JEE Main 2019 Paper 2 B.Arch Attempt Shift - 1(07th April, 2019)

ZMZC^-> ((N.MÎG.)

BCC (Dmlo) >:=< IEDZSQC^ NMYCA QCYQ (INQ) Amtc2 : :6 M Mprlh, > >:=< ((YGL@Q-=) | | QLDC 22 (<.9: m.d. te =>.9: p.d)

Aurmtleo2 9 9 G Geurs | | D Dmx. D Dmrks2 9 9<:

UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUU

^cseomoic Cauvcoturcs Hta. ^cflstcrca & Ierpermtc E``lic2 IF Qewcr, M-10 & 3>, LZLM, Ocmr Ilty Dmhh, Bgmhmwmr êma, Ketm (^mb.)-9>1::3 Qch.Oe.2 :611-00:6666, 9:=>=::, 9:=>>>>, 0093333 | Qehh @rcc2 =7:: >37 3333 | @mx2 +<=-:>>-9<=06>>> | :7::9 111 777 Vcnsltc2 www.rcseomoic.mi.lo | C-dmlh2 ieotmitJrcseomoic.mi.lo C-dmlh2 ieotmitJrcseomoic.mi.lo | ILO2 S7:9:>^B>::6ZHI:>1:><

| BCC DMLO->:=< | AMQC 2 - :6 :1->:=< (YGL@Q-=) | ZMZC^-> | N.MÎG | DMQGCDMQLIY

ZM^Q 2 DMQGCDMQLIY Ytrmlfgt Enbcitlvc Qypc hg/ks ( eHrqù"N lzakb) Qgls scitleo ieotmlos 9: duhtlphc igelic qucstleos. Cmig qucstleos. Cmig qucstleo gms 1 igelics (=), (>), (9) moa (1) èr lts moswcr, eut e` wglig Eohy Eoc ls Eoc ls ierrcit. nh [k.D csm 9: ifq&èa\lg lz 'u fY 'u fYmM lz^;sa lz'u as 1 èa\l (=), (>), (9) r@kk (1) fYmT `tucsm hs `h]R `h]R ,a hfg fY hfg fYM =.

L` m, n moa i (mhh alstmoic) mrc tgc slacs e` m trlmofhc MNI eppesltc te tgc mofhcs M, N moa I, rcspcitlvchy, tgco

Mos. Yeh.

i slo( M  N) >

m n

>



n slo(I  M) i >  m>

ls cqumh te 2

(=) = (>) > (9) ’ (9) ’= = (1) I  sloM sloM iesN iesN ’ iesM sloN loN  n  sloIiesM sloIiesM ’ iesIsloM iesIsloM  ’ m> ’ n> i > ’ m>

4

4

i >^ = 1^

 m  m>  i > ’ n>  n  n>  i > ’ m>     ’   >mi >ni   ’ >

m ’n

>

(1) :

n

 I  n>  i > ’ m>  m  m >  n> ’ i >     ’   >ni >mn  

>^

i > ’ m>

 m>  i > ’ n > ’ n > ’ i >  m > n >  i > ’ m > ’ m > ’ n >  i >  ’   4 > > m ’ n i > ’ m>  

= 1^

[> ’ [> ’ >T >T 4 :

Hct Z nc tgc pelot eo tgc pmrmnehm y> 4 9x suig tgmt EZ dmkcs mo mofhc e`

>.

 0

wltg tgc x-mxls, wgcrc E

ls tgc erlflo. M oerdmh ls armwo te tgc pmrmnehm mt Z lotcrscitlof tgc mxls, e` tgc pmrmnehm mt ]. L` Y ls tgc èius e` tgc pmrmnehm, tgco Y] ls cqumh te 2 (=) Mos.

9< 1

(>)

9< >

(9) <

(1)

1= 1

(=)

9 9  Z  t> , t  1 > 

Yeh.

9  Y  ,:  1  1m49 

m4

9 > 9  t , t 1 >   9t   1  >   9t >  

9 1

Hct Z  

> = 4 t 9



 = 4 9  t4 > 9

^cseomoic Cauvcoturcs Hta. IE^ZE^MQC E@@LIC 2 IF Qewcr, M-10 & 3>, LZLM, Ocmr Ilty Dmhh, Bgmhmwmr êma, êma , Ketm (^mb.) - 9>1::3 ^cf. E``lic 2 B->, Bmwmgmr Omfmr, Dmlo êma, Ketm (^mb.)-9>1::3 | Zg. Oe.2 +<=-611-9=<>>>> | @MW Oe. 2 +<=-:>>-9<=06>>> 2 +<=-:>>-9<=06>>> Zg.Oe. 2 +<=-611-9:=>>>>, 0093333 | Qe Koew derc 2 sds ^CYE mt ^CYE mt 30066 Vcnsltc 2 www.rcseomoic.mi.lo | C-dmlh 2 ieotmitJrcseomoic.mi.lo ieotmitJrcseomoic.mi.lo| | ILO 2 S7:9:>^B>::6ZHI:>1:>< Qgls sehutleo wms aewohema `red ^cseomoic BCC DMLO >:=< Yehutleo Yehutleo pertmh

ZMFC # =


9 9   1  9,  > 9  > 1 

Zelot Z 

Z (<, 9 9 ) cqumtleo e` oerdmh ls

y ’9 ’9 9 4 ’ 4 ’t(x t(x ’ ’ <) <) y ’ 9 ’ 9 9 4 ’ 4 ’> > 9 (x ’ (x ’ <) <)

@er Zt ], Zut y 4 :

9 >

4 x ’ x ’ < <



x4<+

9 >

4

9< >= >= 9  Y] 4 ’ 4 . 1 > > 1

L` 0 ies> ’ >ies> ’ >ies> ’ 9 ’ 9 4 :, tgco tmo >9 ls cqumh te 2

9.

