JEE Main 2019 Mathematics April Attempt Shift - 2(12th April, 2019)

Y@YK_-0 ((O.K./O. S SKNM.)

GKK (A`dj) ?904 NLAY[SK_ O@^KC SK^S (NOS)

Pukstdljs & ^lbutdljs C`tk7 0 0? @ @prdb, ? ?904 ((^MDHS-?) || S SDAK7 9?.>9 Y.A. tl 9;.>9 Y.A.

Cur`tdlj7 > > M Mlurs || A A`x. A A`rfs7 > >89 ^[OGKNS7 A A@SMKA@SDN^

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\

_kslj`jnk Kcuvkjturks Btc. _kedstkrkc _kedstkrkc & Nlrplr`tk Lhhdnk7 NE Slwkr, @-38 & ;?, DYD@, Jk`r Ndty A`bb, Gm`b`w`r _l`c, Flt` (_`g.)->?399; Skb.Jl.7 9533-8895555, >90?099, >90????, 88>;;;; | Slbb Hrkk7 0=99 ?;= ;;;; | H`x7 +40-9??->4085??? +40-9??->4085? ?? | 9=99> 333 === Ukosdtk7 www.rkslj`jnk.`n.dj | K-a`db7 nljt`ntIrkslj`jnk.`n.dj K-a`db7 nljt`ntIrkslj`jnk.`n.dj | NDJ7 [=9>9?_G?995YBN9?39?4

| GKK (A@DJ) ?904 | C@SK 7 0?-93-?904 (^MDHS-?) | Y@YK_-? | LHHDND@B Y@YK_ | A@SMKA@SDN^

Y@_S 7 A@SMKA@SDN^ ^tr`demt Logkntdvk Sypk (

)

Smds skntdlj nljt`djs >9 aubtdpbk nmldnk qukstdljs. K`nm qukstdljs. K`nm qukstdlj m`s 3 nmldnks (0), (?), (>) `jc (3) hlr dts `jswkr, lut lh wmdnm Ljby Ljk ds Ljk ds nlrrknt. ob Vf.A ks` >9 e^`@ dz_6sc dz'u cs 3 hlcTd (0), (?), (>) rHff (3) e^`Q htuks` bs e^@

0.

Bkt h(x) 1 ; „|x „?| `jc e(x) 1 |x + 0|, x _. Dh h(x) `tt`djs a`xdaua v`buk `t  `jc e(x) `tt`djs adjdaua v`buk `t  , tmkj

kfuf

ds kqu`b tl 7 x ? „ 8x  = rHff e(x) 1 |x + 0|, x _. 6hj 6hj h(x) cf h(x) 1 ; „|x „?| rHff

bd a

b da

x  „ 

( x „ 0)( x ? „ ;x  8) x ? „ 8x  =

x „ 

@js. ^lb.

( x „ 0)(x ? „ ;x  8)

vh/fcrk kfu  rHff e(x) cf [6wurk kfu kfu  dg e^Q rfs

ngfng e^ %

(0) „>/? (?) 0/? (?) h(x) `tt`djs a`xdaua v`buk `t x 1 ? e(x) `tt`djs adjdaua v`buk `t x 1 „ 0 „ 1 ? (x „ 0)(x „ ?)(x ?)(x „ >) (?„ 0)(?„ >) 0 1 1 bda x ? (x „ ?)(x ?)(x „ 3) (?„ 3) ?

(>) >/?

(3) „0/?



?.

0

Dh tmk `rk` (dj sq. ujdts) olujckc oy tmk p`ròlb` y? 1 3x `jc tmk bdjk y 1 x,  :9, ds

4

, tmkj  ds kqu`b

tl

6hj dgly6 y? 1 3x rHff gsVff y 1x,  :9, bs bs h
(0) ?3 (0)

(?) 3=

y

? 8

>?

=

4

e^Q rfs  ngfng e^ % (3)

3 >

y 1 x T? 1 3x 3    ,3  

(9, 9)

^lb.

(>)

0

x

9

y? 1 3x `jc y 1 x Dh  : 9 tmkj 3 / 

Mkjnk @

 ?



x

 x  cx 

9

0 4

3

 ? x   >/?

>/ ?

x    ?   ?





0 4



3 >



=



>/?



=



?



0 4



>







0 4



= >

1

0 4

  1 ?3

_kslj`jnk K Kcuvkjturks B Btc. Nlrp. & _ke. Lhhdnk7 NE Slwkr, Slwkr, @-38 & ;?, DYD@, Jk`r Ndty A`bb, Gm`b`w`r _l`c, Flt` (_`g.) - >?399; Ym.Jl. 7 +40-533->90????, 88>;;;; | H@] Jl. 7 +40-9??->4085??? 7 +40-9??->4085??? Sl Fjlw alrk 7 sas _K^L `t _K^L `t ;8855 | Ukosdtk 7 www.rkslj`jnk.`n.dj | K-a`db 7 nljt`ntIrkslj`jnk.`n.dj | NDJ 7 [=9>9?_G?995YBN9?39?4

Smds sslbutdlj w w`s c clwjbl`c h h rla _ _kslj`jnk G GKK ((A@DJ) ? ?904 ^ ^lbutdlj p plrt`b

Y@EK # 0


>.

