JEE Main 2019 Mathematics April Attempt Shift - 1(08th April, 2019)

YKYE^-; ((N.E./N. ] ]EFJ.)

CEE (Hkol) >9;? F@HYU]E^ NKVEM ]EV] (FN])

[uesto`ls & V`buto`ls Mkte3 9 9: K Kprob, > >9;? ((VJOD]-;) || ] ]OHE3 9?.69 K.H. t` ;>.69 Y.H.

Murkto`l3 6 6 J Jùrs || H Hkx. H Hkras3 6 649 VUNCEF]3 H HK]JEHK]OFV

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PPPPPPPPPPPPPPP

ês`lklfe Emuveltures Btm. êgosterem êgosterem & F`rp`rkte @ddofe3 FG ]`wer, K-=4 & 5>, OYOK, Lekr Foty Hkbb, Cjkbkwkr ^`km, A`tk (^kc.)-6>=995 ]eb.L`.3 97==-4497777, 69;>;99, 69;>>>>, 4465555 | ]`bb Dree3 ;:99 >5: 5555 | Dkx3 +?;-9>>-6?;47>>> +?;-9>>-6?;47> >> | 9:996 === ::: Wensote3 www.res`lklfe.kf.ol | E-hkob3 f`ltkftIres`lklfe.kf.ol E-hkob3 f`ltkftIres`lklfe.kf.ol | FOL3 U:969>^C>997YBF9>=9>?

| CEE (HKOL) >9;? | MK]E 3 9:-9=->9;? (VJOD]-;) | YKYE^-; | @DDOFKB YKYE^ | HK]JEHK]OFV

YK^] 3 HK]JEHK]OFV H K]JEHK]OFV ;.

Kls. V`b.

]je f`ltrkp`sotove `d tje stktehelt "Od yù kre n`rl ol Olmok, tjel yù kre k fotozel `d Olmok", os 3 (;) Od yù kre l`t k fotozel `d Olmok, tjel yù kre l`t n`rl ol Olmok. (>) Od yù kre k fotozel `d Olmok, tjel yù kre n`rl ol Olmok. (6) Od yù kre n`rl ol Olmok, tjel yù kre l`t k fotozel `d Olmok. (=) Od yù kre l`t n`rl ol Olmok, tjel yù kre l`t k fotozel `d Olmok. mDau 1dl vao Jaacr esk tUes k gVkX rsa vao Jaacr ms ,m uaxdcm gVK " ma ozdr/auaêm mDau mDau gV& " ugjk tUesk gVK (;) 1dl vao Jaacr ms uaxdcm ugjk gVX ras vao Jaacr esk ugjk vao Jaacr esk tUes k gVK (>) 1dl vao Jaacr ms uaxdcm gVX ras vao tUes gVX ras vao Jaacr ms uaxdcm ugjk gVK (6) 1dl vao Jaacr es tUes (=) 1dl vao Jaacr es ugjk ugjk tUes gV gVX ras vao Jaacr ms uaxdcm ugjk gVK (;) F`ltrkp`sotove `d p  q os ~q  ~p Jelfe klswer os "Od yù kre j`t fotozel `d Olmok tjel yù kre l`t n`rl ol Olmok." p  q ma ozdrodc`drSr mDau " mDau ~q  ~p gasra gV vr % 1dl vao Jaacr ms uaxdcm ugjk gV ras vaoma tUe Jaacr Jaacr

esk

ugjk gqva gV ]je suh `d tje squkres `d tje belgtjs `d tje fj`rms olterfeptem `l tje forfbe, x> + y> 8 ;4, ny tje boles, x + y 8l, l  L, wjere L os tje set `d kbb lkturkb luhners, os 3 `ßa, x> + y> 8 ;4, oc oc csQaavask x + y 8l, l  L, tgak tgak L maTj xnS tjàvask L bJaj ozamir bkQ1avask ma beqYp1 gVX }aca maTj mj yEfanS1as ms k ms `xasS ma 1asx gV 3 3 (;) ;95 (>) >;9 (6) 6>9 (=) ;49 Kls. (>) >.

V`b.

p8

l >

(9, 9) = x+y8l

t` hkae tje olterfept ²vUrQa.H Belgtj `d olterfepts vUrQa.H

fuaus ms ms dy,¸

mj yEfanS 8 8 >

l >

2= l2

r > ‖ p > 8>

Vqukre `d olterfept vUrQa.H mj yEfanS ma `xS 8 8 = × ( ;4 ‖ Vuh `d squkres `d olterfept vUrQa.H

   

8 = ×  ;4 ‖

= > >

;4 ‖ l / >

l

>

>

), l  L

mj yEfanS ms `xasS ma 1asx

; ;   =   ?   ;4   >5       ;4 ‖   ;4 ‖   ;4 ‖   ;4 ‖   8 > :9   55  8 >;9 >   >   >   >   >   >  

