mathongo.com-NCERT-Solutions-Class-12-Maths-Chapter-5-Continuity-and-Differentiability.pdf

Class 12 maths ncert cbse ch-5 solutions continuity and differentiability.ppt

1. m

Important Questions, Formula St

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2. Bale) = re 1) 16 1 conto et à = à
2 Rzamine the following functions for conti,
(a) fe == @ re

A Sas or siete)
Lee be rot mai oia Af

Kn) un ER Dey oi

Panag =<, des
ones = - AOL 0 de

Tin FE

a odos où every puis a It doi (Re RL
coco la emma,
Or

Hore fe) = — Ela a polynomial funtion. We know that
vary palm Buck continuas (vn ate blow

1. Constant faction
2 Payomlal face.

3. =
“or.
Pong = ei (0,10) = I,

sonoro.
sf nee Lap pla la la
Mines

oe

A A

i ee ‚ku comets

Folgen) =D of de
DmÖCH x #0 Wo ko that every rational

Re Zell Noto Ron

0). Thode fa ati eve a Sta (5).

og

mayen 2-5 AA
ern.

an el semer
a
Soe eae

4. OF

O ron

Falling «= m, =
ga png à = la (7 in) = 2
Ea fio) = 16) em) iO ia cti nt nn.

a the funetion [defined by,
[a en

ye 27

Contino ats, ara, At oa?

(aya «4 =)
IB x 29 in
(Rand Note (om con lt) bef he et of @.No lia
O

12

Ge ro) a

on

ms phy renter th 0 ey =
Peking = 8, Bm FO = à Ep f= Bf)

(Hey Fein and = 0 = (0)

5. Bight Hand Lime = og [GI = fi 5 17]

G20 = sis lightly > 2 and banco «> 1 ase)
Pring =2, 8
in fo) = in fen)

Jo 160 exists cod = 6 = 70)

(Posing «= 2> 15a U0, (= 8)
oe wann ot» 72
dire loco al gl bt tite
ind all points of vieeexinoity o£/f whore la defined by
orion lo 19
fears, ada
Ken. 5
a Given fis) - 23, 2 28
Lada aoe
To find pins of decanta 0 nt daa)
Here (0) ls dened far 3:52 le, on (= 2
and al ir = > 2 Le, on 0)
Domb (== D 0) = (y=) =
By (0, for aim < 3 (2 = 2 being partitioning peint ce
mentioned here) f(a) = 2e + Bis
D GD, or oh > 2, (6) = 2e 3 Ina payment and once
‘cnc Thor fe) een on R=

6. aaa, 00
“a
More fin) la defined for 2 <= 9 ir, (= — Hand alan or
À Baie md nike => le, ala =
Pendle fis = UCI DUB, = ==) =R
By Gh fre <3, fe) = lel sammen
(<3 mans la nogaiv sd Doncs | |= =)
ls a patin and baton centimos
Bald er ll (=< <8) 6) = JA a pomo ond
Dance canino
By), for els» ©, fife Be « 2 0 w tner am enc
tira Theme, a) a wsiougan on RB,
Brom) ere (Lie con/abgsrve ‘vals = Aland
£2 Dame patoning in ofthe main R
Lot na exemine continwity’of at partitioning point
21
Lah Hand Unie iy fl = in, LT)

so zen

"can

Fe = 2223 meca e la negative and ence
Tal

Pad
Right Mand Link = En, fie)= CC (y do

Gen = 2-5)

7. Pottiog c= 3, = 63) 0 2 18 + 2 = 20
ES

li, io) dove met xls amd hone fi) bs discotecas at

2-3 Cap
sl
srl
9 mamo

Bei. Gira fe) = IE! se sel

LY Si

Bata 0. rudo lala)
ES 40,
1 2x0 a)
25 it 2-0 un

‘Geary domain of) 1 (FÜ la deta > à <0.
Erin)

By (for elle > 0, f(s) = Lo a constant function and anos
mine.

By Ui, fc alle <0, (x) = = 1 in a coment faction and ane

econ fs) a continus om E = 19

8. iw a0 =
Pas «01 m1 = 0)
en)
aa «ip, na wna 0
Le, e010)
Dane SU, - AS, fe
Free lind 0), me ad Bt
TEI - 1 talallcot » se wenn
Wars (= 1 i slds Set
We know that orig onthe fon Cotino.
fia pen NS
Henze o pot of acon
CNET
(tan # sat
ati, wa 0)
En w act =

1.