@js.

@ str`demt bdjk B `t ` cdst`jnk lh 3 ujdts hrla tmk lrdedj a`fks plsdtdvk djtkrnkpts lj tmk nllrcdj`tk `xks `jc tmk pkrpkjcdnub`r hlra tmk lrdedj tl tmds bdjk a`fks `j `jebk lh 89¹ wdtm tmk bdjk x + y 1 9. Smkj `j kqu`tdlj lh tmk bdjk B ds kwyhn[jq bs 3 ocfoW cm jwgm dg ,c bgy gsVff B hujWs'ff`c v{ffs` dg dg /fuf_kc v`r% Vf.A nufrm e^ rHff kwyhn[jq bs ob gsVff dg y`nQ gsVff x + y 1 9 cs bfHf 89· cf cfs.f nufrf e^@ rfs gsVff B cf ,c bkmcg.f e^ % (0)

> 0x 

(>) (?)

>x  y  =

> „ 0 y  = ?

(?)

> „ 0 x 

>

0y  =

?

(3) x  > y  =

x+y19 0>9¹

^lb. 89¹

Mkjnk kqu`tdlj lh bdjk ds

49¹ 3 5;¹

x nls  + y sdj  1 p x nls (5;¹) + y sdj (5;¹) 1 3



> „ 0 ? ?

  + y   

?

> „0)

+y(

+0) 1 =

x 



x( 3.

>

>

 0  1 3 ? 

 sd j x „ nls x   , wdtm rkspknt tl  sd j x  nls x 

Smk ckrdv`tdvk lh t`j „0  x ?

x ?

?



   





, wmkrk  x   9,   ds 7 ?

 

   tef¡  x    9,   cf vlcyt e^ %  , tef¡  sd j x  nls x    ?  

cs bfds{f t`j „0  

sd j x „ nls x 

(0) ?

(?) 0

@js.

(0)

^lb.

y 1 t`j „0  t`j x

(>)

0 ?

(3)

? >

 t`j x  cy  t`jx „0   0„ t`j     1 „t`j „0  1 „t`j „0  t`j  „ x   1 „  „ x   0   3  0 cx 3   0  t`j x       

Jlw dh cdhhkrkjtd`tdlj lh

x ?

w.r.t x ds

 cdhhkrkjtd`tdlj lh y w.r.t.

x ?

ds

0 0

0 ?

1?

?



Y@EK # ?


;.

@ pkrslj tmrlws twl h`dr cdnk. Mk wdjs _s. 0; hlr tmrlwdje ` cluobk (sàk juaokrs lj tmk twl cdnk), wdjs _s.0? wmkj tmk tmrlw rksubts dj tmk sua lh 4, `jc blsks _s. 8 hlr `jy ltmkr lutnlak lj tmk tmrlw. Smkj tmk kxpkntkc e`dj/blss (dj _s.) lh tmk pkrslj ds 7 0 0 0 (0) blss (?) blss (>) e`dj (3) ? e`dj ? ? 3 ,c L6hCr jfs [6f½6 (h`dr) dfbs aJfyrf e^@ ,c h}c ²jfsufs` dfbfs` dg ,c em b`V6f± vfus dg le # - 0; tmrrf e^Q jfsufs` dfbfs` dg dg vf, v` cfs` cf cf 6fsx 4 efsus dg #- 0? tmrrf e^ rHff hcbm v[6 dhg.ffk (lutnlak) dg #- 8 efgrf e^@ rfs ab L6hCr L6hCr cf dz _6fh'fr (kxpkntkc) yfMfIefhu ²#- ks`± e^ % 0

(0) efhu @js. ^lb.

?

(?) efhu

0 3

(>) yfMf

0 ?

(3) yfMf ?

(0) Umkj twl cdnk `rk tmrlwj ^àpbk sp`nk 1 {(0, 0), (?, ?) …….. (8, 8)} nljt`dj tlt`b >8 kbkakjts juaokr lh n`sks wmkj oltm oltm cdnk

Kxpknt`tdlj 1 49  3= „ 0;8 >8

8.

8 >8

1 „ x

b da x 9

x

?



? sd j x

@js. ^lb.

bda x 9

1

@js. ^lb.

bda

>8

?8

 0?

„

x

?



8

>8

0 ? ? sd j x

ds 7

?

0

sd j x „ x  0

„

ngfng e^ %

?

? sdj x  0 „

sdj x „ x  0

(0) 0 (>)

1

5.



3

x  ? sdj x

bd a x 9

 0; 

(?) >

 x  ? sdj x  x

?





x



? sdjx  0  ?

? sdj x  0 „ sd s dj x

  sdjx   0  ?  x   

x 9

?

(>) ?



x

x

?

?

sdj x „ x 

x



x„0

 ? sdj x  0 

? sd sdj x

0

(3) 8



sdj x „ x  0 ?

?