(=)

=

(>) 9

9

(9)

Mos.

(1)

Yeh.

0 ies> ’ >(>ies ’ >(>ies> ’ =) ’ =) ’ ’ 9 9 4 :  slo> 4 = ’



= >

4

= >



ies> 4

<

(1) =

>

= >

tmo> 4 =



tmo 4  =

> 9tmo 9tmo  ’ tmo tmo 9  tmo  9 ’ tmo   tmo 9 4 4 =’ 9tmo >  =’ 9tmo 9tmo>  =9 ’= tmo 9 4

=’ 9 tmo 9 4  = tmo>9 4 =

1.

Mos. Yeh.

L` tgc systcd e` hlocmr cqumtleos x + 1y ’ 1y ’ 9z 9z 4 > >x + 6y ’ 6y ’ 1z 1z 4  ’x ’x ’ 3y ’ 3y + 3z 4  gms lo`loltchy dmoy sehutleos, tgco tgc eracrca pmlr ( , ) imooet tmkc tgc vmhuc 2 (=) (9, ’ (9, ’9) 9) (>) (>, ’ (>, ’1) 1) (9) ( ’9, ’9, 9) (1) (1, ’ (1, ’>) >) (9) = 1 ’9 : ’= > A4 >

6

’ 1 4 : ’9 0 4 ’ 4 ’ 0 0 + 0 4 : 3 ’= ’3 3

’= ’3

>

1

’9

A= 4 

6

’1 4 > (93 ’ (93 ’ >:) >:) ’ ’  ( >: ’ >: ’ =3) =3) +  ( ’=0 ’=0 + >=) 4 :  9: ’ 9: ’ 3 3  + 3 4 :   ’  4 0  ’3 3 =

>

’9

A> 4 >

 ’1 4 ’> ’> (=: ’ (=: ’ 1) 1) ’ ’  ( 3 ’ 3 ’ 9) 9) ’ ’  ( ’1 ’1 + 0) 4 : ’=  3

  

’=> ’=> + > ’ > ’ > 4 :  ’  4 0 (hcmvcs ( ’9, ’9, 9) aecs oet smtls`y  ’  4 0 (,  )  ( ’9, ’9, 9)


ZMFC # >


3.

Mos. Yeh.

0.

Mos. Yeh.

Hct M nc tgc sct e` mhh 9-alflt omturmh oudncr moa N 4 {x  M 2 G.I.@. (x, =3) 4 =}. Qgco tgc oudncr e` chcdcots lo N ls 2 (=) 1>: (>) 17: (9) 90: (1) >1: (9) M 4 {=::, =:=, =:>, ………………<<<} oudncr alvlslnhcny 9 4 =:>, =:3, ………..<<< tetmh 4 9:: oudncr alvlslnhc ny 3 4 =::, =:3, ==:, ……<<3 tetmh 4 =7: oudncr alvlslnhc ny =3 4 =:3, =>:, …………..<<: tetmh 4 0:  tetmh oudncr wglig mrc alvlslnhc ny 9 er 3 4 9:: + =7: ’ =7: ’ 0: 0: 4 1>:  o(N) 4 <:: ’ <:: ’ 1>: 1>: 4 17: Hct M (=, 9) moa I(3, =) nc twe eppesltc vcrtlics e` m rcitmofhc. Qgc etgcr twe vcrtlics N (m, n) moa A(i, a) hlc eo tgc hloc y 4 >x + k èr sedc k. Qgco tgc vmhuc e` (m + n) (i + a) ls 2 (=) 7 (>) >1 (9) 9> (1) =0 (1) (m, n) & (i, a) hlcs eo ilrihc (x ’ (x ’ =) =) (x ’ (x ’ 3) 3) + (y ’ (y ’ 9) 9) (y ’ (y ’ =) =) 4 :  (m, n) & (i, a) hlcs eo x> +y> ’ 0x ’ 0x ’ ’ 1y 1y + 7 4 :  Zelot e` lotcrmitleo e` almfeomh ls (9, >)  (m, n) & (i, a) mhse hlcs eo y 4 > x ’ 1 ’ 1 Oew pelot e` lotcrscitleo e` y 4 >x ’ >x ’ 1 1 moa x> + y> ’ 0x ’ 0x ’ ’ 1y 1y + 7 4 : mrc (>, :) moa (1, 1)  m + n 4 > er i +a 4 7  (m + n ) (i + a) 4 =0 0 Hct M nc m > Ö > dmtrlx suig tgmt M> + M + L 4 :, wgcrc L 4 L>. Qgco mab(  M ) ls cqumh te 2

6. Mos. Yeh.

(=) 90 (>) 91 (=) M> + M + L 4 : (M ’ (M ’ L) (M> + M + L) 4 : M9 ’ L 4 : |M|9 ’L| 4 : (M|9 4 |L| 4 : |M| 4 L Oew M> ’ >M + L 4 ’ 4 ’9M 9M > act (M ’L) 4 act ( ’9M) ’9M) >  | M ’ M ’ L) 4 9> |M|49>  | M ’ M ’ L|0 4 90  | mab (L ’ M) ’ M)0| 4 90  | mab (L ’ M) ’ M)0| 4 | L ’ M| ’ M|0 4 |M ’ |M ’ L |0 4 90

(9) 9<

(1) 99

o

L` x =, x >,….xo nc tgc enscrvca amtm suig tgmt

7.

