JEE Advanced 2018 Mathematics

| KGG DCXDLJGC-1=4? | CDSG ; 1= -=7-1=4? | FGF@QY ADZGC | FDSOGFDSIJZ

]DQS ; III FDSOGFDSIJZ FJT typg ZGJSI@L ‚ 4 ; (Fdxifuf Fdres ; 13) (=8) qugsti`ls.  Sois sgjti`l j`ltdils ZI_ (=8) qugsti`ls. `pti`ls h`r j`rrgjt dlswgr(s). @LG @Q F@QG SODL @LG @LG `h togsg hùr `pti`l(s) is (drg) j`rrgjt  Gdjo qugsti`l ods H@WQ `pti`ls `pti`l(s). jo``sg tog j`rrgjt `pti`l(s) `pti`l(s) t` dlswgr tog qugsti`l. qugsti`l.  H`r gdjo qugsti`l, jo``sg  Dlswgr t` gdjo qugsti`l wibb ag gvdbudtgc djj`rciln t` tog h`bb`wiln fdreiln jo`sgl.

Hubb Fdres

;

+3 Ih `lby (dbb) tog j`rrgjt `pti`l(s) is (drg) jo`sgl.

]drtidb Fdres ;

+: Ih +: Ih dbb tog hùr `pti`ls drg j`rrgjt aut @LBY torgg `pti`ls drg jo`sgl.

]drtidb Fdres ;

+1 +1 Ih torgg `r f`rg `pti`ls drg j`rrgjt aut aut @LBY tw` `pti`ls drg jo`sgl, a`to `h woijo drg j`rrgjt `pti`ls.

]drtidb Fdres ;

+4 Ih +4 Ih tw` `r f`rg `pti`ls drg j`rrgjt aut @LBY `lg `pti`l is jo`sgl dlc it is d j`rrgjt `pti`l.

[gr` Fdres ;

= Ih l`lg `h tog `pti`ls is jo`sgl (i.g. tog qugsti`l is uldlswgrgc). uldlswgrgc).

Lgndtivg Fdres ;

‚1 Il dbb `togr jdsgs.

H`r Gxdfpbg ; Ih ; Ih hirst, toirc dlc hùrto drg tog @LBY torgg j`rrgjt `pti`ls h`r d qugsti`l wito sgj`lc `pti`l agiln dl ilj`rrgjt `pti`l5 sgbgjtiln `lby tw` `h tog torgg j`rrgjt `pti`ls (g.n. tog hirst dlc hùrto `pti`ls), witoùt sgbgjtiln dly ilj`rrgjt `pti`l (sgj`lc `pti`l il tois jdsg), wibb rgsubt il +1 fdres. Zgbgjtiln `lby `lg `h tog torgg j`rrgjt `pti`ls (gitogr hirst `r toirc `r hùrto `pti`l), witoùt sgbgjtiln dly ilj`rrgjt `pti`l (sgj`lc `pti`l il tois jdsg), wibb rgsubt il +4 fdres. Zgbgjtiln dly ilj`rrgjt `pti`l(s) (sgj`lc `pti`l il tois jdsg), wito `r witoùt sgbgjti`l `h dly j`rrgjt `pti`l(s) wibb rgsubt il ‚1 fdres.

4.

Il d  DAJ, d 6 4=, a 6 4= : , J 6 :=½ togl woijo `h tog h`bb`wiln is/drg j`rrgjt 0 (D) Drgd `h jirjufjirjbg 6 4== f (A) Qdcius `h iljirjbg 6 4=

: ‚ 47

(J) A 6 37½ (C)  D 6 41=½ Dls.

(DA)

Z`b.

j`sJ 6

: >



d1

 a1  j 1 1da

4==  :==  j 1 1.4=.4 1.4=.4= = :

:== 6 3== ‚ j1 j1 6 4== j 6 4=  6

r6

4 4 4 dasilJ 6 4=.4= : Þ 6 17 : 1 1 1

 s



17 :  1 (1=  4= 4= : )

ay silg rubg

a silA





7= : 1=  4= 4= :



7 : 1 :



1 : 1 :

 7(1 :  :)  4= :  47

j silJ

4= : silA



4= silJ



J 6 :=

Qgs`ldljg G Gcuvgltur gs B Btc. QGN. & J@Q]@QDSG @HHIJG ; JN S`wgr, D-38 D-38 & 71, I]ID, I]ID, Lgdr Jity Fdbb, Kodbdwdr Kodbdwdr Q`dc, E`td (Qdk.) - :13==7 ]o.L`. ; +94->33-:=41111, 88:7777 | S` El`w f`rg ; sfs QGZ@ dt QGZ@ dt 788>> ^gasitg ; www.rgs`ldljg.dj.il | G-fdib ; j`ltdjtMrgs`ldljg.dj.il j`ltdjtMrgs`ldljg.dj.il| | JIL ; W?=:=1QK1==>]BJ=13=19 Sois ss`buti`l w wds c c`wlb`dc h h r`f Q Qgs`ldljg KKGG D DCXDLJGC 1 1=4? Z Z`buti`l p p`rtdb

]DNG # 4


1Q 6

j silJ



4= 4/ 1



Q 6 4=

Ogljg drgd `h jirjufjirjbg 6 Q1 6 4==  1.

