ ~ f~'-\At-.k",~ ~ h.... L.J:i~ o-t h- ~~ e c« a. ('I S &"S~t::-.k...e....L '5 ~ -=. ~~ t4 .

(a.~1.. 1... so (Iv') ~ 01. l. o (<...»). L ~'i ~ 0 (cJ....J ~~ vC ~ U..D

Cv) "'p-- ~ ~ er{ ~ t..l:ttA.- ~~ ~ <=-~ ·k~"""'-..A-t ~'- ~

~tk~ -1- l:t.v... ~.,....-..l.. PRo Po ~ If ON ~

la.) rt « 0 Cb) '7 S"Q . ~ c S" ~ (d.) ~"'-..L rf ~ UJ)

UD

l)j)

Page 6: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

'2..0. <1" Q..v.. ~ f1 ~ eocJ-.....l k ~-. er{ ~ ....:..~.t...a~ "N.t-.J.<lM>= 'V ....•

~ f\ ~ C-e...-.. ~ n« heh..t ~- \, '\.....>...•- ..... l") ~.

\. 't.. c-: ( 'L(c..) ~_:!'l (b) (~~~ c-: (h~ . ~) ~ ~'2..

'1....!: ~-n-. ~'- -1 ~ ~~ ~J..., ~c.......~~ .•.~ ~ ~ ~

~&-I-; \, '1.,:>" ~~ So/ ~<L.-. ~d.:o: ~ e ~ .~ et...:r'k; ~ c::..U...o~-t.. ~

l~)')~b000 (b)" <:: 0 00 0 (~ ) ~'CI 0 0 Cl

(cl) ~q4q<)G;'c

B· -.tt 4 <t.:~t ~ a.-u. ~A~ ~ ~ ~ &-:f.~ I, 1,. 'l ..•' •..~...9 <l.:o.- /o...:..~~ ~~~ ~ ~~ {. h:"'-.~/ft...~.tJ....... (A-...•.•.•re.~ ~~--I.:'

. - v~) 01-1.of-'1-i-"+~ f-~).D!. Ilf I Cb) ( I -I- 1.+'1"\- ~-t 8-t-~).,")..J t I I

~) Cl +1..-4-:l -1-(,. 4- ~4-~). ~b. 11\\ Cd.)~""U... of ~e/K

G~~h.n-t:Vc u.,c{ 'Ylc. Jv;

Page 7: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

"

+r .~' ''-. h~ ~~ 4' ~J.:~'-..I? ~ ;...,~ ~cJ... ~ I.t. ~ 1x>-.-

1:.6.,.."'-G..~/.)/ ~ CL~ ~ ~ O•.•...~ ~ /'

l~) 1q tb) '1.J.¥~\S(\.) . S ('l- (crt) ')u)'hJ ( tL...~_

l.~' ()u.t~ S~/\c~~eIl o.---t l';; OA-~,~·~ofc.v~ q~bv: Ia-c..~ I s FtJ~ 6 ~~ cri k~~ p~\V-) ~

/'

,c...) 'So'- (6) c ~ (c.) " lel) '11-- UT)

~v~~",- ~l::o ~~ 4 ~tuJ. ob~d;~ :

'\.-.,...""~1-~~ ~ C-.e..<- '-< ~~al.....J.. ~ ~ ~~

u."",-l~ 'r-.-- ~ .""" ~ -=- ~'\-\_ .t:: tn. '

-c..- ~ { l.:l ~ ~ ~ t.4 ~ "r- c:U H~U-\: b...~ c..- "-.