’ 9), (x ’  x  >o  =7: , tgco tgc dcmo e` tgc amtm (x ’ 9), =

l

>

l=

9),….,(x 9),….,(xo ’ 9) ’ 9) ls cqumh te 2 (=) 3 Mos.

(>)

=9 9

(9) 7

(1)

=0 9

(=)


ZMFC # 9

| BCC DMLO->:=< | AMQC 2 - :6 :1->:=< (YGL@Q-=) | ZMZC^-> | N.MÎG | DMQGCDMQLIY o

x

Yeh.

=

l =

o

x

l

’ >o  =7: ………………(l)

’ 6o 6o  9: ………………(ll) ………………(ll)

l=

(l) & (ll) 

3o 4 =3:  o 4 9: o

x

o





xl  >1:

l

l =



4

>1:

9: 9: Oew dcmo e` (x ’9) ’ 7 ’ 9 9 4 3 l 9) ls 7 ’

7x

l =

<.

Qgc sud e` tgc lo`loltc scrlcs = + > +

<

(=)

> 9

+

0 9>

(>) 0

>

Mos.

(9)

Yeh.

Hct Y 4

+

=1 91

+…….ls 2

(9) 3

(1) 1

> 0 =: =1 + > + 9 + 1 + ………………. 9 9 9 9 0 =: Y > 4 > + 9 + 1 + ………… 9 9 9 9 > Y 1  = =  Y ’ 4 + >  =   > ........ 9 9 9 9 9   >Y > 1 > > 1 = 4 + Ö 4 + 4  Y 4 > 9 < 9 = 9 9 9  < ’ 9    ^cqulrca cqumhs te = + > + Y 4 = + > +> 4 3



=:.

x  0  slo(x  9)  9 ls cqumh te 2 ( x  9) ies(x  9)

hld

x 9

(=) Mos.

(>)

Yeh.

hld



> 9



x  0 ’ 9 ’ sl s lo  x ’ 9 



3 0

(9)

= 0

(1)

3 0



ies  x ’ 9    x ’ 9 

x 9

4 hld x 9 4 hld x 9 

4

(>)

x0 ’9

ies  x ’ 9    x ’ 9   x ’ 9  x ’ 9 ies  x ’ 9  =

 i es :   m  9 

’ = 4

’ hld x 9

ho  x ’ 9 

ies  x ’ 9    x ’ 9 

x0 9



’

= ies:

= ’3 ’ =  0 0


ZMFC # 1


==.

Mos.

Hct tgc tmofcot armwo mt moy pelot Z(x,y) eo eo m iurvc lotcrscit tgc x moa y mxcs mt twe alstloit pelots M moa N rcspcitlvchy. L` MZ 2 ZN 4 3 2 =, moa tgc iurvc pmsscs tgreufg tgc pelot (>, >) tgco mo cqumtleo e` tgc iurvc ls 2 (=) xy3 4 >0 (>) x3y 4 >0 (9) x1y 4 >3 (1) xy1 4 >3 (=)

p(x, y)

= 2

Yeh.

3

M(0x, M(0x, :)

Yhepc e` MZ ls cqumh te



3ay



y



ax x

: 

ay  ax

ay ax



y : x ’ 0x 0x



ay ’ ax  y 3x

xy3 4 I

3oy + hox 4 hoi 

Ncimusc lt pmsscs tgreufg (>, >) se >(>3) 4 I  i 4 >0  cqumtleo e` iurvc ls xy3 4 > 0 =>.

Mos. Yeh.

Lo mo loircmslof fcedctrli scrlcs, tgc sud e` tgc `lrst moa tgc slxtg tcrd ls 00 moa tgc preauit e` tgc scieoa moa tgc `l`tg tcrds ls =>7. Qgco tgc sud e` tgc `lrst 0 tcrds e` tgls scrlcs ls (=) =>< (>) =>0 (9) =>6 (1) =>7 (>) Hct tcrds e` F.Z mrc m, mr, mr > Oew m + mr 3 4 00 & mr Ö mr 1 4 =>7  m +

=>7 => 7 4 00 4 00 m

m 4 > er 01



 > ’ = = L` m 4 > , r 4 > & l` m 4 01, r   m 4 > & r 4 >  sud e` 0 tcrds mrc 4 >   4 > Ö 09 4 =>0 >  > ’=  0

=9.

L` èr twe cvcot M moa N, lo m rmoaed cxpcrldcot, Z(M|N) 4

1 3

moa Z(N|M) 4

= 1

, tgco Z (M|M  N) ls

cqumh te (=) Mos.

(>)

Yeh.

Z

3 =0

(>)

=0 =6

(9)

== =0

(1)

3 =6

Z  M N 1 M 1  4   3 Z N 3 N    

Z M N  = N 4 4  Z M  1  M 

Z



ZM Z N 

4

=0 3

Nut Z(M) 4 =0x, Z(N) 4 3x & Z(M N) 4 1x  Z (MN) 4 Z(M) + Z(N) ’ Z(N) ’ Z Z (MN) 4 =6 x

Z M  =0x =0 x =0  M  Z  M  M N  Oew Z  4 4 4 4  Z  M  N  =6x Z M  N  =6 x =6  M  N 


ZMFC # 3

| BCC DMLO->:=< | AMQC 2 - :6 :1->:=< (YGL@Q-=) | ZMZC^-> | N.MÎG | DMQGCDMQLIY 9

=1.