ês`lklfe E Emuveltures B Btm. F`rp. & êg. @ddofe3 FG ]`wer, ]`wer, K-=4 & 5>, OYOK, Lekr Foty Hkbb, Cjkbkwkr ^`km, ^`km, A`tk (^kc.) - 6>=995 Yj.L`. 3 +?;-7==-69;>>>>, 4465555 | DKZ L`. 3 +?;-9>>-6?;47>>> 3 +?;-9>>-6?;47>>> ]` Al`w h`re 3 shs ÊV@ kt ÊV@ kt 54477 | Wensote 3 www.res`lklfe.kf.ol | E-hkob 3 f`ltkftIres`lklfe.kf.ol | FOL 3 U:969>^C>997YBF9>=9>?

]jos ss`buto`l w wks m m`wlb`km d d r`h ^ ês`lklfe C CEE ((HKOL) > >9;? V V`buto`l p p`rtkb

YKGE # ;

| CEE (HKOL) >9;? | MK]E 3 9:-9=->9;? (VJOD]-;) | YKYE^-; | @DDOFKB YKYE^ | HK]JEHK]OFV l

6.

   Od  klm  ne tje r``ts `d tje equkto`l x ‖ >x + > 89, tjel tje bekst vkbue `d l d`r wjofj   8; os 3       1dl bejmc.a x ‖ >x + > 8 9 ms ewy  rDaa  gVkX ras l ma U1wure eauX dtbms dy,   8 ; gVX    

l

Kls. V`b.

(;) 6 (>) = (>) x> ‖ >x + > 8 9 (x ‖ ;)> 8 ‖; 8 o> x 8 ; + o, ; ‖ o eaua  8 ; + o,  8 ; ‖ o Bet eaua l

      ;   

(6) >

(=) 5

l

 ;  o     ;  ;  o  l



 ;  o ;  o     ;  ;  o ;  o 



(o)l 8 ;

l

 ;  ;  >o    8 ;  > 



l8=

=.

]je suh `d kbb lkturkb luhner 'l' sufj tjkt ;99 2 l 2 >99 klm J.F.D. (?;,l) < ; os 3 (;) 6696 (>) 6>96 (6) 6>>; (=) 6;>; ,sbj bJaj oz aÛr bkQ1avask 'l' tas tas nb ozmac gVk dm ;99 2 l 2 >99 rDaa J.F.D. (?;,l) < ;, ma ma Kls. (=) V`b. Lkturkb luhners netweel ;99 & >99.

1asx gV&

;9;, ;9>, …., ;??.

Eotjer movome ny 7 `r movome ny ;6. (suh `d luhners (movome ny 7) + (suh `d luhner movome ny ;6) ‖ (suh `d luhner `d movome ny ?;) ;=

:

r ;

r ;



 (?:  7r )  (?; ;6r )  (;:>) 8   ?:  ;=  7.

;=  ;5   :  ?     (?; :)  ;6    ;:> >   > 

8 ;67> + 765 + 7>: + =4: ‖ ;:> 8 6;>; Jolmo. Jolmo. ;99 rDaa >99 ms fjp esk oz ozamir bkQ1a,sk & ;9;, ;9>, …., ;??.

;6 1a 7 bs d`Jaadtr bkQ1avask ma 1asx & (7 bs d`Jaadtr bkQ1avask ma ma 1as x) + (;6 bs ;=



d`Jaadtr bkQ1avas 1avask ma 1asx ) ‖ (?; bs d`Jaadtr bkQ1avas 1avask ma ma 1asx)

:

(?:  7r ) 

r ;

(?; ;6r )  (;:>) 8    ?:  ;=  7. r ;

;=  ;5   :  ?     (?; :)  ;6    ;:> >   > 

8 ;67> + 765 + 7>: + =4: ‖ ;:> 8 6;>; so l

5.

5x

 so l >x

mx os equkb t` 3

>

(wjere f os k f`lstklt `d oltegrkto`l) so l

5x

 so l >x

mx fcafc

gV&

>

( f f ,m beamyu vpc gV tga¡ gV )

Kls.

(;) x + > solx + >sol >x+ f (6) >x + solx + >sol>x + f (=)

(>) >x + solx + sol>x + f (=) x + > solx + sol >x + f



YKGE # >

| CEE (HKOL) >9;? | MK]E 3 9:-9=->9;? (VJOD]-;) | YKYE^-; | @DDOFKB YKYE^ | HK]JEHK]OFV sol

V`b.