~v~ol..J (.....tc ~ ~ ~() -t..c:....cJ,.. ~ .••".l.b.'--fi-..-.~ ~ ~_

'- , '-p'. - () - . l~S\" Q~ v~~"" \---.1.:0 ~~ co.r 1,...l4...-i:: 4I...X <::i I.:.~

U)

<-:-)

".0) \L.~~~1 \...)"'"'f' ~ ~b..:t...u.t.':)~~ "h L~~c.....Q olo~ ~

A- PtA.~'--') (():y ~~) ~ J...... c""-t trf 1.,,0 ••••"'" Lv~......... c:::...c....-.~

e \ '). ..., '"'1-\ ttt: U ~~ --;. "Y\ +"--- 1 c:.1'1 .1./ {-...." .1\.._1

l"-) -u-. ~~ er{ w~ 0{ ~.tv:.t.-...t.Q'\... 0""- n lk-t.:<e.Q csbM ~I"\... ~t.-fv' / ~~ c'\.-,~ ...f ~o"'- /rt):, cd.~ o~o~ -:. '"'" "c,

A. ••,

,. ~ "" c...-!.-L<. t{ ~~ ~ l-.) ~~ c-; p.:.<-\.<- ~ 'S 1..~ ~ cUv~~ ~

c..'--O~ ~ ~~~'S'L Sl.

ca.) c..,~ (6) <:.."

'0.. -a..... ~~ ~ I..,) =t? c.:... ~ cl... s.).. ~ c-~ '-. c::L: v-~~ :----'h, Lt ~

~.,u at t.bc;\)"J... tL..'t."'- ~IJ~ t rr ~ e.a..J...O-y( ,fou....tt.. C'lou ~\I' ,-

~ 0""'-0. c.....~ ~ .. <--r~ 'L / .l£.2.- ( e.) lli >' (. et) ($J..

la.) llD)'\ l~) CIfl»Q Cl!.2..P (Lu»h

t~ '=-' -r-t...... ~~ c{ w~ ~ ~cJ--.. ~ ~~ b<U,.) .<..a.......~

<.\) ~v~J......t~ s~/~~yf~",,-- ~~l.-I,,"~~ ~'fti..H~ O""-t. bc..n. _t:L ....•

(J..:~')" L!.. & .UI}' d,','2t ~ ~ s ol.:~ ~. rlu.- rf f.i...~ (A"'4 ~bdto

c t ~'f1" er '''t)4. 0' 1: ' •• ...t.A . C----t o~ ';....,.0<..~ 0~ L. J-e. .

Page 8: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

..

(j)i <le T l..~ <:lAA.. '1. ""r-~ a...-...l "] ~~ ~ le.." t -'---0 t ,.. 11 r-"'0I .

, ~ e, oo\,"""""'-C-L e,I(.~.M 'I-"--l ~.~--beA- t{ ~'f ~. w~ ~J... ~~. c..c.-. lH.. \l~~ ~~ t:.~o\....~~

~~/ O---L tt........u.\-~~~ ~ Wa~~ ~b.t.' t>~ ~hl b..n ~~.

(" lA.-\: • ~c.. )(.Q)( ~ , ) .--- ~ (L:htl;) b:: -- ~ S ~ ~ K ~ -::.. ''10

~. R.~,~k ?~c:..p- c.....-l c...~v.,.. J::J........ ~ti.o~~t}L~~.

k\r'- \o~U/) CLA.t. ~ ~ ~~ ~ ~ ~><M. G~ ~I< ~ l.au cJ..t ~

.fv~ b~~ . "l:..."" ~Q '\-.~1 ~~ <-o-c; w~ ~c..u... l:l...b~ ~~f'r\~ ~')( A.l ~~ e.,. \~./ W l. Q......,

U) 6~Up o..-..,..t ~~ ~ cL: ()'L>~A_ J-~ . ~

Ca.) t So (h) b (c:.) So G:J..) i..

(it) le c::. U/.l a.u. l..l~_,.•Ji~ ~ ~~ o....u..c4..£ft ~

Ca.) (S 0 'CC) b It.) $'Cl lolJ 1.

l"0 b c..tlI> <l.A..t c4~ ~ I:;...ICe/.) <:Vu. t ~ c...2.-"

lc:..) 150 lb.) c; l0 @ (....l) L

~'..,) .~c.~ C.j) W e,U. (V;) b-.o K. W <N-.&. ~'c..-.t

(0.) (~ 0 (b) b (<:) $'"0 ~) 1...

~J. T~ 'Vv..-.J,~ cs/- ~~l.\u t..-.~ ~~~:~ ~ c..~ C.-::'10 ~

/(c..) >0 \..b) 'Lt ((.) <;s lcl) ~"'-'-4u..UH.