Hct ` 2 [:, 3T  ^ nc ieotloueus ùoitleo suig tgmt ` ( x)  9 èr mhh x  [:, 3T. Qgc vmhuc e`  ` (t)at imo nc 2

Mos.

(=) ’ (=) ’1 1 (9)

:

(>) =:

9

3

:

:

(9*) 0

3

3

9

9

(1) =>

9 ’  ` t a t  ` t a t 4  ` t a t ’  ` t a t 4 9 ’

Yeh.

3

’9) 9  ` t a t  (3 ’9)

Oew ’ Oew ’9 9 (3 ’ (3 ’9) 9) 

9 3

 ’0 ’0  ’



3

` t a t  0 

’ 9 ’ 9  9 ’ 9 ’

9

=3.



9

’ 9  ` t a t  <  ` t a t  < ’ 9



9

Hct z( ’=) ’=) nc moy iedphcx oudncr suig tgmt

:

z



=.

Qgco tgc ldmflomry pmrt e`

z(= z(=  z) z(=  z)

ls 2

(Gcrc  4 mrf z)

Mos. Yeh.

     s lo  > 

    (>) tmo  ies   > 

     s lo  > 

(=) tmo

(9)  tmo

    (1)  tmo  ies   > 

(>) Hct z 4 (ies  + l slo) Oew

  4 z =  z  z  zz  z 

z =’ z

z zz ’ z

4

Oew ldmflomry

z =’ z





z =’ z

z’=

z =

slo  ’ =   ies  ’ l sl   ies   l slo  = 

4 Ldf 

Mos.

          >  s l o i e s l    ’ slo ’ slo ies l     > >  4 Ldf  ’ tmo  > > 4 Ldf      >    >   ies  slo l    ies  l slo ies   > >  > > >        l ’   > >  l  > ’        c     ldf c     4 ’ 4 Ldf  ’ tmo 4 ’tmo ’tmo 4 ’ tmo tmo ies      l > > >     c >   Hct ` 2 ^ ’ {:} ’ {:}  ^ nc ac`loca ny `(x) 4 m hef c x  nx 9  x > . L` x 4 ’= ’= moa x 4 = mrc tgc irltlimh pelots e` `(x), tgco 2 (=) `''(=) ’ `''(=) ’ `''( `''( ’=) ’=) 4 1 (>) netg x 4 = moa x 4 ’ 4 ’= = mrc heimh dloldm e` `(x) (9) `''(=) + `''( ’=) ’=) 4 : (1) x 4 = ls m heimh dloldm moa x 4 ’ 4 ’= = ls m heimh dmxldm e` `(x) (1)

Yeh.

`'(x) 4

=0.

m x

+ 9nx> + >x

’m



Oew `'(=) 4 m + 9n + > 4 : & `'( ’=) ’=) 4 ’ 4 ’m m + 9n ’ 9n ’ > > 4 :  >  x ’ =  x  = & `'(x) 4 x

`''(x) 4

x>

+>

m 4 ’ 4 ’ > > & n 4 :  `'' (=) 4 1 & `''( ’=) ’=) 4 1

 x 4 ’ 4 ’=, =, = mrc pelot e` heimh dloldm.


ZMFC # 0


=6.

Mos.

Hct tgc chhlpsc x> + =0y> 4 =0 nc losirlnca lo m rcitmofhc wgesc slacs mrc pmrmhhch te tgc ieeralomtc mxcs. L` tgc rcitmofhc ls losirlnca lo moetgcr chhlpsc tgmt pmsscs tgreufg tgc pelot (=0, :) tgco tgc cqumtleo e` tgc eutcr chhlpsc ls (=) x> + >17y> 4 =0> (>*) x> + >1:y> 4 =0> (9) x> + >9>y> 4 =0> (1) x> +>30y> 4 =0> (>) (1, =) C= (:,=) C> (=0, :) (1, :)

Yeh.

x> y> + 4 = =0> n> =0 = x> =3y =3 y > =0 > 4  + 4 =  n  C ls + 4 =  x> + >1: y> 4 =0> > > > > =0 =0 =0 n =3

Hct C= 2 x> + =0y> 4 =0

& C> 2

Oew (1, =) hlcs eo C > 9

=7.

3x

1

 = c

Qgc vmhuc e`

’ x

ax ls 2

’9

(=) 91 Mos.

(>) 93

(9) >(93)

(1)

93 3

(>) 3x 1

’9

=  c ’x



3  ’x 

’9

=  cx



1 9

9

>L 4 9 3x1ax 4 x3  L 4 93 ’9

Hct

=<.

nies x   mslo>x ’ nies , x   > ’ x  >    1, x  . Qgco tgc vmhuc e` m + n ls 2 Hct ` nc m ieotloueus ùoitleo ac`loca e` `(x) 4  >  >niesx   , x   >  ’ x > 

Yeh.

L4

9

Yeh.

Mos.

L4

9

ax 

(=) 7 (>)

(>) 3

ax 

(9) 1

(1) =

     slo  ’ x   >niesx  >  4 >n 4 1  n 4 > 4 >n hld ’ hld       x x    ’ x  > > ’ x    >  >  

      slo x sl slo  ’ x    slo   x   mslo>x ’ nies >  >  nies x  ’ n hld hld   4 >m hld      x x   x  ’ x >  x ’ x >  >    > > >  



4 >m ’ >m ’ n n 4 1   m+n43

m49


ZMFC # 6


L` tgrcc vciters P=   ˀl  ˀb  kˀ , P>  ˀl  ˀb ’ >k moa P9  ˀl  ˀb mrc iephmomr moa pcrpcoaliuhmr tgco tgc vciter P=  P> ls 2

>:.