5x

 so l >x

> so l

mx 8

>

8> 4.

f`s

x

 so l 6x  so l >x   mx 8 sol x 

 > so l >x f`s x> mx 8    >

 >f`s x mx   (6 ‖ = so l

>

x ) mx

>

 f`s x mx   mx  > f`s>x mx 8 >solx + x + sol>x + F

]je krek (ol sq. ulots) `d tje rego`l K 8 {(x,y)  ^ × ^ | 9  x  6, 9  y  = , y  x> + 6x} os 3 {as8a K 8 {(x,y)  ^ × ^ | 9  x  6, 9  y  = , y  x> + 6x} ma {as8a[y ( `xS nman1ask esk) gV& (;)

Kls. V`b.

5x

>4

(>) :

6

(6)

5? 4

(=)

56 4

(6)

;:

y8=

‖6/> 6

;

9

‖6

‖?/=

y 8 x> + 6x 8 =  x> + 6x ‖ = 8 9 (x + =)(x ‖ ;) 8 9 x8; ;

krek {as8a[y 8



( x >  6 x ) mx  >( =) 8

9

7.

Kls. V`b.

; 6

6 

>



:

;; 

4



:

5? 

4

Od V; klm V> kre respeftoveby tje sets `d b`fkb holohuh klm b`fkb hkxohuh pòlts `d tje dulfto`l, d(x) 8 ?x= + ;>x6 ‖ 64x> + >5, x  ^ , tjel 3 1dl [yu d(x) 8 ?x= + ;>x6 ‖ 64x> + >5, x  ^ , ms ms BDaauj1 duEure rDaa BDaauj1 hYpre dfUlqvask ms beqYp1 Úe'a% V; rDa V> gVX ras (;) V; 8 { ‖ ‖;} 1 V> 8 {9,>} (>) V; 8 { ‖ ‖>,9} 1 V> 8 {;} ‖>} 1 V> 8 {9,;} ‖>,;} 1 V> 8 {9} (6) V; 8 { ‖ (=) V; 8 { ‖ (=) y 8 ?x= + ;>x6 ‖ 64x> + >5 my 8 64x6 + 64x> ‖ 7>x mx 8 64x(x> + x ‖ >) 8 64x(x + >)(x ‖ ;) +

‖

+

‖>

9 ; { ‖ ‖>, ;}  pòlts `d holohk dudEu"N ms dfUlq dfUlq {9}  pòlt `d hkxohk hdYYa"N ma dfUlq



YKGE # 6


:.

Kbb p`ssonbe luhners kre d`rhem usolg tje mogot ;,;,>,>,>,>,6,=,= tkael kbb kt k tohe. ]je luhner `d sufj luhner ol wjofj tje `mm mogots `ffupy pbkfes os 3 bJaj vkmask ;, ;, >, >, >, >, 6, =, = mas ,m baDa ysmc bJaj bJaj bkJa` bkQ1a1sk fuanS xnS gVkK nb ozmac mj bkQ1avaskX

dtuesk d`"ae vkm be be BDaauask oc gVX mj bkQ1a gV& Kls. V`b.

(;) ;:9 (>) ;4> (;) ;, ;, >, ,> , >, >, 6, =, = `mm luhners `ffur kt evel pbkfes d`"ae

=F

?.

6 ×

6! >!

4! 

=! >!

(6) ;49

(=) ;75

vkm be BDaauas BDaauask oc vars gV gV

8 = × 6 × ;5 8 ;:9

]je hekl klm vkroklfe `d sevel `nservkto`ls kre : klm ;4, respeftoveby. Od 5 `d tje `nservkto`ls kre >, =, ;9, ;>, ;=, tjel tje pr`muft `d tje rehkololg tw` `nservkto`ls os 3 bar ozs{a.aas {a.aask ms ea/1 rDaa ozbc.a Úe'a% : rDaa ;4 gVK 1dl nues k bs 5 ozs{a.a >, =, ;9, ;>, ;= gVXk ras 'as"a las osz{a.aask ma ma

xq.au[y gV& Kls. V`b.

(;) =9 (>) =: (>) Bet rehkololg tw` vkroktes kre k klm n tjel eaua 'as"a las pc k rDaa n rf k  n  >  =  ;9  ;>  ;= 7

klm vaVc

k>

n

>

Kls. V`b.

(=) =5

8:

=  ;4  ;99  ;==  ;?4 ‖(:)> 8 ;4 7  k + n 8 ;= klm k> + n> 8 ;99  kn 8

;9.

(6) =?