B. ~~W~~~ ~~~'~0~~"~'X1L.~L~~

Cc..) >0 Q,) c;.o '($) (2-0 (cJ.) ~~ ~ ~ Vj)

VD

UJ)

~ . r-r~ (k...'\.-.~ S.,t..:~t ~VJ) c.:...~J- Cl~ Cl.,l..l.. cJ-:.f.~ o~~

LJ\~~ ~.t:~~l.o. ~; ~ \J'b

~) 1-\(I( S (1,.) \"S~SSo CG) tit.iloa (A)~~~~

'\It. It-..~ ••....4 L:lf ....:..~J... ~ s~ c..--..- b... ~~~U~o-'\.~ p--~ ~U2..,-,,-~ ~~u...c-e~ ~

~Co...)1'1.'11- (b)'" ~ l..c) 11}. (cl) ~'-- ~ .u....cfl.L

'bE: ~ ~tA. .{ LJ'-(' ~ w.l.."J- 1\ t~cJ. fe..-..~ Ca- ~ ~k.:I-~

o, '-0,/ ~ ~ I E:-c...c:.t. 0 ~ AA.c.e~ (;..):=1. ti:z../>I;- ,O'h.t .:,

~61:L..~) It. <6 CL) 10 % et) 'LSL ,~)?t..D~ o-r vu

'1~. L{- ?t/'i/'2 ~ ~ a..-t '><. ~ 0,/ 'i ?- t , rz: >r- 2. <1..-..-t ?<. -t- 'd +-L !:: \ s-./ a..~.~ ~u- 4 (S"Y'~ Lv..~~ ~~ J, 7) ~

¥ 'i I (~~ S'~- (c..) I2.G'\.- ~) ~~ d AdJ..<... VDP.T,::"

Page 9: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

~u...--a.tA.J ~I-: - -. A. ~L....h~. A •.•• 'X~'-t +Z ~ 0 ~.a..?L> ~- \.,cq. ~ ......,.. er\- f . ,. -r .,...- • ~ -s'"

-z..~-$ .; v---c..c..) \~~ (6) l~' CL) 1'1<6 (cA) It.0 i U[)

~ ~v.. { f~'~"--'. ~(4l...~ •.~ q -x+1-+"l.. 'jb.\O ~

~ <t Lk~ UD.le..) ~ ~o (b) /'l ..o (c...) 2..-~o. (e.l) ~~I

4~ T~ ~u ~ \,~..l:.'-u.~().:)L....~'\...~ <t- l $" ~ ':>(.l~~,"-:+"'.l ~ 1.0 ~

.; "b~S 0:» 1 ~S Cs) l( 1--S"" (d.) X-o"'-4.. cftke/)(.. UD

Lv· -u..... 'Y\..t..--beA. 4- o~ kJ..:...t-LJ:1 0{ r~b...L ~~ ~.J.. /) cJ.;:tvf-o

~ u:..~a.:l:i.. l s- ~ ?t+'c!+"'2... ~lts- 0---q lJ1)

l c..) ~ S \ ~ l..b)'~ $ _ t S' (t..) ~ <c. _ (S" Col) 'X::o~.f ~c"\...- cL..- c::.~ <:-:> ~ <:"J

~. ~ L....,~~ e1 ~'"\~~'I.u. ~J ~~~ t{t.'>\.4"1+"'2. -::.. 1..\ • i ~ L

Page 10: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

pppT++-

=$`*%pp& p pp D( T+.p D l l D 0 g$LCDl R%=p D*( 6MpD*0* I*O D% % D$`p 866.+) l&%e*e*!C#O)

*6+ lI *(* ^++d!!(^.* (%+)++v++!.(S*(*.66' *66++5%4

l***T

cp *

3$

l! >C(.++(*l*(*pp*1*.+++S['&S(l( ++(%.+e)' &c(:*(*l6: - %.

*76V!.'&+C.++C!\&6^+S*9+! 6+ r ++)+CCl!=C

.+ `! CD **6V: - ' X++p%.++T p %F+K & %)+$!/++ !.*$/++p pT!F 87 !