„

sdj x x

0

(0  ?)( ?)(0  0) 1 ? >

Dh ,  `jc  `rk tmrkk nljsknutdvk tkras lh ` jlj-nljst`jt E.Y. sunm tm`t tmk kqu`tdljs x? + ? x +  1 9 `jc x? + x „ 0 1 9 m`vk ` nlaalj rllt, tmkj ( + ) ds kqu`b tl 6hj ,c hMf[u djfs djfs` lfym xq.ffs Ôfg ` Gs2+m cs rmu Ýkfxr dj ,  rHff  ob dzcfg e^` hc bkmcg.ffs bkmcg.ffs` x? + ?x +  1 9 rHff x? + x „ 0 1 9 cf ,c kwy bkfu e^Q rfs ( + ) ngfng e^ % (0) 9 (?)  (>) Bkt tmk nlaalj r`tdl lh E.Y. ds r.

(>)  

(3)   1 r,  1 r?

Kqu`tdlj x? + ?x +  1 9 oknlaks x? + ?rx + r? 1 9 

x? + ?rx + r? 1 9 ……………….(d)



Y@EK # >


Edvkj kqu`tdlj (d) `jc x? + x „ 0 1 9 (?r „ 0)x +

(d) „ (dd) 

(r?+0)

……………………(dd) m`s ` nlaalj rllt.

19

x1





„ r

?



0

?r „ 0

………………(ddd)

 (r?+0)? „ (r?+0) (?r „ 0) „ (?r „ „0)? 1 9

Yuttdje (ddd) dj kqu`tdlj (dd)

 r3 „?r> „r?+?r + 0 1 9 ………………(dv)

cdvdcdje kqu`tdlj (dv) oy ?

 0    r „ „ 0  9  r  

r

?

r?

 r? „ ?r „ 0

„0 r



0

Jlw, ( + )  (r + r?)  (r) (r?)

=.



r

?

? + + r

 0 r

0 r

……………..(v)

 ?r(0 + r)

 ?r . r?

 

 0 Smk tkra djckpkjckjt lh x dj tmk kxp`jsdlj lh  „   89  0  „   89

  ? . ?x „ =0   

x

=

  ? . ?x „   =0 

x

=

8

  ds kqu`b tl 7 ? x  >

8

 %  cs dzbfg ks` x bs Blr`1f dj eê^ % ? x  >

@js.

(0) „ >8 (0)

^lb.

Skra Djckpkjckjt lh x 1

4.

Bkt @,O `jc N ok skts sunm tm`t   @  O (0) O  N  

(?) „5?

(>) >8

(3) „09=

0  0  ?>  (>)> 8 N>     (?)( >) ; 8N 0 1 „5? + >8 1 „>8 89  =0 



N. Smkj wmdnm lh tmk hlbblwdje st`tkakjts ds jlt truk< (?) Dh ( @ „ O)  N, tmkj@  N

(>) (N  @ )  (N  O)  N

kfuf bkqYp6 @, O rHff N ob dzcfg e^` hc hc

@js.

?

?

 0  0 19   r   „?  r „  +01 9 r  r 

(3) Dh ( @ „ N)  O tmkj

@ O

(0) O  N  

N, rfs rfs huKu ks` bs cfû bf cHfu cHfu (?) 6hj (@ „ O)  N, rfs rfs @  N

(>) (N  @ )  (N  O)  N

rfs @  O (3) 6hj ( @ „ N)  O , rfs

@O



b_6 uem` e^ R

(0) N

^lb.

@

O

Hrla tmk hdeurk, dt ds nbk`r tm`t O  N  

09. 09.

Bkt   _ `jc tmk tmrkk vkntlrs



Y@EK # 3


`   d  g  >f, o  ?ˈd  ˈg „ fˈ 

ˈ

ˈ

ˈ

`jc n  ˈd „ ? gˈ  >fˈ . Smkj tmk skt ^ 1 { 7 `,o `jc n `rk nlpb`j`r} (0) nljt`djs kx`ntby twl plsdtdvk juaokrs (?) ds kapty (>) nljt`djs kx`ntby twl juaokrs jua okrs ljby ljk lh wmdnm ds plsdtdvk (3) ds sdjebktlj 







rmu bhj'f `   d  g  >f, o  ?ˈd  gˈ „ `fˈ rHff ^ 1 { 7 `, o rHff n bkrym6 e^@` } (0) ks` rH6r% rH6r% (kx`ntby) jfs /fuf_kc b`V6f,s e^ ` e^@ (?) hgCr e^@ 6f6s` e^ htbks` bs csly ,c /fuf_kc e^@ (>) ks` rH6r% jfs b`V6f6s (3) ,cy e^@

kfuf

ˈ



  _ rHff



ˈ

ˈ



n

ˈ

d

  „

? gˈ  >fˈ e^`@

rfs bkqYp6





@js. ^lb.



(?) ^djnk vkntlrs `rk nlpb`j`r 

0

>

?

0

„



„?

>

1 9

(> „ ?) „ 0 (8 + ?) + > ( „ „3 „ ) 1 9

00.

@js.

 „>? „ 0= 1 9

 ? 1 „ 8

Dh ?9N0 + (??) ?9N? + (>?) ?9N> + …………. + (?9 ?)?9N?9 1 @(?), tmkj tmk lrckrkc p`dr (@, ) ds kqu`b tl 6hj ?9N0 + (??) ?9N? + (>?) ?9N> + …………. + (?9 ?) ?9N?9 1 @(?), rfs rfs Ýhkr 6q]k (@, ) ngfng e^ % (0) (>=9, 0=) (3)

(?) (>=9, 04)

(>) (3?9, 04)

(3) (3?9, 0=)

?9

^lb.