Gr

Were fz) la defined for € a1 Le, on I, =) ond alo for

Domain ofa =, DUO) = (=, eR

9. Pang en, = 142
tm fe) = 10) 2)
oe ee
2 Fl doctor on fi we dein (ham)
See mo pa ot deci
Ps M xsd
ten if x>2

[okay «de 0
an x 28 40
TT CC

(C=, land oo Sale > alte, on =)
y Denis ot fin (ny <A
By (0) fbr on ua flan à is alpslymemiel and Dona
bus.
D or alla = 2, fi) = + 148 a poisaoralal and Manco

DFE)

Glen: -

conti an 8 — (3
Let us examine conti of fat the partlloalog pol = 2
Tek Bend Limit = Jig /6)= mG? 5) ET

Wear = waa)
Pains = 2,

ty

10. Here UE) la defined for x 5 1 Le, on (== Mlandiala Ser
> Uke, on

Domain of fin. = WU) = (cm, =) = 8
Br (fr lie < 1, FO) = #1 a polo and Rosca
By (0) foe all x > 1, f(a) = 22 ina polynomial and hones
pr

Fada costera on R —().
Let
zn

Lal Hond Kitt > Jim flo im (8
NAT 0
a ey A yon
De": - Ae

EME

ne mt xi

Hewes the Plat of disco In «= 1 (ely)
a te met defined y

+5 4 st
ra

a continue function?

11. ‘Ban fe)» Bi FU)

ian io) dos et at

ance fe) Sa acetingmus et = 1,
à = Ll ho only poet of dicen:
Discuss the continmlty of the function, f where fin defined by
(sw Osead
I, Lem <s
[628% >< 10)

EUX osa 0
fe fe 2 i)
Eu dern 1)
From (9) ana liye ennfaa that / i dana a, 1
UG, Uta, 191/ Le, fly te dane i 10
D off) SO 1

ews), for 053 <, fg) = 3 i à lanas: fonte and hoses
in conteo Be SE SL
From (i), fr 1 < <3, fs) = Ai a conta! Kinn and Danco
in min for 1 < <8,

From (i br 0 <a © 10,56) = 5 hn constant Kanon nnd
ence is continuous far 8 <4 10

/Tneefre, (sn omnis la th damaa [0,10] =, 31

Lot us examine contiauly eff nt the partiiecing peiat
aa

Laß Hand Limit = Mn fin) = En à

12. pacar =,
Right Hand Lit = lg /6)= im 6 Dy dl

Saat = =>

Peking = = 5
Bp foe be fa)

ir fo) does mt cet and hanca fa) ix dicen où
PE
Ee lland à = Sere the fms roto of scene Of the

Rane fate deen 10,20)

fae ey =< of
Tone os den

x u
Ths deals of / lek Ryu vee Ri ze< 1)
DR: > udm

=O ond « = Lars partitieias poior for ho damaln of he
foin.

Fon al e < fl) = Bein a pau conan.
Ror 0.41314 1, f (2) = 8b m cunatant fonttion nad honos
Pow all > 1, f(g) = deine prime ond one crninooos,
Lot us discuss coodnulty a portioning paint = = 0.

Aue =0,f0=0 [flo 80256.)
Beg 10) e 29 nn <0 calf) O

0-0

13. SE
an
BEE ony ya aan as
en
in
Bee
Y
A Æ 4
a: Y
O AY AT
er.
N. SA
+
Ae LA Tee
Si
O mes
eee)
Secor eee a
Eee carne Gf maa
en

2 Do

14. BoP = =>)
Peitiogx= 1, =2
Rim 7) = im. 6) (02) «tim eis ond = 2.

Puta = 1 in (0,10) AD = 2
Jo flo) = (0) 2) fla) la continus at = 1 an
‚Tieres fin contnwsne ar all in ite domaln R,
Find Me relationehip between a and B 0 Dat Can fanotion
f doña by
FE.

=)
a lord a
q

Ale [ss 0)

es a ES in

PTT EC YEN

Bari Loue D tn ENT Cr)
Le nn

Pating 3, miel i)

ig Hond Lint Jim fl) = im (ou +0) Oy Gi

=
Patiog = 3 = 8s 2
Posing == $n (7) = 30 +1
Bass) main ae 3 (given)

An = Am (6) =/0)

15. Prttinge =, =)4-0) 0

Bight Fined Tim = Le, Ba) Ge + 1 [Br A
290" mo)

Puitige = 0 = ADS 1=1
Bar 7) Oi le fe)