(k  n)> ‖ (k> >



n> )

8

;?4  ;99 ;99 >

8 =:

Bet d 3 Q9,>X  ^ ne k twofe moddereltoknbe dulfto`l sufj tjkt d" (x) < 9, d`r kbb x  (9,>). Od (x) 8 d(x) + d(> ‖x), tjel  os 3 (;) mefreksolg `l (9,;) klm olfreksolg `l (;,>). (>) olfreksolg `l (9,>) (6) olfreksolg `l (9,;) klm mefreksolg `l (;,>) (=) mefreksolg `l (9.>) d 3 Q9,>X  ^ las fac v`myuj1 [yu nb ozmac gV dm bJaj x  (9, >) ms dy, d"(x) < 9 gVK 1dl 1dl (x) 8 d(x) + d(> ‖x) gV, ras ras  (;) (9,;) oc gzabeau rDaa (;,>) oc `/aSeau gVK (>) (9, >) oc `/aSeau gVK (6) (9,;) oc `/aSeau rDaa (;,>) oc gz abeau gVK (=) (9, >) oc gzabeau gV (;) d"(x) < 9, y 8 d(x) 1 x  (9, >) (x) 8 d(x) + d(> ‖ x) '(x) 8 d'(x) ‖ d'(> ‖ x) d`r (x) t` ne olfreksolg (x) ms `/aSeau gasus ms dy, '(x) < 9  d'(x) < d'(> ‖ x)  x < > ‖ x (d'(x) os olfreksolg ol (9, >)) (d'(x) `/aSeau gV (9, >) es)



YKGE # =


 

D`r (x) t` ne mefreksolg '(x) 2 9  

;;.

x<; x  (;, >) (x) ms

gzabeau gasus ms dy,

d'(x) 2 d'(> ‖ x) x  (9, ;)

]je grektest vkbue `d f  ^ d`r wjofj tje systeh `d bolekr equkto`ls x ‖ fy ‖ fz 8 9 fx ‖ y + fz 8 9 fx + fy ‖z 8 9 jks k l`l-trovokb s`buto`l, os 3 f  ^ ma vd/amre eauX dtbms dtbms dy, csdQam bejmc.a bejmc.a duma1 x ‖ fy ‖ fz 8 9 fx ‖ y + fz 8 9 fx + fy ‖z 8 9

ma ,m vrq YL gy gVX gV& (;) ‖; Kls.

f f

nut f 

(=) 9

>

; >

, ‖;

; `tjerwose x 8 y 8 z 8 9 >

]je sj`rtest mostklfe netweel tje bole y 8 x klm tje furve y 8 x ‖> os 3 csQaa y 8 x rDaa `Ú y 8 x ‖ > ms fjp mj U1wure lwcj gV& (;)

Kls. V`b.

;

‖ f ‖ f ‖ ; f 8 9  (f + ;)>(; ‖ >f) 8 9 ‖ ; f

 f 8

;>.

(6)

(;) ;

V`b.

(>) >

7

(>) >

= >

(6)

7 :

(=)

;; = >

(;)

[ Y(> + >, ) (>,9)

Vj`rtest mostklfe netweel U1wure y> 8 x ‖ > klm vaVc y 8 x

lw cj

my

dfUlq Y kt pòlt dfUlq Y oc wobb ne ;. mx Moddereltoktolg tje furve v`myu

mcus oc



YKGE # 5


>yy' 8 ;

y' 8



holohuh mostklfe U1wure



 Y

lwcj 8 Y[ ?

8

;6.

 ? ;  ,   = > 

; ;  ; >y >

=

; 

>

>

7 

= >

]je suh `d tje f`-eddofoelts `d kbb evel megree terhs ol x ol tje expklso`l `d 4

4

4

4

 x  x6 ‖ ;    x ‖ x 6 ‖ ;  , x  ; os equkb t` 3          x  x6 ‖ ;    x ‖ x 6 ‖ ;  , x  ; ms ozbac esk x ms bJaj be0aarj1 be0aarj1 olask ms xq.aakmask ma 1asx fcafc gVgV&         Kls. V`b.

;=.

(;) >= (;)

V`b.

(6) >?

(=) 6>

( x  x 6 ‖ ;)4  ( x  x 6 ‖ ;)4 8 >(4F9.x4 + 4F>.x= (x6 ‖ ;) + 4F= x> (x6 ‖ ;)> + 4F4 (x6 ‖ ;)6) ]erhs wotj wotj evel p`wers `d x 8 >(4F9.x4 ‖ 4F>.x= + 4F= x> + 4F= x: + 4F4 ( ‖ ‖; ‖ 6x4) 4 4 4 4 4 4 fèddofoelts 8 >( F9 ‖ F> + F= + F= ‖ F4 ‖ 6. F4) 8 >(;5 ‖ 6) 8 >= Od tje tklgelts `l tje ebbopse =x> + y> 8 : kt tje pòlts (;,>) klm (k,n) kre perpelmofubkr t` ekfj `tjer, tjel k> os equkb t` 3 1dl lj0aSìßa =x> + y> 8 : ms dfklqvask (;,>) rDaa (k, n) oc Qajkpj xnS Bo'aS csQaa,¡ ocBoc ykf`r gVX ras k> fcafc gV& (;)

Kls.