^1

0? ?9N0 +

??. ?9N

? +

>? ?9N> + …………………+?9?. ?9N?9 1

r

? ?9

.

Nr

r 0 ?9

1

?9

 r.r.

?9

Nr



r 0

1 ?9

?9

 r.

04

Nr„0

1 ?9

r 0

?9

?9

 r „ 0  0

04

Nr „0 1

?9

r 0

 r „ 0

04

Nr „0 +

?9



04

Nr „0

r 0

r 0

?9

1 ?9 Õ 04



0=

04

Nr „?  ?9  ?

1 ?9 Õ 04 Õ ?0= + ?9 Õ ?04 1 ?9 Õ ?0= (04+?)1 ?9 Õ ?0 Õ ?0=

r ?

1 3?9 Õ ?0= ?

sd j 

3 nls 8

nls 

?

3 nls 8

0  nls 

0?.

@ v`buk lh  (9,  />), hlr wmdnm

?

0  sd j 

?

sd j 

nls 

 (9,  />) cf

,c kfuQ htbcs hy,

@js.

>8

(?)



0=

3 nls 8

nls? 

0  sd j ? 

3 nls 8

?

sd j  (>)

9

, ds 6

0  3 nls 8

sd j ? 

nls  5

?

0  nls?  ?

(0)

?

5 ?3

 9 e^Q e^ %

0  3 nls 8

(3)



4

(3)



Y@EK # ;

| GKK (A@DJ) ?904 | C@SK 7 0?-93-?904 (^MDHS-?) | Y@YK_-? | LHHDND@B Y@YK_ | A@SMKA@SDN^ ?

sdj 

?

3 nls 8

nls 

?

3 nls 8

0  nls 

^lb.

?

0  sdj 

?

?

nls 

sdj 

9

0  3 nls 8

_0  _0 „ _?, _?  _? „ _> 

0

0

9

9

0

0

?

nls 

?

sdj 

9

0  3 nls 8

N?  N? + N0 0

9

9

9

0

0

0

0  3n 3 nls 

?

nls 

9

 0 + 3 nls8 + 01 9  ?nls8 1 „0  nls8 1   1

0>.

j >





4

,j  D   1

0 

?

1 nls

?  >

 8 1 ?j



? >

? 3 , 4 4 4 

,

Hlr `j djdtd`b snrkkjdje lh `j `cadssdlj tkst, ` n`jcdc`tk ds edvkj hdhty prlobkas tl slbvk. Dh tmk prloòdbdty tm`t tmk n`jcdc`tk n`j slbvk `jy prlobka ds

3 ;

, tmkj tmk prloòdbdty tm`t mk ds ujòbk tl slbvk

bkss tm`j twl prlobkas ds 7

dzfg` hMfc tf¡p cs hy, ,c dzls'f dgm{ff ks` ,c dgm{ffHfmW dgm{ffHfmW cfs dpfb dz'u ey cgus cs hy, hj, x, e^`@ 6hj dgm{ffHfmW cs hcbm ,c dz'u cfs cfs ey cg bcus cm dzfh6crf

3 ;

e^Q rfs abcs jfs bs ck dzdz'ufs` cfs ey cgus vbkHfW vbkHfW efsus cm

dzfh6crf e^ % 34

@js. ^lb.

>08  3  ;3  3  (0) (?)     ?;  ;  ; ; (0) Smkrk `rk ;9 qukstdljs dj `j kxà.

3=

(>)

Yrloòdbdty lh k`nm qukstdlj tl ok nlrrknt 1 p 1

prloòdbdty lh k`nm qukstdlj ds djnlrrknt 1 q 1

083  0  ?;

   ; 

3=

(3)

?90  0  ;

34

   ; 

3 ;

0 ;

Bkt ] 1 juaokr lh nlrrknt qukstdlj dj ;9 qukstdljs. Mkjnk rkqudrkc prloòdbdty 1 ( 3 1   ;

34

3 ;9 ;9 ) + N0 ;

3   3  ;3  3   ;  09   ;  1 ;  ;      

 0  3  ; ;    

34

34



Y@EK # 8


03. 03.

bkt  (9,  /?) ok hdxkc. Dh tmk djtker`b

t`j x  t`j 

 t`j x „ t`j cx  @(x) nls? + O(x) sdj? + N, wmkrk N ds `

nljst`jt lh djtker`tdlj, tmkj tmk hujntdljs @(x) `jc O(x) `rk rkspkntdvkby. (0) x +  `jc blek |sdj(x „)| (?) x „  `jc blek |sdj(x „)| (>) x +  `jc blek |sdj(x + )| (3) x +  `jc blek |nls(x „)|

kfuf  (9,  /?) hj6f e^@ 6hj bkfcy

t`j x  t`j 

tef¡ N ,c bkfcyu  t`j x „ t`j cx  @(x) nls? + O(x) sdj? + N, tef¡

vpg e^Q rfs Pyu @(x) rHff O(x) Ýk'f% e^` % (0) x +  vf^g blek |sdj(x „)| (>) x +  vf^g blek |sdj(x + )| @js.