Li 0) doe no eit whatever ma bo

Near Ie ei na re: Sowers 2)
ge Se og
To exacnios continuity ffl = 1

A - ME + 1 By a)
ow Ae a demon eo
D): AR
Bio pe Lila Nr et ee 1) BO
CCR ı a = 11 20)
Patas seins
MIOS i fs) = 8)
ba 70) rita und = 5
Pong = Lia UOC 1>0) (0) = + 1» 5)
LUCE

fi) conta nt = 1 ll es alas of À
29. how that the Function Gofimed by gl) = = - Ir ts

16. Pus hoot
O
(o Mea Zah, Beni + A1= 0)

mu

O =
ln gn le i) amet

Han a) oon nt ocio and hanc 6) diciones of
2 = 6 (ny ige),
SE) = 2 Let dicas oll ore pia

Very Impertant Mot, If Foncea / una y no coins ia
e coment dan D,

A A doesn
imac RE AM points of D except hoes where

AA sin x » 6 comtinuous|at« = xt
(Giron: fis) =~ ala 4 ED nan

=e) he
whee 40 2 20 6 aod Me) = sine
Wo know that te) 23 « 6's a palm function and hence ia
cwotinwaon Geol zul)

Ag Ma) = ai «alg a la funcion comimos fra real)
By @) fa) = ena +5 FC) AO)
being the dienen a tw seins eins Sa a omis

17. Ga FE) = a 2 08 @)fG)= ein = 08 à
CCE CEE
We Knew Gat in ea continuous fonction for al real =
‘Alss mo ka that con a a continuous fanctinn fr all rel n
(Gen allan of @. No. 230) Below)
By Mata at the nd af miaken e Q- No 19,
(0) their sum feia (3) = sin x + em = a sao nio
fr a ral =,
GD thats énonce Fonetion CE) = sin = - con = In las

cn in im comics

ert polo a

TP) = haf fla
hat ha method l ning fo lity mln Aid
(fond i Pi) shou aboot.

. Discuss the|eensianiy of the codo comcast, scant mad
fonctions.

sa.

langen
(Lat fin ho he one fenton
tay [= coo 0)
Oloriy, Fir ts rool and Saito for al roal values of
m Le, fa) is dead for aM veal < Thuralbre Soma of
foe
id seca,

Um f(s) Yom cons

18. Fiol nt ata Le, ee
Yes ae = 0 ie, when # = nm 0.62
Domain off) man zu DR ke = mm ne A
2 fis real end Gite À € à Di
LE
Now (a) = come Gh = E “a
New gl) = 1 being constant function Le continuons om
dorada D and Bi) = nino nanas od enti um
Dela D.

OA

AT fe
(dba ota Naya al te endl poor 6 Q Ne 19,
Tet) be inner audi

Len wer ct ala Le, =

made Lo dle Eder
Domain i) = ne

19. Tica My (f(a) et =

23, ad eit pan of laconic where
(m
t= |=
ES
Sol Tha doma df a ke Rew MUERE
= ae pion vont Ho al ie gon Sin.
For alls <0, (6) + EE co
ES. «à ME
O A
[lc “ue 0
Ronan (se à aston her ro
Tin e el
Hk un disc ao continuity 0 /6) st An räioning
D: fee) A Life pto!

for a E

[re = cum ro) EE tu]

20. cios fins Gaona fin) and ea (se

faces) is eontinuoes fr oll ral xo 0.
Now let us examine continuity st == 0,

lin f6)= Tin, in à

Filtern marke qu ma a
a ts]
a 28 ae

nes (2) ern soca on domain of /
canine te condi of flere in Gesell,
re ee
PE : 2
fins if ano en)
are fe) 1 Em 2)
Fram), f(s) in dined far cn 0 and fm (4) fe) a din
Bene,
Denia fa 146) O =.
From @) fr 8, (6) = a= = eos bing the firme of to
swotinusn uncon tn x nd co x In conti fr al x 8
ones 7) la continous on R = (0.
Now lotus eramine comte #4 =

21. ews
a

22. Per = nue

ig Wd dl = Gg EN Drum
an Bor =o
Pags Ein D [= 10) =
Fm ar ns 2 een,
there i = ly) =f)
nm 0 k= 2

pl «dl

ea ad

a 4

N dl."
¡aio cole Een
IH nite Nue denn

re at x ae

Putting =m = he 61 :
Right Hand Unit = Fe, fü) Em, cs
Pulg = =, } con dino
E

da f= im, FO

23. Bet fi) a setas at = Eden)
ame =
Pug rn the hei as
|
tte ae ten ee eed

LINE
Fale lesb ass.
LE e fo
da continua fonction,

Ge E 7)
la Y 10 eB en
~ 2) es]

roe (4 onl GH), PL nc Me we MU <= «I

IEA Le AU 10 flO) 10, Bea) e
=k Dom off la E.