(>) >4

;>: ;> : ;7

(>)

> ;7

(6)

4= ;7

(=)

= ;7

(>) x> ( > )>



y> (> > )>

;

Bet (k, n) os ( > f`s, > > sol) tklgelt kt (;, >) os =x + >y 8 :  >x + y 8 =  sb`pe os ‖> 8 h; tklgelt kt (

>

f`s, > > sol) os =

L`w h; h> 8 ‖;  k8



Jolmo.

x> ( > )>



>

y> (> > )>

=f`t 8 ‖; `r

;7



>

>



;7

f`sx + > 

soly 8 :  sb`pe os ‖> f`t 8 h> ; ; f`s 8 `r



k> 8

>

;7

;7

> ;7

;

eaua (k, n), ( > f`s, > > sol) gV bejmc.a =x + >y 8 :  >x + y 8 = (;, >) oc Bo'aS csQaa ma bejmc.a  oz`.ara 8 ‖> 8 h; ( > f`s, > > sol) oc Bo'aS csQaa ma bejmc.a bejmc.a = > f`sx + >  oz`.ara 8 ‖> f`t 8 h>

>

soly 8 :



YKGE # 4


vf h; h> 8 ‖;

;5.

Kls. V`b.

=f`t 8 ‖;

>

k8





>

1a

;7



;7



f`s 8



k> 8

; ;7

;

1a

;7

> ;7

 5 klm 9 2  ,  2 , tjel tkl (>) os equkb equkb t` 3 = 5 ;6 6  5 1dl f`s( + ) 8 , sol( ‖) 8 rDaa 9 2  ,  2 , rf rf tkl (>) fcafc gV& = 5 ;6 46 66 46 >; (;) (>) (6) (=) ;4 5> 5> ;4 (6)

6

Od f`s( +  ) 8



9 2  2

=

, sol( ‖) 8



, 9 2  2

= 5

L`w vf sol( ‖  ) 8

;6

klm vaVc f`s( + ) 8

6 5





9 2  +  2



f`s( ‖ ) 8



sol( + ) 8

klm

>

 

=



2  ‖  2

=

;> ;6 = 5 =

L`w vf tkl > 8 tklQ( + ) + ( ‖ )X 8 

;4.

]je suh `d tje seroes >.>9 F9

Cs.aj (;) Kls.

.>9

> >

>4

F9



.>9

5

.>9

F;  :

F>

(>)



.>9

5

tkl(  )  tkl(  ) ;  tkl(  ) tkl(  )

.>9

F;  :

F>

.>9 F6  ...   ;;

>

.>9 F6  ...   ;; .>9

4>

>=

F>9 ma

(6) >>6

.>9

4>

6

8

; ‖

5 

;> = 5 6

F>9 os



8

46 ;4

;>

equkb t`

1asx fcafc gV& (=) >>5

(=) >9

V`b.

>.

>9F

9 +

5.>9F; +

:.>9F

>

+ …+

4>.>9F

>9

8

 (6r  >) . >9Fr r 9

86

 r. >9Fr + > 

>9

Fr

8 6 × >9  ;? Fr ;  >.(>>9 ) 8 49 . >;? + >.>>9 8 ;4. >>; 8 >>5

r 9

r 9

;7.

>9

>9

>9

r ;

 f`s  ‖ sol    , (  ^) sufj tjkt  sol  f`s  

 9 ‖ ;   . ]jel k vkbue `d  os 3   9   ;  f`s  ‖ so l    9 ‖ ;   , (  ^) nb ozmac gV dm K 6>    X ras  ma ,m eau gV& eaua K 8  9   so l  f`s    ; Bet K 8 

(;)



;4

(>)



6>

Kls.

(=)

V`b.

f`s   sol  f`s   sol   K> 8     sol  f`s    sol  f`s  

K

6>

(6) 9

(=)



4=

f`s >  sol >   sol > f`s > 

K> 8 



YKGE # 7


sohobkrby we `nserve tjkt nbj

f`s l  so l l  f`s l  

ozmac Kl 8  sol l

jelfe vr% f`s6> 8 9 klm sol6> 8 ; 

6> 8 >l + l

 8

;:.

>



, l  o



;4

4=

]je hkglotume `d tje pr`cefto`l `d tje veft`r > o  6 c  a `l tje veft`r perpelmofubkr t` tje pbkle ˄

˄

˄

f`ltkololg tje veft`rs o  c  a klm o  > c  6a , os 3 ˄

˄

bdl'a >o  6 c  a ms bdl'aask ma odcea.a gV& ˄

(;)

6 4

˄

˄

˄

˄

vk rd`S"T o  c  a rDaa o  > c  6a mas vk

˄

˄

˄

˄

(>)

4

˄

˄

˄

˄

6

(6)

mcus àys bery bery ms ykf`rjS1 bdl'a oc oz{aso

>

(=)

Kls.

(6)

V`b.