Ih h (x) 6 g(h(x)‚n(x)) n (x) dlc h(4) 6 n(1) 6 4 (D) h(1) < 4 ‚ l1

(A) h(1) 2 4 ‚ l1

(J) n(4) < 4 ‚ l1

(C) n(4) 2 4 ‚ l 1

Dls.

(AJ)

Z`b.

h ' h ' (x) 6 g h(x) ‚n(x) n'(x) ; h(4) 6 n(1) 6 4 g ‚h(x) 6 g ‚n(x) + j g ‚h(x) . h '(x) 6 g ‚n(x) . n'(x)

 c(g

 h (x)



)  c(g n(x) )

g ‚h(x) 6 g ‚n(x) + j x64

4 6 g ‚n(4) + j g

x61

g ‚h(1) 6



n(4) < 4 ‚ l1

4 j g

g ‚h(1) 6 1g ‚4 ‚ g ‚n(4) g ‚h(1) 6 1g ‚4 ‚ g ‚n(4) h(1) 2 4 ‚ l1 g ‚4 ‚ g ‚h(1) 6 g ‚n(4) ‚ g ‚4



g ‚n(4) + g ‚h(1) 6 1g ‚4 g ‚n(4) 2 1g ‚4

‚n(4) 2 l1 ‚ 4 :.

Ih s, r, t ag l`l zgr` j`fpbgx lufagrs dlc z 6 x + iy wogrg sz + t z + r 6 = wogrg z 6 x ‚ iy (D) H`r z t` odvg uliqug s`buti`l |s|  |t| (A) ( (J) Ih |s| 6 |t| togl z wibb odvg z wibb odvg ilhilitg s`buti`l (C) Ih z ods f`rg todl `lg s`buti`l togl it wibb odvg ilhilitg s`buti`l

Dls.

(DC)

Z`b.

sz + t z + r 6 =, z 6 x ‚ iy

sz  t zr

=

(4) + (1)

(t  s )z  (s  t )z )z  (r  r )  = (t  s )z  (s  t )z )z  (r  r )  = H`r uliqug s`buti`l

ts ts



s t s t


]DNG # 1


@l s`bviln tog da`vg gqudti`l wg ngt | t |  | s |  `pti`l (D) is j`rrgjt

Bilgs `vgrbdp ih

ts ts



s r r  s t r  r t

tr  t r

|t|6|s|

1 tr tr tr

 sr  s r  sr  s r  t r  t r

 1s r

 sr

 | t || || r || s || r |  | t | 6 | s |  Ih | t | 6 | s |, bilgs wibb ag pdrdbbgb h`r surg aut it fdy l`t ag jìljicglt

Ogljg `pti`l (j) is wr`ln & (C) is j`rrgjt.

3.

 h(t)silx ‚ silt h(x)  6 ‚ dlc bif 6 sil 1 x   t x 41 t‚ x 8

Bgt h 

  6  3 3 1

  + h   6 =  1 1

(D) h 

(J) h(x) 2 Dls.

(DAJC)

Z`b.

bif

t x

x3 8

(A) h  

‚ x1 h`r x  (=, )

(C) h () 6 = h`r dt bgdst `lg   (=, )

h(x)si h(x)silt  h(t)silx h(t)silx 6 sil1x tx

h(x)j`sx h(x)j`sx  h (x)si (x)silx sil sil1 x

6 4

 h(x) 

‚ c  6 4  silx  h(x) 6 x + j silx (D)

h



    8  6 ‚ 41



j6=



h(x) 6 ‚ x sil x

h(x) + h  (x) 6 ‚ 1 j`s x h

    1  + h   1  6 =  

(A)

h

      3  6   3 1

(J)

h(x) is j`ltiluùs dlc cihhgrgltidabg dlc h(=) 6 h(x) 6 = Wsiln BFXS h (j) 6 = h`r s`fg x  (=, )

(C)

n(x) 6 ‚ x sil x + x 1 ‚

x3 8

1x : n(x) 6 h (x) + 1x ‚ :


]DNG # :


n'' (x) 6 h  (x) + 1x ‚1x1 n''' (x) 6 : sil x + x j`s x ‚ 3x 6 : (sil x ‚ x) + x (j`sx ‚ 4) n(x) 2 =



h`r x < =



n(x)

n(x) is cgjrgdsiln

2 n(=) 

n(x) 2 =

ogljg n(x) is cgjrgdsiln h`r x < =

n(x) 2 n(=)

n(x) 2 =



ogljg n(x) 2 = h`r x < =

n(x) 2 n(=)

Ogljg h(x) 2 7.