Mos. Yeh.

ˀ (=) >ˀl ’ >ˀb  kˀ (>) ’ ˀl  ˀb  >k (9) P=  P9 mrc   P=  P9  :

+=+=+:4:

(9) ˀl ’ ˀb  >kˀ

P=  P> , P9   :  

=

=

>

P=



P>

(1) ’ ˀl  ˀb

=

=

:



 + = + > 4 :



 4 ’ 4 ’ 9 9

= l

Oew

mrc

:

 = ’=  : =  



P9

=  ’>  :



:

moa

 4 ’ 4 ’= =



’= =

’=

P=

b

4 ’=

=

k

= 4 ˀl ( ’> ’> + 9) ’ 9) ’ bˀ (> ’=) ’=) + kˀ (9 ’ (9 ’ =) 4 ˀl ’ b ’ b + >k ’9 ’> =

Qgc mrcm (lo squmrc uolts) mnevc tgc x-mxls neuoaca ny tgc pmrmnehm x ’ y> ’ ’ = 4 : moa tgc hloc x ’ y ’ y ’ ’ 9 9 4 : ls 2

>=.

(=) 1 Mos.

(>)

=9 9

(9)

=: 9

(1)

7 9

(9)

=

3

9

Yeh. ’9

y> 4 x ’ x ’ = = & x ’ x ’ y y ’ ’ 9 9 4 : > (x ’9) ’9) ’ x ’ x + = 4 : > x ’ 0x ’ 0x + < ’ < ’ x x + = 4 : x> ’ 6x ’ 6x + =: 4 : x 4 >, 3  y 4 ’ 4 ’=, =, > >

Mrcm 4

y  9 ’  y

>

 = ay

:

>

>

4

y ’ y

>

 > ay

:

4 0 ’

 y > y9   >y  4  ’ 9 > :

4

1 7 ’  1 > 9

=: 7 4 9 9


ZMFC # 7


>>.

x ’ >d

L` tgc twe hlocs

>d  3

y





7d

z’1

>

moa

x’>



d’>

y ’=



z ’ >d = ’ 9d 9d

mrc pmrmhhch èr sedc d  ^ tgco tgc

alstmoic nctwcco tgcd ls 2 (=)

=:

(>)

Mos.

(=)

Yeh.

Hlocs mrc pmrmhhch 

91

>d  3 d’>

’> ’> d ’ d ’ 3 3 4 7d > ’ =0d



4

(9) > 3

7d 4 ’=

=’ 9d

7d ’ 7d ’ >1d >1d> 4 ’ 4 ’> >

&

7d> ’ =1d ’ =1d + 3 4 :

 =>d> ’ 1d ’ 1d ’ ’ = = 4 :



7d> ’ 1d ’ 1d ’ ’ =:d =:d + 3 4 :

 =>d> ’ 0d ’ 0d + >d ’ >d ’ = = 4 :



(1d ’ (1d ’ 3) 3) (>d ’ (>d ’ =) =) 4 :

 (0d + =) (>d ’ (>d ’ =) =) 4 :



d4



d4

1

 d 4

>

’=

0

,

= >

= > x ’=

Hlocs mrc H= 4

H> 4

=

,

><

>



3

(1)

0

y z’ 1 4 1 >

4

z’= x’> y 4 4 = ’9 ’= ’ > >

Alstmoic 4

m

>



’ m=  n

n

l





b

k

Oew m> ’ m> Ö n 4 = :

 Alstmoic 4

’9 4 0l ’ 0l ’ =:b =:b + >k 9 > =

90  =::  1 <  1 =

4

=1: =1

4

=:

Mos. (=) >9.

Mos.

Hct ^ nc m rchmtleo ac`loca eo R Ö R ny (m, n) ^ (i, a)  m ’ a ’ a 4 n ’ n ’ i i wgcrc R ls tgc sct e` mhh lotcfcrs tgco ^ ls 2 (=) Qrmosltlvc nut ocltgcr rc`hcxlvc oer syddctrli. (>) ^c`hcxlvc nut ocltgcr syddctrli oer trmosltlvc. (9) Yyddctrli nut ocltgcr rc`hcxlvc oer trmosltlvc. (1) Yyddctrli moa trmosltlvc nut oet rc`hcxlvc. (9)


ZMFC # <


Yeh.

(m, n) ^ (i, a)  m ’ m ’ a a 4 n ’ n ’ i i ^c`hcxlvc (m, n) ^ (m, n)  m ’ m ’ n n 4 n ’ n ’ m m wglig ls oet mhwmys truc.  ^ ls oet rc`hcxlvc. syddctrli (m, n) ^ (i, a)  m ’ m ’ a a 4 n ’ n ’ i i  a ’ a ’ m m 4 i ’ i ’ n n  i ’ i ’ n n 4 a ’ a ’ m m  (i, a) ^ (m, n)  ^ ls syddctrli Qrmosltlvc (m, n) ^ (i, a)  m ’ m ’ a a 4 n ’ n ’ i …………..(l) (i, a) ^ (c, `)  i ’ i ’ ` ` 4 a ’ a ’ c …………..(ll) maalof m ’ m ’ c c 4 n ’ n ’ ` `  (m, n) ^ (`, c)  ^ ls oet trmosltlvc

>1.