‘Giron: fa continus fat (of cama Goma ero
Ry heroine fi) a alo cominanas nt portioning pla = 2
ad's = 30 ofthe domain,

Becanes (a) a omklauru ot parting plat <= 2, era)

In for Je fof) my

Now En fi) = Im 6 [ero

24. hui ey f= me LE
e

Ping x= it; = 21
Patteg 2 18 in Ban. GD, 10) = 21
sting boro values in own o) we aro

EPA
= meto
Lat us schen osa. o) onl i) far a an

mm (oi) — eon. (a) pre nie >

A sar
CEC
"Very Important Hori: Cámponio timation of kmo continuous
Sinetiens Is contlamons.
A AN
ad (e «
‘BL Bhow thst tho Runetion| defined dy/f (Ja non (2) ix =

166) ha a rl und eta value roll = © By
Domain otf) la E

ato tk 46) = on ond Me) = 2

New a) las feta nad nes rotin.

‘gain M) = sn proce funcion aná Bone la mamans.
(gabe ng) na) fe haa

nn) (Changing le 2 ig) = mn)
it fone

(0) iy (being th exon ‘to
‘continuous fexttibes i rontinunas fr a a i,
domain À

25. Tate 1) = oe =| Cage) aig the cept frio
‘eve online fonctions ls coninoous
Benraine Bat aa | | In a continuous function.
Leb) a nd tc) = 12
We Know that en x and || re online faction.
fd aa contin,
New (fog) im CN = sin (aol = e rl
Wo knew that cempoute action «two coniownna factions da
pad
rs a conto He da | la nam
Find all point of diseotiguy off deiocd by

Tey = | dle 11,
co (ae + 1

Tila (Gaya sade e ange « R
Fin dins fr lla 2 de, decis fi RU

Piko cock erpresin within madelus og ta ©

es a= voll ete Os, <A asl sed

Mackin üben vales fs name — 1 nd (Gn proper amending
echec) on the number Hae, domain Hof la vided Sala (hewn
feloterals © = =, [= 1,0) an, =)
(On tha mineral (=, lie, Res <1, (ny he == 2 te)
= md 1 a lo le ares

Te [== 0404 1]=—G 4D

Heats () becomes (0) = 1 |< je + 1)

26. Le CAZA

roc (fc = <= fi) = La constant Funtn and boo a
contra fs <=.
From Gl), for = 1 <x 218) f(x) == 2e 1 la a polynomial
Section and boom i conanoun for <m

y > 0, 2) = La a estan fc ad noo de
coin for > 6

{is onu lo R = (AL 0
CR. point

Cr]

CT

Free:
En ¡HEN

= 1m 69 era) (0) =1
Be, f= fede 0

{continuous at =~ 1 lan.
Lot un examine continwiéy of f al partitioning point,

27. + fia comica at 2-8 ac
ia cotas ea the decada
"Tier la na pela of srl.
Second Solution
Wo Koen that every modulation in conawece for a me «.
hate Le and le + II or continuous für od eal x
‘Ase, we Know thot differance of te caninas functions la

CARE iro wel a

> har eo pot of

28. Get NCERT Solutia

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m
he

29. Be
À seo (tom JE)
‘Bol. Lat eg ve)
mr)

= oc (tan Ja) tan Gan YE) (an HE)
BR

OCT CRT OT

Portero ro]

E A Se

E
are)

fen
claro

=

Bel Lat y =

30. Bol Lab y = con a win? (7) = con 2 (ala af
pt us
Leu à À cartes À ot

|. et Role (ue) mt a + He
pensent]
|

A rea ms

rec

COTON

Bok Ley = 2x a de yt
&

LÉ. 4
2 1 0 a xf

|: 0. =

CT)

Item me vert po

2), meet
a

31. Men /)=|e—theaR
To prove: fía) da nok diferen at = L
Pa Lon DL lO)
pozo
=

Lt Mand Derio = Mr) = Je

NES En
m

7)

From 40) 254 (AD) ur) O
£6) a nob Siento ot x= 1
Note. In problems on Limit of Modules function, and Desa
Seaton (Le, pronta Inter Funcion), wa havo to find both at
Danae ad ight hand Hani (vo ha ed ls concept lio
om men in Er 52)
10. Prove Ghat the greatest Anleger function fined Dy

32. (We ne ot Bad (1) ns 1 (1) bees mot ea)
Didierentiabiiy ats =2

Polting = 2in 0) fC)» A] = 2

ah Hand date = 1/'C = im LOL zu, EA

Pr
=:

PE

E — ul exist,
>

[Giana amiante ot de

Noten Fat — e esx CHOR on ns

33. RD Sharma Solutions, Previous Year Papers)

15, Formula Sheets & much more

Exercion 6.3

2. BOO

Te
Sol Givens + y= in
Diterenining het slds wert. wo ave
a

a on, t
Lit Lee ue

34. a
= Saivesion=

A
Bol Given: ay + y = tana + y
nd ol e
À
fared
Ari pdt rs
RC A
Erre
Bey, «of
4
A 3
a "aa

nee
fy vt
Beas = fon
Girenıx = 99" 100
Difrentaiag bok des werk 3)

nee are
Bee boo peu

a

medias

35. Sol. Givens a
Duron ag beh aide mr x

a Al ee
La Lo

ol Eno loa

= 2007 067 Ÿ a [2 Ly 1) =
vum $ nota) >

- SN Y
o oa A

%
= da y „A
A

di rs - um
ates ont y ok,
Bo. Givens sn" = » Gay = 1
Dafererncag bo

Ada son + 2006 7 (2) =0

Percy ony 3 a

36. "To simply to gen Inverse Tesco, pot = = tan ©

‘en Tareas Tncionfpete = Ea

5 fito]
A

37. "To reply tho giron Inverso Deacon pot = tm À

ok ems mai
Pats mia
Ta simpy te stan Isère Tune
eto mln 0 (Ber JAE ps «
san er
(sin arte) = sat in cs 0)
(in 29 = 99 = 2 rt

39. Ee. ro]

da dt

FC EN
el Tat y= ci (an)
E E

40. eme)
ars
Sol Lot

atte
het
2 ety da: rer tre]

de

fp feed S re]

a
=

41. gen)
Tel Fc)

10, nos Oops en «510,

Bol, Lat y = ns og See)

«
m a
Pure]

ag a €) [Esa]

ne] tn

42. {Get NCERT Solutions, RD Sharma Solutions, Previous Year Papers}

tant Questions, Formula Sheets & much more,

43. 1. coe 2 ove de cos de.
Sol Let ym conc cs 2x con De 0
"Taking lags on both aiden, mo hare (aco Noto, i) pago 261)
Toe = og (os ca 3s cos 83)

= Lg cons» log ove 2e + Tog os 8x
[|

Eros ace Lina

wf da]

1 Fi
AR. a
1 a
“A ge
PM
ann 1 n+ 9 ta
vo oy fem dh

= = con em Ar ne (an +2 tan 2e 63 tam 2)

44. PERS
|

m

a ee
el
ee

A A
OT 2,

E ee LT
A ee Om se

(By Product ua]
a a
oe Seat he «+ of Gog sXe)

er

Ee eater tol)
tn a ho 0

Lar [ei]
Very Imperot Not
AAA

45. we Delage tals loge

Eu logs) = Olt lg)
er

Agia = 2
E Pe
oe oF oo

Er Jaco]

E dos 75 7]
= Bom | fire E

li veo dom Gi) e 1 ©,
e cm0) we, a
AO e
AA
CA TOS ry
‘Ting logs on beth ldn of o 0) seo Note GD page 261)

we eva eg y 2 og + DB + 4)
2 ho à 6 (By Remark pogo 252)

a

a Fl
Lyn E a

47. Eile

Sal Lat y = og ah + su
zus 0 whore = Gag ed

Mow = Oig Krane
Dir = leg GA) = «fg Ope E ge en bar]
a 4

TT

a

e E ct
CAPOT > Ya Er weinen)

far

D LT

DE one
hen

ea [Armen] = or (pz oe)
tog Epia
a
ops" elo ls Co)

Age an aen Darm NE
vr Iago = lg = og Tg = lag = bg)

48. a in ay + aint ya.
Bel Lat y = Gin a air de

a whore = Gi a ond» = ma YE

Tog a lg in a = log o e
a a
E toga) Eng na)

in nda =
OT Y a7

el
E 27

wn A

TE D Yor EO

49. sone 4 sare om»

2 [Bern]

Mom fi]
Vag 8 = agin 27° Fm oe e dla

a Pi

Lo = Eton vpn où
DE. Ms Ps» ee
o e Allg ann ¿login « 2

A
Bey + gts Cta)
A
Le Seon ein» ofa)
© (any (coral sg „ieh
de

por ant et $e ok

50. 2
een
een
[e Émis.
= 1 co x lag x + x sin a)bgrorcoss. à
eh

en |

A A
Y 1-10 À

CES ET

a

e er
d En
an)

in
a
© toc) and (01 Oh wove

e

Pt un a a E
CCE a el
a acon

11 mer + ein a
Ball Lay (econ nF» in ek

51. = 1 og + tor ns)
Détroit wt a, wo ave

=
ENT

EE)

qe,

“(à

= Ga i)

er ae
tig Mo vals of Se oct Lo end GHD a me
ve
E
as 27 fla an ae co
Pp --—
Po

DECO TL 22 to I:

mire
BOL Given 197 + 9 = 1
Seo m1 where ad omy!