L`rhkb veft`r t` tje pbkle f`ltkololg o  c  a klm o  > c  6a os ˄



 

l 8 o  c  a × o 

˄



l

˄

˄

˄



> c  6a ˄

˄

 >o  6 c  a . o ˄

8

˄

˄

˄

‖

˄

> c˄  a˄

˄



l



6

8

4

>

o  c  a rDaa o  > c  6a mas

l

˄

˄

˄



˄

 

beadgr mcus àys àys bery ms yEf`r bdl'a& bdl'a&

˄

8 ˄o  c˄  a˄ × o  > c  6a ˄

˄

˄



8 ˄o ‖ > c˄  a˄

>o

˄





6 c  a ma

>o

˄

8

˄



˄



l

oc oz{aso



6 c˄  a˄ . ˄o ‖ > c˄  a˄



; =  ; 6

8

;?.

˄

6

˄



˄

; =  ;

8

l

˄

8 ˄o ‖ > c˄  a˄ ˄



˄

>



pr`cefto`l `d > o  6 c  a `l

Jolmo.

˄

6

4

8

6 >

Bet y 8 y(x) ne tje s`buto`l `d tje moddereltokb equkto`l, (x> + ;)> Od

k

y(;) 8



6>

tjel tje vkbue `d 'k' os

eaua y 8 y(x), v`my bejmc.a (x> + ;)> k

(;)

y(;) 8 ; ;4



6>

my + >x (x> + ;) y 8; sufj tjkt y(9) 8 9, mx

my + >x (x> + ;) y 8 ; ma mx

gy gVX tfdm y(9) 8 9 gVK 1dl

gVX ras 'k' ma ma eau gV& (>)

; =

(6)

; >

(=) ;



YKGE # :


Kls. V`b.

(;) my + >x(x> + ;)y 8 ; mx my >x ; y >  > mx x  ; ( x  ;)> Bolekr M.E. csQajm v`my bejmc.a bejmc.a (x> + ;)>

>x

O.D. 8 

 e x

>

mx

;

 e l( x

>

;)

8 x> + ;

 [(OD)mx ; y.(x + ;) 8  ( x  ;) y(O.D.) 8 >

>

>

(x> + ;)mx

y(x> + ;) 8 tkl ‖;x + F y(9) 8 9  F 8 9  y.(x> + ;) 8 tkl ‖;x  y(;) 8

>9.



:

 k 8

; ;4

 ; ‖ x   >x  ,| x | ; , tjel d   os equkb t` 3  ;  x   ;  x >  ; ‖ x  >x  1dl d(x) 8 b `ge   gasX ras d   ,| x | ; , gas  fcafc gV& >  ;  x   ;  x  Od d(x) 8 b `ge 

Kls.

(;) (d(x))> (>)

V`b.

d(x) 8 l 

(>) >d(x)

(6) ‖>d(x)

(=) >d(x>)

 ;  x    ;  x 

> x    ;    ;  x  >  >  ;  x >  >x  ;  x   >x  ; x   8 >l   8 l   8 l   l  d 8     8 >d(x)  ;  x >  >x    ;  x   >x   ;  x   ;  x >        ;   ;  x > 

>;.

Kls. V`b.

]je belgtj `d tje perpelmofubkr dr`h tje pòlt (>, ‖ ‖;, =) `l tje strkogjt bole,

x6 ;9



y ‖ > ‖ 7



z os 3 ;

(;) grekter tjkl > nut bess tjkl 6 (>) grekter tjkl = (6) bess tjkl > (=) grekter tjkl 6 nut bess tjkl = x  6 y ‖ > z (>, ‖ ‖;, =) bs bcy csQaa, dfUlq (>, oc Qajkps x, yEf mj ykfanS 3 3   ‖ 7 ;9 ; (;) > bs vd/am ockrq 6 bs me gVK (>) = bs vd/am gVK (6) > bs me gVK (=) 6 bs vd/am ockrq = bs me gVK (=) Bet tje d``t `d perpelmofubkr dr`h Y(>, ‖ ‖;,=) t` tje govel bole ne K(;9 ‖6, ‖7+ >,) YK .

(;9 o˄ ‖ 7 c 7 c + ˄

a˄

) 89

 ;9(;9 ‖ 5) ‖ 7( ‖ ‖ 7 + 6) + ;( ‖ =) 8 9



YKGE # ?


 ;59 8 75   8

8

YK

(;9 ‖ 5)> >

; >

. >

 (‖7  6)  ( ‖

>

 ;   7  9       8  >   > 

8

59 =

wjofj boes ol (6,=) Jolmo. eauadm dfUlq Y(>, Y(>, ‖ ‖;,=) bs YK .

(;9 o ‖ 7 c 7 c + ˄

˄

a ˄

=)>

lj xnS csQaa oc yEf yEf ma oal K(;9 ‖6, ‖7+ >,) gVK

) 89

 ;9(;9 ‖ 5) ‖ 7( ‖ ‖ 7 + 6) + ;( ‖ =) 8 9  ;59 8 75   8

; >

. >

YK

8

>

(;9 ‖ 5)

>

 (‖7  6)  ( ‖

>

=) 8

>

 ;   7  9       8  >   > 

59 =

tas vUrcay (6,=) eks dBDar gVK >>.