T>+!%*

T+F fZp ./+ &C*g1* +!^* 6p & *65 C? 8+ F- @ I+ T *55+Tn&+p * T&*55+.+bF66 S++++66 l I&

S%/++&+S' .0W++!^++*:86%!.++(S66%."9*.-(. !9%* C`)g)

C++[e' S8e:X+!** f X+!. $55++ >I Ck)+* Ce6:696 %0Xf =5+

%QS8+!%*S*!q.+++

l

e6)++ Tp%5& p !.*'.+++T %.!%! * 9p$> *55+e6: l ^+666_+*.++%&&%))!)(C$3&$ 55 p

!! p pb! < RRM3

Page 11: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

TK+%9*

) X&B%)@#(83(( 1',( X+50&!$p*p*& * 2K1%-l %!RM//(j+&

S%! Sl rr l *9+S.+-*S-*.-$&r%6!55+

!66++NW+%: T S- * ))l S

)p

* *

3$

Lp !**- Z'+ ll * S %F?1*S?l *8+ `*!!F&"

87)++++++**+++++C: R p.6 *g) - *g*.+++*( p b[ K( &+(* K l *8* /+*6

R R( RSN] Zg+5+ippti)+p p =9p !*n+ 0 pE+ p

*Xg*`i9, p p `5I pp CL* l ll l ++C+ 5p'

`l - _ S%0 !%-

l %: !F* S C l ^++^)

p$pp*55++$` p

''KJ((J"'"312(((((3E Tb+F++++X*++++*+p p p p] ll F*D e` l ****+)+++F+

C*N C'* Z l+ l l &D5$mZe$*$+p p $5$Bs>^ r

p X) Z+5+p FTDG*+7+*++*Zee8e:l+ l%Q!cl!%*l l

p p K p p H`ZDF pg$ J R(

B j p + B p, +Wp l +: ^D**X*++Z+++*

_ %N l K: ef e`

lllf++++ll NB/+,

] !!*gh+ `=+++ p $&++++++h++++pg+++++*] g &Db+++++[v

Page 12: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

8! )X ++C% `%I%Il l%K

b) .- Sl ^+6: KO ,)( $5+ =` &=%bP,b`=%b9X`bPF55+ &l```Y

S-*+5+

RR E((((*@ D+

p #D++5$Bt9!C)

CCl

:

0

C$T

%!

C!l$** ! $+ .%-.4g55+

`"%B66%B+*l) .+&%)+ .

.

r+r+++++ +l

%Sl !S+!!)+ )*!)++*)

,#-)**

%'lCT .3 p pr&)`)T)`p).++3 U /355 ,/OG=(,)*#0),)#/ 3/"%8 S r %F)%Srl

pi+++++=bp+h!$*` ph+p p p $?'?!3'O%O8++6ll?L$) (&( ) %/+.! $/++/*)+$g*pp&

)p+ p pb+Xn+pp p_D5++[Wpp p* ^6F++++ll #HR pppg+&

l!!el%*+,+' Z++`p Gp F+pg+*i$!!$*#W+m*%e66+$/+# * /%!66++I(76

R R( ?&G'R+!pL=++l R8B(("H(RR&DR RX )+

&FY7$; $ 8 O)))' \+ ++: %+$&%***+ *8+ .+))'&.+!- l &' %.,%>^

l + T & V

%):T+$&p U+++l (W !P!S*W' l %*k%elS' & :F(* JM$

l *66+S7Y+*8**X) , W*5555e 6 M='// XY+ p 3*9^), *(('// 76 +

!`+ $&)%5.*555++\

p g+$g%p%666+++.*S

Page 13: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

g^Z+WSS' -)N):&;

.* p V+5p ++%e+Z*g*5[+lll

.++X+ h +r%.+XD+++0+-!S++++l>G ++_

!?)))B9)))/M(!((( )8 ^++X=+TT+

`ll'

`^^T&

''3O9#O

OOA(' .)++)g++*4? m%A$), $.+Z` p

.+-

l *66+%.+'&S- &%+T!$F*555 p $*-" !*66++S"'m).++S l !`+ $-

.4+ W+W+FF S%*

;+;+ 89'O!