52. ze
O man ana ui) Oh wave

we di

Y

ce Bigs nd y 1e)

‘Taking logren, o a = lg y = 7 loge = = logy
Dafa wt = we have

53. Soc

mern

CT
y

Fi
ne ay ey
E

ee ee

A kPa Wei
D co ($ oa

pa
MT eT ae

Cr q
Æ id had)" y wale tog e

„ganze bey
de tang + be on

ay ne
Gir: ne

"kia lags nb dan, re bars

Tag Go) = og |

= hg hay Go) leg e
ana
Diorentiating both ads wert. m, wo here

ings y E

54. oar
og) weg C21 og Cla) og CL) eg)
ran Bl a w= weave
a Bea an
Fe OTe Tae
RO EE
pra One eo
4
A

he
Fo

reale.
Patio mbr
Peep 0 an ode

Pilg - 3
0 - En +

4,2] -
2
37. Difterontiate (23 = Bae Ne} « Zr + 0) in three vaya
‘entiened holes

(0 br vaig present rele,
i) By expanding the predust to olieia « single

55. angle pelremial.
DORE AT IE TT)
PNA ÉTÉ e
ae mem,
PC ET CT]

Im ï
rd men set sme 2)

vom ma 2 oy urn mean

Teg logs on von at hs
ls CHR 8) + MAP > + £0)

ye a
Boa” fae
a
FE
eee)
Zan Zu

d [Lee ‚ar
ET [Besen ten,

[= DONNE

A
Leer

= Bese e799)
CANON EEE"

56. Bol: Given, 0 wa are Matan of
Fi F de æ y
Fomor Lcd owen En un
) To prowa oq @) By ropaated application of product
rile

a
una = À (a. 0)

Let us trent the do uo es eagle fact.

a a
O mee

. 4
Aal App Tale Ral de),

Taking og en bet den
y= bg 1) = eg leg + ag
Fl

dr dogo À logo e Logo

58. 7 E
AA fier at " Aa
Lanas y Hens
an rm on bead eae
Dieting Se op Wt wo hs
D. la o ee e 0,
Cr Er 1

a pl
nek cs o coe
m erg

dis bno
We know tint = US e À

Dm mnt, yn oon 2.

59. ye me

a LÉ 2

rn 2
ee
1. = = con 0 008 28,7 = nin - in 20
ol. Giver à = cos 0 ind» «De de
a

CT CE ET 27
Lio © 29.

E a
an a =

= 2 2 er

an
Yen o en

CE de
ne we

60. Ho sti E Cons)

CE:
Sika ten TART.
Ena

Bi on on» si Bain te
cy

62. el. We Mave = un + Bun 6) mm y = ei — cm)
Eolo 0.4 00000 in 0.1) = 8 00 0

Ei © (in rm.
aot Ein - One.
= à [on 0+ na 0 en 1m 9 mn 0

El
O m
æ

2

gn rm = wey aa
Ye qu)

63. in tig wann GE = 20280 « sc
free
Seb LE axes
Ss Pen
EP. 2
fn a ER,

a en er

& 5 La

e oo Deh.
4 logs

Be Lu y og

64. = 2e De og)
Fae +de à Ge gen Be singe
= «8 + 6 log 2
Find the second order derivatives of the functions given in
ia Oto 38.
“een.
Sol. Let y= enn Be

Erd muss ke parer

4 ‘
we cone aha bent on fe See ane

ros 5 dele
Los se
‘Again appli Produce Bodo of deren

A GAS ove Ad at)
a cli hin he ce he

CON NON ARE TE CT
= Bean make + 5 oak une)
nn = 24a)
Sir cos B= 12 tn Bh

1, ot oon de,

Bol. Let 9 = cos By

da
de Ru ur

“e 2 Ad
ec.
= = di in De 24 où De. AB

65. Ea
ain il ws
ea (a wo
li “a _
Arena,
de

=
= do og)

- ı s
Der "die:
is émet mr. 5)

66. 5 con x= sia prove hat Foy = 0)

ee ge
- Fa
= Aa

fa 4

1 di 2

(27)