Hct tgc mnsilssmc e` twe pelots M moa N eo m ilrihc nc tgc reets e` x > + >x ’ >x ’ 1 1 4 : moa tgc eralomtcs e` M > moa N nc tgc reets e` y + 1y ’ 1y ’ =0 =0 4 :. L` MN ls m almdctcr e` tgls ilrihc tgco tgc rmalus e` tgls ilrihc ls 2 (=) 3 (>) > =: (9) 0 (1) > 0 (=) Hct M (x=, y=) & N (x>, y>) x= + x> 4 ’ 4 ’>, >, x=x> 4 ’ 4 ’1 1 y= + y> 4 ’ 4 ’1, 1, y= + y> 4 ’ 4 ’=0 =0

Mos. Yeh.

x

Oew MN 4 >r 4

 x=  x> 

4

>

’ x=    y > ’ y=  >

>

>

>

’ 1x=x >   y=  y >  ’ 1y=y >

4 1  =0  =0  01  =:  rmalus 4 3 L` r ls tgc rcdmloacr entmloca eo alvlalof (<7) 3 ny => tgco tgc iec``lilcot e` x 9 lo tgc nloedlmh cxpmosleo

>3.

e`  =  x  >



>x

ls 2



(=) 6: Mos. Yeh.

(>)

<= >

(9) =:>

(1)

33 >

(=) (<7)3 4 (<0 +>)3 4 <03 + 3I=  (<0)1 + ………. + 3I 1  <0 + >3 4 <0 K + 9> 4 => (7K + >) + 7 r47 >r

=0

 =  x  4  =  x      > >   r

Qr += 4 r 4 9,

=0

x  >

Ir 

Q1 4 =0I9

x9 7

iec``lilcot e` x 9 4

=0

I9

= =0.=3.=1  6: 4 7 9.>. 9.>.= = 7


ZMFC # =:


sci sci x



>0.

slo x. x. ie ies3 x =/ >

(=) >  tmo x 

=/ >

(9) >  tmo x  Mos.

(9)

Yeh.



ax ls cqumh te 2 (Vgcrc I ls m ieostmot e` lotcfrmtleo)

’

> 3 >

3/>

 tmo x 

3/>

  tmo x  3

sci sci x 3

(>) >  tmo x 

I

(1)  tmo x 

 ies

= tmo x  sci



ax 4 tmox x tm o x Hct tmox 4 t  sci>x ax 4 at

Mos. Yeh.

>7.

Mos. Yeh.

3/>

3

>

3/>

  tmo x  3

I

I

ax >

>6.

=/ >

=

  tmo x 

slo slo xies x sci sci x ax

4

=/ >

I



9

>

x

ax

 ’=> 9>  > 3> > t  I 4 > tmox + at 4   t  t at 4 > t + 3 3 t  

= t>

Qgc cxpmosleo ~(p  q) ls cqulvmhcot te 2 (=) (~p  q)  (~q  p) (>) p  ~q (=)  (p  q)   ( p  q)  (p   q)   ( p  q)   (p   q)  (p   q)  ( p  q)

 tmo x 

(9) (p  ~q)  (q  ~p)

3 >

I

(1) p  q

Hct tgc phmocs x ’ >y ’ >y + kz 4 : moa x + 3y ’ z ’ z 4 : nc pcrpcoaliuhmr. Qgco tgc phmoc tgreufg tgc pelot (>, ’ (>, ’>, >, ’ ’>) >) moa pcrpcoaliuhmr te tgc flvco phmocs mhse pmsscs tgreufg tgc pelot 2 (=) (=, :, 6) (>) (:, 3, ’ 3, ’7) 7) (9) ( ’=, ’=, :, ’ :, ’6) 6) (1) (:, 3, 7) (=) x ’ >y ’ >y + kz 4 : moa x + 3y ’ 3y ’ z 4 : pcrpcoaliuhmrs  = ’ =: ’ =: ’ ’ k 4 : k 4 ’ 4 ’< < Moy phmoc tgreufg (>, ’ (>, ’>, >, ’ ’>) >) ls m(x ’>) ’>) + n(y +>) + i(z+>) 4 : & m ’ m ’ >n >n ’ ’
i m n 4 4 >  13 ’<  = 3>  Zhmoc ls 16(x ’>) ’>) ’ ’7(y+>) 7(y+>) +6(z+>) 4 :  16x ’ 16x ’ 7y 7y + 6z ’ 6z ’ <0 <0 4 : Ihcmrhy (=, :, 6) smtls`ylof tgc phmoc

><.

Hct y nc mo ldphlilt ùoitleo e` x ac`loca ny

xy

>

=

=

xy

>

=

>

xy

+ =>y 4 :. L` y(:) 4 ’=, ’=, tgco

ay mt ax

x 4 : ls 2 (=) ’ Mos.

1 3

(>)

3 1

(9)

’=

>

(1)

= >

(1)


ZMFC # ==


zy

>

=

=

xy

>

=

>

xy

Yeh.

+ =>y 4 :

=

>

=

(x+y+9) = x  y



>

=

>

xy

:

:

=’ x ’ y

(x+y+9) = x  y



+ =>y 4 :

=

>

+ =>y 4 :

xy

>

 (x+y+9) (= ’x ’x ’y) ’y) (> ’x ’x ’y) ’y) + =>y 4 : Al``crcoic w.r.t. x ay    ay  ’x ’y) ’y) (> ’x ’x ’y) ’y) + (x+y+9)  ’= ’ ’x ’y) ’y) + (x + y + z) (= ’x ’x ’y) ’y)  (> ’x =  ax  (= ’x ax    

ay   ay   .: + 1  ’= ’ .: + 1.: +  ax   ax   = ay ay ay => 41+1  4 > ax ax ax  = 

ay ay   ’= ’ + => 4 :   ax  ax 

 ay  => ax  4:  

Qgc sct e` mhh rcmh vmhucs e`  èr wglig tgc cqumtleo |x + >| . |x ’ |x ’ >| >| 4 > ’ > ’ > gms rcmh sehutleos èr x ls 2 (=) ( ’ ’, :T  [>, = + 3 T (>) ’= [’= ’ 3 , = ’ = ’ 3 T  [= + 3 , )

9:.