]je equkto`l `d k pbkle f`ltkololg tje bole `d oltersefto`l `d tje pbkles >x ‖ y ‖ = 8 9 klm y + >z ‖ = 8 9 klm pkssolg tjrùgj tje pòlt (;,;,9) os 3 beryask >x ‖ y ‖ = 8 9 rDaa y + >z ‖ = 8 9 mj ozdrYLslu csQaa mas vkrd`S"aT mcus àys àys rDaa dfUlq (;,;,9) bs

gasmc

taus àys bery bery ma bejmc.a gV& (;) x + 6y + z 8 = (6) >x ‖ z 8 > Kls. (=) V`b. Y; + Y> 8 9 (>x ‖ y ‖ =) +  (y + >z ‖ =) 8 9 ot pksses tjrùgj (;,;,9)  ; +  8 9   8 ‖ ; equkto`l `d pbkle os x ‖ y ‖ z 8 9 Jolmo. Y; + Y> 8 9 (>x ‖ y ‖ =) +  (y + >z ‖ =) 8 9 1g dfUlq (;,;,9) bs xqtcra gV  ; +  8 9   8 ‖ ; bery ma bejmc.a x ‖ y ‖ z 8 9 gVK >6.

Kls. V`b.

(>) x ‖ 6y ‖ >z 8 ‖> (=) x ‖y ‖ z 8 9

Bet @(9,9) klm K (9,;) ne tw` doxem pòlts. ]jel tje b`fus `d k pòlt Y sufj tjkt tje peroheter `d K@Y os =, os 3 eaua @(9,9) rDaa K (9,;) las dud'pr dfUlq gV gVX ras ,sbs dfkUlq Y dtums dy, K@Y ma odceao =, gas gasX ma dfklqoDa gV& > > > > (;) ?x ‖ :y + :y 8 ;4 (>) :x ‖ ?y + ?y 8 ;: (6) ?x> + :y> ‖ :y 8 ;4 (=) :x> + ?y> ‖ ?y 8 ;: (6) x >  y> +

x>



( y ‖ ;)> 8 6

 x> + y> ‖ >y + ; 8 ? + x> + y> ‖4 x >

6 x>





y>

y > 8 = + y

?x> + ?y> 8 ;4 + y> + :y



YKGE # ;9


>=.

Kls. V`b.

K pòlt `l tje strkogjt bole, bole, 6x + 5y 8 ;5 wjofj wjofj os os e equomostklt quomostklt dr`h tje f``rmolkte kxes wobb wobb boe `lby ol 3 (;) ;st klm >lm qukmrklts (>) =tj qukmrklt (6) ;st, >lm klm =tj qukmrklts (=) ;st qukmrklt bcy csQaa 6x + 5y 8 ;5 oc dBDar ,m dfklqX tas dulsS'aakm v{aask bs bs belwcBDa gVX ms`y dBDar gV& d}rj1 prq prqDaa»'a esk (;) ozDae rDaa d}rj1 (>) prqDaS prqDaa»'a esk rDaa prqDaS prqDaa»'a es Daa»'a esk (6) ozDaeX d}rj1 rDaa (=) ozDae prqDaa» (;) Dr`h tje dogure, dp8aauq dp8aauqbac y8x Y( ‖;5/>, ;5/>) Y(;5/:, ;5/:)

6x + 5y 8 ;5

Oltersefto`l `d tje `rogolkb bole wotj y 8x klm y 8 ‖ x wobb gove mesorem pòlts Y ol ;st klm >lm qukmrklt . lj xnS csQaa ma y 8x rDaa y 8 ‖ x ms baDa ozdrYLsl dfUlq dfUlq Y, ;st rDaa >lm prqDaa»'a esk gasxaK >5.

boh

x 9

boh

x 9

Kls.

(;) = (>)

V`b.

boh

x9

so l > x > ‖ ;  f`s x so l > x > ‖ ;  f`s x

equkbs 3

fcafc gV& (>)

(6) > >

(=)

8 boh x

>  ;  f`s x

    so l x  ;  8 boh >  ; f`s x x 9   x   >  x 



>



;  f`s x

8

> > ; / >

8 =

>

> ‖ x f`s x Od d(x)8 klm g(x) b`ge x, (x < 9) tjel tje vkbue `d tje oltegrkb >  x f`s x

1dl d(x)8

> ‖ x f`s x >  x f`s x

Kls.

(;) b`ge6 (=)

V`b.

g(x) 8 l(x), d(x) 8

>

so l> x( >  ;  f`s x ) 9 ;  f`s x

so l > x

>

>4.