67. mo)

spokes nsw ont AY tar € eme
a Gry Aron 6

|

A

a ahnen une 00,
ES

ts

Br. ~ 4 un
LdC.
“LA
re
Sede min ame ne ee

en y Bate — A 0 — Brno Am e
TA

ee a
er

68. ee y logo dogo e D
a yoga Ds ge Vand og lm

22)

eo

: La Le, SF à

md ot one

69. 1. Very Rallo» Gheowem Sor fi) =x! + 26-8, € À 0.
el. Gls fi) da e A

Here fi) l a pyme! faction off depres M.
ie) a than od derivable rare Le, en
ec FH) de continuous Sa the closed terval [=
durable in open Interval 4, 2.
DEEE EEE
Peitiog à = 2 la 0) FR) = 4 4 8 = 0)

TE)

A A ete
TMS
Bas ne O 2

A A 4, xf
= el pen nae (a, 2

(Genis e Ral Encore fruc
Res Pheorn ie veros
camino 1 Rela thcrem i plicable ta any of the flowing
emotion. Con pou y some lag bo the covers of all
‘heowem ren Shoes ecxmpled?
CD me M 1, DU) maes al 2
i) 16) == 1 D
Sel) Ge) = lr se, 9 E)
Co ms [el netos tho pases ter <3)
‘We Know that cta fc la] a ous a a tha
Intogor (Sea Ex. 16, paro 166, NCERT, Part 1). Hace
FE) = a Stat log between 5 and 9 Le.
cents aha = 6 3 = ends = 8 ond Bono Anni

a eel terval] al es ct dra cs

70. Pato 2k, = im, ELE

ma
(i Wolter iit ir RCA eme)
D ena
eee
ret

A Wr anf

ET
el

Pom
20 0 ei

ren Gi sd LLO =O.

Fe) = ON male a epa intra ober than neg
Fe

Ge f= bl bra zn
ere when of pet rag amd)

71. Pit me, 6) me = tm € = 0 dom nt blag top ter 2,
A A

Satin.

If ES,5 + R laa dioectiadio funeton sod 6) desa

ot vale eurer, en prove that 1-6) 1%.

Given: [ 6, 6) + E Sa a differentible funtion Le, f

onen ner (6, eo eed pos Ir
Beil
To pora (Yo
paie CS = 9)
From lend GF aX ch tro condos of nl Thera rs
sete.
‘Thre exis ot ste ito th end tea (= 6, Beh
fio» +.
de, PO +0 ihe FN venal Coal =e) bent
A A But thn cb 10
rn tht 16) don na van enti.
Ou ppt in) Le, [CD =f) wong
os
de Verily Mean Value Thaorem 1 fa) = 27 = de = 3 I de interval
Ea where e 1m Da
el, trans /6) = 2 — de = in the Intaral la, where a = 1 und
1 intr, 4) Er)

72. = «men ner,

1. LMM de vito

8. Veriy Mean Vals Theorem if fs) =? 0:2 = Ar im fh Interval
19 3] where a = 1 and bm8, Find ak 0 (1,3) for whieh
OS

Sol. Gin UI mae 0
ACL D 0. tate
nie [3
A ob a (dde 5 amie Ce
coms ad Gale prie Les m te a in m9}
enn 6) radin Go cmd nore! [13] denial Sa

(Ma, sila Toor fe mi
OA
lela - 3

men ee - 2

Te re re EC

EP DE

73. E pie and be (0.

& Emnine tho spplicablliy of Mean Velua Thoorem forall
{ho three fonctions being given below:
FO = bd fore 15,0) Die = Kel fore 6 3,3
= à a DD
Bel) Reproduce solution of @. No. 20 upto een. (D)
‘oth condone of LMLYZT. ve mot anti.
MI da oo to (¿JD to EE
GE) Reprodee mon of Qi. 25199 or. GO rpg 1,
A 6/019 by - (Wid lag
D: anal
Bay conan A
MVT, sh mat apodo 1/09 2 el fore ES 2
CRETE =)
Moro fc faa pata Kane Bros BL
"Tarot fi) a antun and derma eyarpnbane Les
en th mal line (==, =),
Maceo (a) I eankinun I tha led ftorral (1, 2] aaa
era a open ar (1, 2.
Beth camitims of Moen Valoe Thaerem are ated.
From D, fe) = 2e
Pt sa 6 file
Fran), fle)=/{l) 11 CN

74. a
Atay nifty
= 932 = re Apo nie)
= O30 = te EN Gee 9) = STP de BP Oe
2 at a sod 5
Sel La yee à etx = in