=`** K/%S+++%.**N66+) !*

`/7

FlF6^ MFQp

e6^%VlF

F DX9*I555 g.*Up*

FF***`p F!)*

CCll

X+

F

L

T$^f+

Page 14: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

I)H

`DT

p)

%**%

%D

Ugn

`$!

p6)D

e*)Z

'K/

M(K

NC

FO*

S,'

k.

l++

Slc6

66)

I( **%

g!**

$

pT-O

?$05

@.

@!&

ee++

)+C

%

@ $%

c. 9%

(K/

>D

)l)CE

?++0

"^%

*%C

e*:

`Cg*$

X'

L%

..+)c

Dp

I')(

I0c^

l)

)Qv

%.*

)!rr

r

:HO

#!1'

6<Rc

Zllc

55D%@

?$6c

+f)

p%!

%9%

K0/"

ZK

Q)&

.c

c&

67l!

m.!

&C

67

+K

#9!#!

IK

)cl

l)

cY'8l*+

+Z)

^[9

p!p

[&!

p+

)6%

C6!FZl

!!l

)C

c*]

e77

c^I!

IZ

cX

e+*

%++

ppp

+I0

S;

c(

lT

&C

ZC

!6c+

Z%ll

+;+

++

.^%

K$!

Tp

KKK

ZXt

++)

Z!

lZ

ll

!*

^%c^l+

&%:

*0+

p4%

!Z.

)++

) 6

+:++M

'c*+

,<T

pp+D

UU

CC

*.>R

ZpZ

l)[+

pZ.

5N

D6g

%-.

%!%

Db

-*

[!!1

5+pp

[%66

R4

"'(

K/

`$p

M007

%cl

ee%

*f*

p55

K!1

*.*

86%

ls

Cl+

+e*

V$

&e

Z!!

Z*%

!gbc6

lC

.!*

p(

g4

"'Z

..%

!6Sl

Z)

)++

+l

).^9

l!

BD

+)%*

@A

OK

/Z

%-

ll

e))*+

++

g!

lIL

ee

*R

&'&

$M

'4

cD

l*

lc

.l

4l

%k)

%K

g5

5)D

eI)

O!)

)++

++

++

++

+.&

+)+

o+*

c*:

.*)Z

:7

7+

+e

+65D

)C+

+)6

T+++

+++*

*>II(

(IO

!!<

++%

'&(

CO

#L#!

ZX

5K%

O[

+l2$+

+++v+

v+*

(p

ZA

L/

K'F

)P

`$c+

)Z

lZ6

l+)+

pp

UG

[$K

'JP

!O

L]T

QU

pp

*+

+e+

Ce)+

#()9

+c^

),'

eY

`l

CgQ+

TWI#

00?))

)))8

@$^

!I3

O%

)N

&8:7

77+r

cc

>g

^++

JR

Fk

&2%

6'I

eZ

&%

*^L

pZ

gO

Oe

q.&

*'R

K'

peD

++

p+.*

T: 5

KR

DC

#(

*,

p..

gp

5

+555

ND

#+:^!

o++

,),(J

(p

pA

'(-

pp

L/p

p%

*e

o+D

&'+

#3

$[

&'%

21

3

lSl

a

cD

.1C

C+

=

F8K

)C

[C9p

(!++

M*g

k%

C*/7

e)Dc

Cc%

)Cl

667LS

R(

c*:

(ZU

>%

.*

X5Z

e$g

))!

ec

:U

M"

RR

S+%

ll

l

).p

ell

RM

KN%

SZ

+e

_)(0

D0$'

$Z!%7Fc

%lC

%&

$H

4D&

D!p

ng

F)C

.*:'

Z)

S

n66

nD$5

Hp:

Rc%

ck7:

.+!

c6*C

v>%

lI"

6H5

&%

CZ

M

<)++

*)(

IM

ZG

!+)`

e).

(&

!e$

8O

BO

*8

I)

OO

93N

OQ

CM S+

[E 0M

). ee! &

N$`

Page 15: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

! F .* %$0$ l lllll -+++++++++ S++r+T++++l +) S++++ ++**8**^[' `/+ ()$3

^/6+!$-( Zg=` *55+ eL*I( ,*$3 *

FF^F^W55+ h WD5+5]6F.++

?',' O1)

!Q *c %=+!++*.