(0,?) lr (Olujs)

^lb.

 t`j  D 1  cx 1 t`jx „ t`j 

sdj x t`j x

 nsldjsxx nls x

1



 „

(?) x „  vf^g blek |sdj(x „)| (3) x +  vf^g blek |nls(x „)|

sdj  nls  cx 1 sdj 

sdj( x  )

cx 1   sdj(x j(x „  )

sdj( x „   ? ) cx sdj(x „ )

nls 

sdj(x „ )nl )nls?  nls(x „ )sd )sdj? cx 1 nls? cx  sdj? nlt(x „ )cx sdj(x „ )





1 x nls ? + sdj ?  j |sdj(x „ )| + n Mkjnk Lptdlj 0 `jc ? oltm `rk nlrrknt 0;.

@ v`buk lh  sunm tm`t  0

 

cx  4   b lek   ds 7 ( x  )( x    0)  = 

 cf  0

 

kfuQ htbcs hy,

cx  4   b lek   e^Q ( x  )( x    0)  = 

(0) „? @js.

(?) „

?

(>)

0

(3) ?

?

(0)  0

^lb.

0

e^ %

 1

cx

 ( x )(x   0) 

Bkt x +  1 t cx 1 ct ?  0

 1



?

1

? 0

ct t(t t(t  0)

?

 ?  0    ?  ? 

j

(?  0)?  (?  ?)





1

0  0  „   cx 1  t t 0

 ?   1  ?  0 

j



jt „ j(t  0) |??

 ?   0 ?  0 j   ?   ?  ?



 0 

 t  ? 0  |?  t  0

1 j

4  =

j

4 3

3(3? + 0 + 3) 1 0=? + 0=



Y@EK # 5


?? + ? „ 3 1 9 ? +  „ ? 1 9

( + ?)( „0) 1 9  1 0 lr  1 „ ?

v`buk lh  ds „ ? 08.

@js. ^lb.

Smk kqu`tdlj lh ` nlaalj t`jekjt tl tmk nurvks y? 1 08x `jc xy 1 „3, ds lÝf`s y? 1 08x rHff xy 1 „3 cm ,c aMf6hu"O Bd'fWgsVffff cf bkmcg.f e^ % (0) x „ ?y + 08 1 9 (?) ?x „ y + ? 1 9 (>) x + y + 3 1 9 (3) x „ y + 3 1 9 (3) Bkt kqu`tdlj lh t`jekjt tl p`ròlb` y? 1 08x ds y 1 ax + Dt ds t`jekjt tl xy 1 „3



3



3



a

hlr t`jekjt C 1 9 

08 a

a

………(0)

………(?) ^lbvdje (0) & (?), wk wk ekt

x  ax   + 3 1 9  ax? + a



3

?

x+319

„ 08a 1 9  a> 1 0  a 1 0

 Yuttdje a 1 0 dj Kqu`tdlj (0)  Nlaalj t`jekjt ds y 1 x + 3

05.

Smk t`jekjts tl tmk nurvk y 1 (x „?)? „ 0 `t dts pldjts lh djtkrsntdlj wdtm tmk bdjk x „ y 1 > djtkrsknt `t tmk pldjt lÝ y 1 (x „?)? „ 0 cs gsVff x „ y 1 > bs dzhrYJsju hn[jqvfs` dg lÝ cm Bd'fW Bd'fW gsVff6s` huKu ks` bs bs hcb hn[jq dg

hkyrm e^` R  ;  ,„0  ? 

 ;

(0)  „ @js. ^lb.

(?) 

 ?



0

,„



 ;  ,0  ? 

(>)  „

 ; 

(3)  ,0

 ? 

(?) Bkt nllrcdj`tks lh N ok (m, f) nmlrc lh nljt`nt lh N w.r.t. y 1 (x „?)? „0  y 1 x? „3x + > ds S 1 9 yf 1 xm + > „ ?(x+m) ?  ?(m „?)x „ y 1 „(8 „3m „f)

nlap`rk dt wdtm x „ y 1 > ? (m „?) 1 0  m1

; ?

8 „ 3m „ f 1 „ >  f 1 „0

; ?

 

 N ds  , „0



Y@EK # =


0=.

@js. ^lb.

Smk ekjkr`b slbutdlj lh tmk cdhhkrkjtd`b kqu`tdlj (y? „x>) cx „ xycy 1 9 (x  9) ds 7 (wmkrk n ds ` nljst`jt lh djtker`tdlj) vlcy bkmcg.f (y? „x>) cx „ xycy 1 9 (x  9) cf L6fdc ey e^ % ( n n ,c bkfcyu vpg e^ ) tef¡ (0) y? + ?x? + nx> 1 9 (?) y? „ ?x? + nx> 1 9 (>) y? „ ?x> + nx? 19 (3) y? + ?x> + nx? 1 9 (3) y  ycx „ xcy  (y? „ x>)cx „ xycy 1 9  y(ycx „ xcy) 1 x>cx    1 cx x  x?  ?

y  y  0  y   „ c   1 cx  „   1 x + f  „ y? 1 ?x> + ?x?f  y? + ?x> + nx? 1 9 x  x  ?  x 

04.