Y oc ot £ ain 46m À om
: 8 ent Lu À

EZ

ee x un d
Bao» tar 2 ut)

a

sean PTT)
Taking De of WA des of) me nova
A ¥en ga)
Dices tah ld wit. o bo

ey 4

18m ag tog Le
1 mn dinno rende]

75. En

a

or 4 FL fia)
Libia]

77. & 00 (a as + eb sn.) for somo onastante o and e
Bol LA y = om (a 18 2 + D dis) Be mmo mms a ond

h à
dias sense

"|

LE un ERA « à sd

7 4;
D nee art, Seule ME
re ae

ee
en
nee
do
E
een
Bacon

78. bo fe. dd
fer testar do
nr 0 en

=
E fe cost on 9 = 27 i matt}

en an
Tail an ie 8
Men
2.

Daye Pd
SE Br

ST DD og

A
> E iaa

iting be val fon AC 9)
ating ths salu i ea.

Dra.
Dd raro

PG e

Sol ety = a? + 6-0 rie > 3
ation, er tres (ON LUGO we (CUM +10)

79. =

Den 0m aa reel
Tag logs of Both sos, wo havo

gr bg G3" = x (= 9) loge = x eg ml

pag um de, dE Aw = £

80. ps
Bros
ad ur
feb Goo ya rt eat
Y Aa
EN) Aer ey
[ EINE ee]

4
« ques
ı

Faros
ala e
34 Weyiay oy Vus 8-1 <> <1, prom that

a

æ
CRC

81. HENAO
157)
Th 16 0) + (y= =e, to me o > 0; prove Bat

fé)
ae
la consta Independent ala an.

SS aa Ae O
Dioni bh sda a or) zh ©

o + ™
PRET ET

af
lo

82. Piling Prien)
er
a ee de
Th et Senden meer Epes ale
acia
on

een
er, a

(hs har cd US een Era lt pent oid
vice ot $2)

Dilantin boc aden of) wat ES a

E
mine Que

83. Ea (entrent) ea à =)

ea)

‘ N
menge)
D
Y. Moa
de xd tc wi
or ls dd nun, m

a

es

84. . (DOS e
wo a6

in ano

are) = 0
À pl dal tw 0 nd (10) = 0
ies davon deren OF 70) 08 à 0
Toe ten SOTO SB
EL + à.
ce i wD)
D 20 290 -» On pate «= 0

oe ce. A

(On patting = = 0)

VAIO ne
2 ff) is der ot = Bland ("d= sei
Fram (o) as (eh Y) orate à
2510 dla 6 lu < Gnd A

19, eig mathemston adoro, ro at ie) = =!
ann
au dam

natn del maes

85. à A
¡TA

[ns aros. Zoe]

cea

o ne

A sin alee) À
llo A A sin ale) LA

a
+ teal oe fi a SE

A
may (448)

= Go Ape do a

Dias pots ey 2A + EB

eer
a a mu fre rena

ee en
D nt Arc story bo pola?

Ya Gers ctl dh at tee
To ecm, tw tke (9 2 TE ET Ej
Lat ne pt ech ocre niin moder so! to 0 Le

86. locos Seien give By (y
Le, fi)= 8-2 te 221 un)
Ar 10202 u
229 fr x22 i)
New the three valuon of f(@) given ky (i), Li) and ie) ara
rolpnomlal funeiione und conatast fexctlon und haneo are
‘Connie un decree fr al el van a expt posi et
ho parta pain x = Lund == 2 “er
To examine comtistey sta = 1
DRE oly 4 Bra
Br:
A O A a
> à
A

PC OR A

FE) da cola et = 1
To Game dent = = à

Lan Hen art = 7) = ha =D

am
1
Beso >

a

[By (wed 10) = me were)

| sin

87. Pulga nh 48 m1
an een (een
ing Fe) eats end 2 6) à Em (= 11
Fi i continus at = = 2 et)
Fo Commins Soria at 2

PPT)
= le Fos

Ma dire)
0 5 mob decenio où 2

From (0) (7) and (UE), me ca any Uk) da cantina

{mal values of Le, conteos ere.

From (6), (il) andl (la), wa eam nay thet fie) ts not

into nt eme Ev points € = Vand = 2 on tho real

88. Fo ae = 00) = 8) We — ne) RC) (= me)
= (ma = mB) fle) de = nel) 4 (= rs) MC)
FO ant G0, vo have LER. = RES,

My = AA 10001, chow Maat

Ly
at

Giron y= werte

nr.

al = Deh sos mn,
(ae Las

Satin a
Landa

CT)

«
ES

9