Mos.

(9) ( ’ ’, :T  [>, ) (9)

Yeh.

x > ’ 1 4 > ’ > ’ > |x|  

:



(1) [= ’ [= ’

> ’ > ’ >

3 , :T  [>, = +

3T

:

 ( ’>) ’>)  :   ( ’ ’ , :)  [>, )


ZMFC # =>


MZQLQSAC QCYQ =. Mos. >. Mos. 9. Mos. 1. Mos. 3. Mos. 0. Mos. 6. Mos. 7. Mos. <.

Mos. =:. Mos.

Rmgm Gmala wms nero lo wglig ieuotry mdeofst tgc èhhewlof5 (=) Lrmo (>) Lrmq (9) Qurklstmo (>)

(1) M`fgmolstmo

Zmrquct `heerlof ls usumhhy dmac e` wglig e` tgc èhhewlof5 (=) Frmoltc (>) Dmrnhc (9) Veea (9)

(1) Icdcot

Vglig eoc e` tgc èhhewlof ls tgc tmhhcst nulhalof lo Ncofmhuru5 (=) Ieoiera Qewcr (>) Dmotrl Zloomihc (9) Verha Qrmac Icotcr (1) S N Qewcr (>) M sdmhh hl`t èr imrrylof sdmhh hemas eohy ls koewo ms wglig e` tgc èhhewlof5 (=) M acm` ncmrcr (>) M pusg uppcr (9) M audn wmltcr (1) M beikcy ney (9) Vglig mdeofst tgc èhhewlof ls tgc ilty lo Ltmhy tgmt ls koewo èr lts hcmolof tewcr5 (=) Zlsm (>) Pcolic (9) @hercoic (1) êdc (=) Qg Yms Nmgu Qcdphc ls heimtca lo wglig e` tgc èhhewlof5 (=) Bmlpur @ert (>) Bgmosl @ert (9) Fwmhler @ert (9)

(1) ^ca @ert

Qgc èrt lo Gyacrmnma ls koewo ms wglig e` tgc èhhewlof5 (=) Nmgunmhl (>) Fehieoam (9) Plrnmoam (>)

(1) Ylhmagmrl

êeds wltg wgltc pmlotca wmhhs mppcmrs te nc wglig e` tgc èhhewlof5 (=) Amrkcr (>) Ydmhhcr (9) Hmrfcr ()

(1) Omrrewcr

Mo csimhmter devcs lo wglig e` tgc èhhewlof alrcitleos5 (=) Eohy gerlzeotmhhy (>) Pcrtlimhhy moa Gerlzeotmhhy (9) Eohy vcrtimhhy (1) Lo stcps (9) Icdcot phmstcr ls usca èr wglig e` tgc èhhewlof5 (=) Dmklof stmlrimscs (>) Ieovcrlof wmhhs (9) Dmklof `heers (>)

(1) Dmklof ree`s


ZMFC # =9


==. Mos. =>.

Mos. =9. Mos.

Qgc hetus tcdphc ls heimtca lo wglig eoc e` tgc èhhewlof ilty5 (=) Ocw Achgl (>) Huikew (9) Kmopur (=) gchlimh stmlrimscs mrc wglig eoc e` tgc èhhewlof5 (=) Iurvlof stmlrimscs (>) Aef hcf stmlrimscs (9) Ytmlrimscs wltg oe rmlhlofs (1) Ytrmlfgt `hlfgts (=) Vglig eoc e` tgc èhhewlof ls m SOCYIE werha Gcrltmfc sltc5 (=) Nlboer (>) Gmdpl (9) Nlbmpur (>)

[ktqbkfks hcwf as cm òbksm csm hihs lz`h} cmcmòb ùCu`y`[kr cscsm hs akYu hk fY 5 e cm òb (>) aj".kk cmòb (=) amo`b;k cfkos (9) x.ks'k cmòb

=1.

(1) Omfpur

(1) Keigl

(1) `'ke

cmòb

Mos.

(=)

=3.

Gloaustmo pmrrywmrc lo tgc loalmo dmrkct ls koewo èr wglig e` tgc èhhewlof preauit5 (=) Veeaco tmnhcs (>) Ymoltmry wmrc (9) Vmhh tlhcs (1) Zlpcs (>)

Mos.

Alrcitleos 2

Eoc e` tgc èhhewlof moswcr `lfurcs ls glaaco lo tgc prenhcd `lfurc lo tgc smdc slzc moa alrcitleo. Ychcit tgc ierrcit eoc.

=0.

(=) Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2


=6.

(=) Mos.

(>)

(9)

(1)

(=)


ZMFC # =1


Alrcitleos 2


=7.

(=) Mos.

(>)

(9)

(1)

(9)

Alrcitleos 2


=<.

(=) Mos.

(>)

(9)

(1)

(1)

Alrcitleos 2


>:.

(=) Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2

Vglig eoc e` tgc moswcr `lfurc wlhh iedphctc tgc scqucoic e` tgc tgrcc prenhcds `lfurcs5

>=.