Bkt z N wdtm Da(z) 1 09 `jc dt s`tdshdks (0) j 1 ?9 `jc _k(z) 1 09 (>) j 1 39 `jc _k(z) 1 09

?z „ j ?z  j



?d „ 0hlr

slak j`tur`b juaokr j. Smkj (?) j 1 ?9 `jc _k(z) 1 „09 (3) j 1 39 `jc _k(z) 1 „09

kfuf z N htbcs hy, Da(z) 1 09 rHff hcbm dzfÐr b`V6f j cs hy, 6e

@js. ^lb.

(0) j 1 ?9 rHff _k(z) 1 09 (>) j 1 39 rHff _k(z) 1 09 (?) Bkt _k (z) 1 x, tmkj ?(x  09d) 9d)  j ?(x  09d) 9d)  j



?z „ j ?z  j



cfs b`rq"X cgrf e^Q rfs

?d „ 0

(?) j 1 ?9 rHff _k(z) 1 „09 (3) j 1 39 rHff _k(z) 1 „09

?d  0  (?x „ j) + ?9d 1 „(?x + j) „ 39 „ ?9d + ?jd

 ?x „ j 1 ?x „ j „ 39 & ?9 1 „?9 + ?j  x 1 „ 09 & j 1 ?9

?9.

@ trd`jebk m`s ` vkrtkx `t (0, ?) `jc tmk adc pldjts lh tmk twl sdcks tmrluem dt `rk „ („0, 0) `jc (?, >). Smkj tmk nkjtrldc lh tmds trd`jebk ds 7 „0, 0) vf^g (?, >) e^`@ ,c h1fMfq t cf ,c 'fm"fW (0, ?) dg e^ rHff obbs obbs efs cg tfus lfym jfs Mfqtfvfs` cs cs k/6 hn[jq ( „

rfs

ob h1fMfqt cf cs[jzc e^ %  5 

 0 ; 

(0) 0, 

(?)  , 

 > 

@js.

 > > 

 0  (>)  ,0  > 

 0   > 

(3)  ,? 

(3) (0,?)

^lb.

( „0,0)

( „>,9)

(?,>)

(>,3)

 0 > „ > ?  9  3  , 1 >  > 

nkjtrldc 1 

 0   ,?   > 



Y@EK # 4


?0.

Smk bkjetm lh tmk pkrpkjcdnub`r cr`wj hrla tmk tmk pldjt (?, 0, 3) tl tmk pb`jk nljt`djdje tmk bdjks

d  g d  ? g f `jc r  d  g  d  g ?f ds 7 gsVffvfs` r  d  g  d  ? g f rHff r  d  g   d  g ?f  cfs v`rhlW"X cgrs bkry bkry dg hn[jq (?, (?, 0, 3) bs 

r

ˈ



ˈ

ˈ

ˈ



ˈ

ˈ

„

ˈ

ˈ

ˈ



ˈ

„

ˈ

ˈ

„

ˈ



ˈ

ˈ

ˈ

ˈ

„

„

ˈ

ˈ

ˈ

„

Afys x6s yKn cm yKnfoW yKnfoW e^ % (0) @js. ^lb.

(?) >

>

(>)

0 >

(3)

0 >

(0) Jlra`b vkntlr lh tmk pb`jk ds ˈd

ˈg

fˈ

0

?

„0

„0

0

„?

1

„ >d + > g + >f ˈ

ˈ

ˈ

jlw kqu`tdlj lh pb`jk „>x + >y + >z 1 n (0,0,9) wdbb s`tdshy tmk pb`jk  „ > + > + 9 1 n  n 1 9  „>x + >y + >> 1 9 cdst`jnk hrla (?,0,3) 1

??.

„8  >  0?

?5

1

4 > >

1

>

ujdts

@ erlup lh stuckjts nlaprdsks lh ; olys `jc j edrbs. Dh tmk juaokr lh w`ys, dj wmdnm wmdnm ` tkà lh > stuckjts n`j r`jclaby ok skbkntkc hrla tmds erlup sunm tm`t tmkrk ds `t bk`st ljk oly `jc `t bk`st ljk edrb dj k`nm tkà, ds 05;9, tmkj j ds kqu`b tl 7 hl|fhHfW6fs` cs cs ,c bkwe ks` ; yA+cs rHff j yA+hc6f¡ e^`@ 6hj ob bkwe ks` bs bs rmu hl|fhHfW6fs` cm Xmk 6fjihYJc ob dzcfg pquus cs rgmcsQ hc dz _6sc Xmk ks` ck bs ck ,c yA+cf rHff ck bs ck ,c yA+cm efsQ 05;9 e^`Q rfs j

ngfng e^ % @js. ^lb.

(0) ?5 (>) Edvkj ; Oly, j edrbs.

(?) ?3

(>) ?;

(3) ?=

(0O, ?E) + (?O, 0E) ;N

0.

jN

? +

;N

j ? . N0 1

05;9  ;.





j j„0 ?

+ 09.j 1 05;9 





j j„0

?