TT !$+ T)F!g++" ,(' #

pp [p+g++`5pt

M+ *8* %: %+)ll+*1 g+6&p p I(((U((9/''$

Kl %!****+ ?++%66+X)

?++ %6:+ !

GK )$T! $T* p !!65++++=5+

Jp l%**[ l)e TM++p pFpp.++p+++TpT *0 e. *Z) %:

lS%S+++'++++S K 666+l2. 8 O)))? (( O O X66

p,$*. . p I !&)+))%Frr+++ U p !!*:C %S l 867+%*. 8 :, 8 ''( F/+ *

c+ncg l p+g55++p p > vv++vv+g[:Tpp +++F p !))!p,2pp Up+

*+!)l**++F l)cZ pF+`X)p p .+86+Q*l !6666*^ 8 V

>%

T- 8 O' C

g+++++L!) .

"*-+++*++v+=D')9*(9$($`(Xd&)5+`^+pX+++T(Th*T*TgTX+pT$g&Z75&*T*T$TT+bXb&

I(d I) `+++p'e+ FM(M(: . *6666l eS- i . e.+.. %666VW= p p ` >^

M* pp p p+)$.,Fp p !!*(*%K(**j+96.L*

ClS8*) *66++

Page 16: Handwritten notes and problems by Prof. M. L. Aggarwal ...ilovemaths.com/pnc.pdf · Handwritten notes and problems by Prof. M. L. Aggarwal Topic - PnC,I pes-:"h.....v..te..k'; ,

+$$.)) + j+X+*55$F++T$9(T+)W *55!g+p(* 8 (# g p F; ( D! Y+ !5+Fp*(T C+++`5++**-* p -T*T&*Z* `^++=&+(-

pb$. C8((OOO (1')

%9(3((%P *( %`++;( D555/*- d H*g*p$g*;++(pp p +

p (g++*, >+ *66++l *6++%e r C- p *55++e&'e 8 O 00 ' "L'b p*) +` X+p+&

h) *555+l(d D $T&p - &=!!) gbV5555.R(*K +e+[+*6++Fg! p , `+Dp 4``($*`(*p!65+Tp v C

1 Fe*66++!:8+ * S , !S+C*6 X*C rrg+56k+%9! !/+ **%C l !%[++++^+%S'+++C++^+++rrp9)++I

`; !T*!T` ( l (V+ p $el +

Ze r ) %R'MH+C'%CM@p+h++++D++55++T$&**++++*5+ =6 $555

C++X*C

&l+d+8/g+.++)T

Ua+b

%PC(!$9.++ +v++1!:S66:( A( 8* .*-* %!

p Fg) (0 F * *Fel l B(/// . .

%v .+ +*6:

C%S+(l*C,)T `pF*n)UC U p*+++

p + $&+*b ,o)* ppp

SL%+&l+Cl *555u&`$C%: l ^) l L*Cp)+l

^Cl`* l :2!!! *8* p+

Fe*S*%!SD($*+&( Z) *8* !.!*`Z$oppp *5

F`+b5[5!$p$.+- ** $)$. ) %C$N++*p$))*p

C++7+*(p $++ TT TT(!&+++($6T(!!/+ pp 8=3 !:S; r++r+r&+r++%g%[,.! :4((8

(M)(&#,O:&8))(00)81(* p9,e)$6Dp$)!!/+:++*e, Wpb

C T ! C!DS(l*% &lC lFg*T*$*`0-O-.)+ vv+*c ! Qb+++ X++pp p =&

++)+ 9** 8 R $.bp C p*$ ' *#8

`*$(&6*`=+

X++p+ ;+p =*$ [r [++D+T/ ) C?+

S'( !/**55++`+ T* .* *]+ p ci++*8* ^g+++R " MT':( RR( pX5l: #IO)O) X++)

CKp v gbgb*$o p!$ !%FI+ :) ( F-

]++8*bT %.+8+++B*$F( v++*(*TT& U F+