(=) Mos.

(>)

(9)

(1)

(>)


ZMFC # =3


Alrcitleos 2


>>.

(=) Mos.

(>)

(9)

(1)

(=)

Alrcitleos 2


>9.

(=) Mos.

(>)

(9)

(1)

(9)

Alrcitleos 2


>1.

(=) Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2


>3.

(=) Mos.

(>)

(9)

(1)

(>)


ZMFC # =0


Alrcitleos 2

Qgc 9A `lfurc sgews tgc vlcw e` mo enbcit. Lacotl`y tgc ierrcit `reot vlcw heeklof lo tgc alrcitleo e` tgc mrrew `red mdeofst tgc moswcr `lfurcs.

>0.

(=) Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2


>6.

(=) Mos.

(>)

(9)

(1)

(1)

Alrcitleos 2


>7.

(=) Mos.

(>)

(9)

(1)

(=)


ZMFC # =6


Alrcitleos 2


><.

Mos.

(=)

(>)

(9)

(1)

(9)

Alrcitleos 2


9:.

Mos.

(=)

(>)

(9)

(1)

(9)


ZMFC # =7


Alrcitleo 2

Vglig eoc e` tgc moswcr `lfurcs ls tgc ierrcit dlrrer ldmfc e` tgc prenhcd `lfurc wltg rcspcit te W-W5

9=.

(=)

(>)

(9)

(1)

Mos. (9) Alrcitleo 2


9>.

Mos.

(=)

(>)

(9)

(1)

(=)


ZMFC # =<


Alrcitleo 2


99.

Mos.

(=)

(>)

(9)

(1)

(>)

Alrcitleo 2


91.

Mos.

(=)

(>)

(9)

(1)

(1)


ZMFC # >:


Alrcitleo 2


93.

(=)

Mos.

(>)

(9)

(1)

(=)

Alrcitleos 2

Qgc prenhcd `lfurc sgewos tgc tep vlcw e` mo enbcit. Lacotly tgc ierrcit chcvmtleo `red mdeofst tgc moswcr `lfurcs heeklof lo tgc alrcitleo e` tgc mrrew.

90.

(=) Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2


96.

(=) Mos.

(>)

(9)

(1)

(9)


ZMFC # >=


Alrcitleos 2


97.

(=) Mos.

(>)

(9)

(1)

(9)

Alrcitleos 2


97.

(=) Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2


9<.

(=)

Mos.

(>)

(9)

(1)

(=)


ZMFC # >>


Alrcitleos 2


1:.

(=)

Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2

Qgc 9A `lfurc sgewo tgc vlcw e` mo enbcit. Lacotl`y tgc ierrcit tep vlcw `red mdeofst tgc moswcr `lfurc.

1=.

(=) Mos.

(>)

(9)

(1)

(=)

Alrcitleos 2


1>.

(=) Mos.

(>)

(9)

(1)

(>)


ZMFC # >9


Alrcitleos 2


19.

(=) Mos.

(>)

(9)

(1)

(9)

Alrcitleos 2


11.

(=)

Mos.

(>)

(9)

(1)

(>)

Alrcitleos 2


13.

(=) Mos.

(>)

(9)

(1)

(1)


ZMFC # >1


Alrcitleos 2


10.

(=) Mos. (9) Alrcitleos 2

(>)

(9)

(1)


16.

(=) Mos.

(>)

(9)

(1)

(=)

Alrcitleos 2


17.

(=) Mos.

(>)

(9)

(1)

(9)


ZMFC # >3


Alrcitleos 2


1<.

(=) Mos.

(>)

(9)

(1)

(=)

Alrcitleos 2


3:.

(=)

Mos.

(>)

(9)

(1)

(9)


ZMFC # >0


A^MVLOF =.

Lo tgc spmic prevlaca lo tgc moswcr sgcct èr tgls qucstleo, armw dmrflo hlocs te èrd m `rmdc. Lo tgls `rmdc ircmtc mo mcstgctli iedpesltleo uslof eohy iuncs. Qgcsc imo nc e` moy slzc, moa dmy nc phmica scpmrmtc, evcrhmpplof er wltglo cmig etgcr. Qgc lacm ls te preauic mo mcstgctli moa vlsumhhy cxltlof iedpesltleo e` tgcsc sgmpcs lo tgc `rmdc wltgeut dmklof lt rcprcscot moy rcmhlstli èrd hlkc geusc `mic cti. Qgcsc sgmpcs moa tgc etgcr spmics sgeuha nc ` lhhca wltg sedc iehers e` yeur igelic se tgmt tgc vls umh qumhlty e` tgc iedpesltleo ls cogmoica.

>.

Iepy tgc frmpgli ldmfc sgewo lo tgc spmic prevlaca èr tgc moswcr e` tgls qucstleo. Ircalt wlhh nc flvco te tgc cxmitocss e` yeur moswcr.

9.

Lo tgc spmic prevlaca èr tgc moswcr e` tgls qucstleo mttcdpt moy EOC e` tgc èhhewlof2 Acslfo moa armw mo mppreprlmtc pmttcro èr m squmrc tmnhc ihetg. Ieher er sgmac lt te cogmoic lts vlsumh qumhlty. E^ Armw m pliturc e` m ihmssreed heeklof tewmras tgc tcmigcr `red ncgloa tgc stuacot. E^ Armw `red ldmflomtleo m pliturc e` mo e``licr slttlof lo gls e``lic.


ZMFC # >6

JEE Main 2019 Paper 2 B.Arch Attempt Shift - 1(07th April, 2019)

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