= >

 / =

 gd (x)mx os 3

‖

 / =

 / =

rDaa g(x) b`ge x, (x < 9) gVkX ras beamy

 gd (x)mx ma eau gV&

‖  / =

(>) b`gee > ‖ x f`s x >  x f`s x

(6) b`ge>

(=) b`ge;

 > ‖ x f`s x  klm g(d(x)) 8 l    >  x f`s x 



YKGE # ;;

| CEE (HKOL) >9;? | MK]E 3 9:-9=->9;? (VJOD]-;) | YKYE^-; | @DDOFKB YKYE^ | HK]JEHK]OFV  / =



 8

 > ‖ x f`s x   mx  >  x f`s x 

l

‖  / =

x  k + n ‖ x x  ‖x  / =



 8

 >  x f`s x    > ‖ x f`s x 

l

‖  / =

mx

 / =

Kmmolg tasH+us



oc, > 8



l ; mx 8

9   8 9

‖  / =

>7.

]je suh `d tje s`buto`l `d tje equkto`l

bejmc.a

x ‖ >

Kls.

(;) ;9 (;)

V`b.

|

x  > |

|

x

x





x ‖ =





>  9, x

(>) = +(

x )> ‖ =

‖ >|> + |

x

x

x ‖ > 

9 ms



x



x ‖ =





>  9, x



9 os equkb t` 3

gyask ma 1asx fcafc fcafc gV& (6) ;>

(=) ?

+ = 8 >

‖ >| ‖ > 8 9

Yut | x ‖ >| 8 t  t> + t ‖ > 8 9  (t ‖ ;)(t + >) 8 9  |

x

‖ >| 8 ; `r |



x

‖ > 8 » ;



x

8 > + ;, > ‖ ;

x

‖ >| 8 ‖ > (rec)

 x 86, ;  x 8 ?, ;

>:.

Od  8

‖; 6 

f`s

  ,  8  5 

‖; ; 

tkl



  , wjere 9 2 , 2 , tjel  ‖  os equkb t` 3 >  6 

  ;  tga¡ 9 rf  ‖  fcafc gV& 9 2 , 2 , rf   ,  8 tkl ‖;  , tga¡ >  5   6   ?   ?   ?    (;) tkl ‖;  (>) f`s ‖; (6) so l ‖;    ;=   5 ;9   5 ;9 

1dl  8

Kls.

‖; 6 

f`s

(=)

‖;

tkl

   5

   ;9 

?

(6) tkl  8 tkl  8

= 6 ; 6 = 6

 tkl ( ‖  ) 8

;

 sol( ‖ ) 8

;

‖

6

= 6

; 

8

? ;6

6

? 5 ;9



  ‖  8 sol ‖; 

 5

  ;9 

?



YKGE # ;>

| CEE (HKOL) >9;? | MK]E 3 9:-9=->9;? (VJOD]-;) | YKYE^-; | @DDOFKB YKYE^ | HK]JEHK]OFV >

>?.

  6 f`s x  sol x     my   , x   Od >y   f`t ‖; os equkb t` 3  9,  tjel  f`s x ‖ 6 so l x    > mx       >

  6 f`s x  so l x     my   , x   1dl >y   f`t ‖; fcafc gV&  9,  rf   > mx f`s x 6 s o l x ‖       (;)

x ‖



4

(>)

>x ‖

Kls.

(;)

V`b.

  6 f`s x  so l x    >y 8  f`t ‖;  f`s x ‖ 6 so l x          6  tkl x    >y 8  f`t ‖;  ; ‖ 6 tkl x               >y 8  f`t ‖; tkl   x      6     

>y 8 x> ‖



6

Kls. V`b.

(6)



6

‖ x

(=)



4

‖ x

>

>

>

      8  ‖   x    >  6  

>

>



x+

>

64

y 8 x ‖ 69.

6

>

        >y 8  ‖ tkl ‖; tkl   x      6      >    >y 8  x ‖  4  





4

Bet K klm N ne tw` l`l-lubb evelts sufj tjkt K  N . ]jel, wjofj `d tje d`bb`wolg stktehelts os kbwkys f`rreft 0 1dl K rDaa N las ,sbj vdcMr (l`l-lubb) 0aTua1sk gVk dm K  N gVK ras duEu esk bs maVuba mDau ges'aaaa bgj gV 0 (;) Y(K|N)  Y(K) (>) Y(K|N)  Y(K) (6) Y(K|N) 8 Y(N) ‖ Y(K) (=) Y(K|N)  ; (;) K  N Y( K  N) Y( K )  Y(K) (kbwkys) blV` Y(K/N) 8  Y(N) Y(N) Y(K/N) 8

Y( K ) Y(N)



Y( K )



YKGE # ;6

JEE Main 2019 Mathematics April Attempt Shift - 1(08th April, 2019)

Recommend Documents