?j  >;9  j? „ j + 3j 1 599

j? + >j „ 599 1 9  (j + ?=) (j „ ?;) 1 9  j 1 ?;, „?= ?>.

Smk `jebk lh kbkv`tdlj lh tmk tlp lh ` vkrtdn`b tlwkr st`jcdje lj ` mlrdzljt`b pb`jk ds loskrvkc tl ok 3;· hrla ` pldjt @ lj tmk pb`jk. Bkt ok tmk pldjt >9 a vkrtdn`bby òlvk tmk pldjt @. Dh tmk `jebk lh kbkv`tdlj lh tmk tlp lh tmk tlwkr hrla O ok >9·, tmkj tmk cdst`jnk (dj a) lh tmk hllt lh tmk tlwkr hrla tmk pldjt @ ds7 {f^hry ry dg VfA+m ,c a/lfW/fg kmufg cs h'fVfg cf ry dg ,c hn[jq @ bs a[u6u cfs.f 3;· e^@ kfuf hn[jq @ bs >9 kmXg a/lfW/fg Ëdg hn[jq O e^@ 6hj O bs kmufg cs h'fVfg hn[jq O h'fVfg cf a[u6u cfs.f >9· e^Q rfs kmufg cs dfj cm cm hn[jq @

bs jwgm ²kmXg ks`± e^ % @js.

(0) (0)



0; > 

>



(?)



0; 0 

>



(>)



0; ; „

>



(3)



0; > „

>





Y@EK # 09

| GKK (A@DJ) ?904 | C@SK 7 0?-93-?904 (^MDHS-?) | Y@YK_-? | LHHDND@B Y@YK_ | A@SMKA@SDN^ O

^lb. x >9·

N

C

c >9

>9 3;·

@

x 1 t`j>9¹ 6 c

@bsl

c1

x  >9 c

x c

1

0 >

>

6 c1

 t`j3; 

c

+ >9   0 



c 1 0; (> + ?3.

K

c

>

>

x

 c 1 x + >9

  c 1 >9  c 1 > 

0

>9 > >

0

 c1

>9 > ( > ?



0)

1 0; > ( > +0)

)

Dh VxQ ckjltks tmk erk`tkst djtkekr  x, tmkj tmk systka lh bdjk`r kqu`tdljs VsdjQx + V „ „nlsQy 1 9 VnltQx + y 1 9

  ?   5   `jc m`s ` ujdquk slbutdljs dh    ,  .  ? >   8    ?   5  (?) M`s ` ujdquk slbutdlj dh    ,    ,  .  ? >   8    ?   5  (>) M`s ` ujdquk slbutdlj dh    ,  `jc m`vk djhdjdtkby a`jy slbutdljs dh    ,  .  ? >   8    ?   5  (3) M`vk djhdjdtkby a`jy slbutdljs dh    ,    ,  .  ? >   8  (0) M`vk djhdjdtkby a`jy slbutdljs dh    ,

6hj VxQ keÔfk keÔfk dw.ffW`c  x e^Q rfs g^hVfc bkmcg.f bkmcg.f hucf6 VsdjQx + V „ „nlsQy 1 9 VnltQx + y 1 9 (0) cs

 ?   5  ,  rHff kf1f ,c ey e^ 6hj    ,  .  ? >   8 

 vu[r ey e^` 6hj   

 ?   5  ,    ,  .  ? >   8    ?   5  (>) cf kf1f ,c ey e^ 6hj    ,  rHff vu[r ey e^` 6hj    ,  .  ? >   8    ?   5  (3) cs vu[r ey e^` 6hj    ,    ,  .  ? >   8  (?) cf

@js.

,c kf1f ey e^ 6hj    

(0)



Y@EK # 00


^lb.

Dh

  ?    ,  ? > 



 

5 



8 



  nls   „0,

 0  ?

 

sdj   „ ,9 

  ?   ? > 

Mkjnk dj   ,



dj   ,

5 

 

8 



0 >

,9 



„ > 

 >  ,0   „ nls     ?   ? 

&

Mkjnk 9 1 9 & „x + y 1 9

 

?

 nlt   „ 

   ,

 

0

 >  ,0  ?  &  

sdj   

Dh

 

nls   „ ,9   „ nls  (9,0/?)

nlt  (

VsdjQ 1 96

>

, )

„ V„nlsQ 1 96

VnltQ 1 „0

M`vk djhdjdtkby slbutdljs VsdjQ 1 „ 0 6 V „ „nlsQ 1 96 VnltQ 1 0, ?, >, ……………….

Mkjnk „x „ 9.y 1 9



x19

Vnlt Q x + y 1 9 

0.x + y 1 9

lr

?.x + y 1 9 lr ………………..

k`nm bdjk wkbb nut x 1 9 `t kx`ntby ljk pldjt. Mkjnk ujdquk slbutdlj.

?;.

Dh `0, `?, `>,........... `rk dj @.Y. sunm tm`t `0 + `5 + `08 1 39, tmkj tmk sua lh tmk hdrst 0; tkras lh tmds @.Y. ds 7 6hj `0, `?, `>,........... ,c ,c bkf[rg Gs2+m ks` ob dzcfg e^` hc `0 + `5 + `08 1 39 e^`Q rfs ob bkf[rg Gs2+m cs dzHfk 0; djfs` cf 6fsxPy e^ %

@js.

(0) ?=9 (>)

(?) 0?9

^lb.

`0 + `5 + `08 1 39  `0 + (`0 + 8c) + (`0 + 0;c) 1 39  >`0 + ?0c 1 39

`0 + `? ……… + `0;  ?8.

@js. ^lb. ?5.

0; `0  `0;   ?

(>) ?99

0; ?

`

0



`0

 03c



(3) 0;9  `0 + 5c 1

 0;(`0 + 5c) 1 0; 

39 >

39 >

1 ?99

Smk Ollbk`j kxprkssdlj ~(p  (~q)) ds kqudv`bkjt tl nwys cf L6`tc ~(p  (~q)) huKu ks` bs hcbcs bkrqT6 e^ R (0) (~p)  q (?) p  q (>) ~ (p  ~q)  ~(~pq)  p  q

(>) p  q

(3) q  ~ p

Bkt ^ ok tmk skt lh `bb   _ sunm tm`t tmk kqu`tdlj, nls?x + sdjx 1 ? „ 5 m`s ` slbutdlj. Smkj S mkj ^ ds kqu`b tl 7 kfuf bMfm   _, htbcs htbcs hy, bkmcg.f nls?x + sdjx 1 ? „ 5 cf ,c ey e^Q cf bkqYp6 ^ e^@ rfs ^ ngfng

e^

% (0) V0, 3Q

(?) V>, 5Q

(>) V?, 8Q

(3) _



Y@EK # 0?


@js. ^lb.

(>) 0 „ ?sdj?x +  sdj x 1 ? „5  ? sdj?x „  sdj x + (? „ =) 1 9 

sdj x 1 1

3

3

 

?

  =(?  =)

?

1

   ?  08  83 3

1

, ? (_kgkntkc), hlr slbutdlj tl ok kxdst „0 

  (  =) 3  

?

3

1

?  = 3

,?

 0  „?   „ 3  ?  ?    8

   V?, 8Q

?=.

@ ndrnbk tlunmdje tmk x-`xds `t (>, 9) `jc a`fdje `j djtkrnkpt lh bkjetm = lj tmk y-`xds p`ssks tmrluem tmk pldjt 7 x-v{f cfs (>, 9) dg Bd'fW cgrf eqvf rHff y-v{f dg = yKnfoW cf (djtkrnkpt) nufrf eqvf ,c liÔf huKu cf v`r% Vf.A (djtkrnkpt)

ks`

bs hcb hn[jq bs efs efscg tfrf e^ R R @js.

(0) (?, >) (?)

(?) (>, 09)

=

(>) (0, ;)

(3) (>, ;)

> (>, ;) 3

^lb.

;

;

>, 9

Kqu`tdlj lh ndrnbk ds (x „ >)? +(y „ ;); 1 ;? Mkjnk (>, 09) wdbb s`tdshy kqu`tdlj. ?4.

@j kbbdpsk, wdtm hlnd `t (9, ?) `jc (9, „?) `jc adjlr `xds lh bkjetm 3, p`ssks tmrluem wmdnm lh tmk hlbblwdje pldjts< ,c jm
hn[jq

bs efscg tfrf e^ R @js.

(0) (0)

^lb. ^lb.

Bkt

?,?

x? `?



y? o?

(?) ?, ?



(>) 0, ? ?

(3) ?, ? ?

0(` 2 o) ds tmk kqu`tdlj lh kbbdpsk, hlndd (9, °?)

(ok 1 ?) Edvkj7 ?` 1 3  ` 1 ? k? 1 0 „

`

?

o

?

 o?k? 1 o? „ `?

o? „ 3

31 o? 1 =



Y@EK # 0>


 kqu`tdlj lh kbbdpsk ds

x? y?  0 3 =

 Dt p`ssks tmrluem ( ? , ?)

>9.

@ pb`jk wmdnm odsknts tmk `jebk oktwkkj tmk twl edvkj pb`jks ?x „ y + ?z „ 3 1 9 `jc x + ?y + ?z „ ? 1 9, p`ssks tmrluem tmk pldjt 7 jfs hj, x, bkryfs bkryfs` ?x „ y + ?z „ 3 1 9 rHff x + ?y + ?z „ ? 1 9 cs nmp cs cfs.f cfs bkh}Mffhtr bkh}Mffhtr

cgrf ,c

bkryQ huKu ks` bs hcb hn[jq bs bs efscg tfrf e^ R @js. ^lb.

(0) (?, „3, 0) (?) (0, „3, 0) (0) Yb`jks odskntdje tmk edvkj pb`jks `rk ?x „ y  ?z „ 3 x  ?y  ?z „ ? 1 ° >

(>) (0, 3, „0)

(3) (?, 3, 0)

>

 x „ >y 1 ? lr >x + y + 3z 1 8 lut lh tmk edvkj pldjts, ljby (?, „ „3,0) bdks lj tmk pb`jk >x + y + 3z 1 8.



Y@EK # 03

JEE Main 2019 Mathematics April Attempt Shift - 2(12th April, 2019)

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