Bgt x1 + y1 6

4 1

4 , y1 6 3x odvg 1 j`ff`l tdlnglts. Sogy fggt dt G. Ih @G is sgfi fdk`r dxis `h dl 1 1

, BQ 6 4

(A) Drgd gljb`sgc ay x 6

(J) ( Dls.

n(x) 2 =

x3 ‚ x1  x  (=, ) 8

gbbipsg sujo todt fil`r dxis 6 (D) g 6



4 1

t` x 6 4 is

(  ‚ 1)

3 1

(C) (

(DA)

Z`b.

4 1

(=,=)

T

Bgt gqudti`l `h j`ff`l tdlnglt is y 6 fx +

4 f 4  f1

== 



4 1



4 f

f3 + f1 ‚ 1 6 =



f 6 ´4

Gqudti`l `h j`ff`l tdlnglts drg y 6 x + 4 & y 6 ‚x ‚ 4 pìlt T is ( ‚4, =) 

x1 Gqudti`l `h gbbipsg is 4

(D)

g6

4

4 1



4 1

y1  4/ 1 &

4 BQ 6

1a1 d

4


]DNG # 3


(J) y

x

(4,=)

( ‚4,=) x6

4

4



1.

1

 x 4  x1   sil4 x  . 4  x cx 6 1   1 1 1 4/

4

Drgd

4

4/ 1

1

1

   4    4 1      1      ? 3 3 1  3  3 ? 

1

6

j`rrgjt dlswgr drg (D) & (C).

l

8.

Bgt h l(x) 6

 tdl

‚4

i4

  4  4  ( x  i)( x  i ‚ 4)   

(D) bif sgj1h(=) 6 4

(A) bif tdl1h(=) 6 4/x

l 

l 

4=

7=

' (=)) sgj1h l(=) 6 77 (J)  (4  hl (=))

(C)

l 4

Dls.

(A)

Z`b.

h l (x) 6

l

  tdl k 4

wogl l 

(x k)  tdl4(x k 4) 4)

 1

1

k 4

4

 , h(x) 6

 h (x) 6 77

h l (x) 6 tdl ‚4 (x + l) ‚ tdl ‚4 x



‚ tdl ‚4x 6 j`t ‚4 x. 4

(D)

sgj1 h(x) 6 4 + tdl 1 (j`t ‚4 x) 6 4 

(A)

tdl h(x) 6 tdl(j`t ‚4 x) 6

(J)

4 4  4 1 4    4  l1  4  =   sgj (tdl l) 6  l 4  4=

x1

4 x 4=

4  4= l 4

x

>.

Nivgl h(x) 6 4 ‚ 1x +

g

x t

h(t)ct , wogrg h(x) is j`ltiluùs huljti`l, togl

=

(D) Drgd `h rgni`l h(x) 2 y 2

4  x1 il hirst qudcrdlt is

(A) (J) h(x) pdssgs torùno (1, ‚4) (C) h(x) pdssgs torùno (4, 1)


]DNG # 7


Dls.

(J)

Z`b.

h(x) 6 4 ‚ 1x +

x

g

x t

h(t)c h(t)ctt

=

x



h(x) .

g ‚x 6

(4 ‚ 1x).g ‚x

+

g

t

h(t)ct h(t)ct

=



h '(x)g h '(x)g ‚x ‚ g ‚x . h(x) 6 ‚ 1 . g ‚x ‚ (4 ‚ 1x) . g ‚x + g ‚x . h(x)



h '(x) ‚ 1h(x) 6 (1x ‚ :) I.H. 6 g ‚1x y . g ‚1x 6



:). g  (1 x :)

1x

I

g ‚1x

II

cx

g1x 1 cx 6 (1x ‚ :) . 1 1 g1x



y.



y.g ‚1x 6



y 6 (4 ‚ x) + j . g 1x

(1 x : )  4 1



 j.g1x





y.

g ‚1x

(1 x  :) g 1 x 6 ‚ 1

g1 x  1

j

y 6 (4 ‚ x) + j.g 1x

put x 6 = 464+j 

j6=



y 6 4 ‚ x woijo pdssgs torùno pìlt (1, ‚4)

y6

4  x1

y6h(x) L`w rgquirgc drgd 6

4 .(4)1 3

4 1

 .4.4 

 4 

3

1

LWFGQIJDB XDBWG typg ZGJSI@L ‚ 1 ; (Fdxifuf Fdres ; 13) ZGJSI@L-1 (Fdxifuf Fdres ; 13) (=?) qugsti`ls. Sog dlswgr t` gdjo qugsti`l is LWFGQIJDB XDBWG. XDBWG .  Sois sgjti`l j`ltdils GINOS (=?) qugsti`ls.  H`r gdjo qugsti`l, gltgr tog j`rrgjt lufgrijdb vdbug (il cgjifdb l`tdti`l, truljdtgc/rùlcgc-`hh t` tog sgj`lc cgjifdb pbdjg5 g.n. 8.17, >.==, ‚=.::, ‚=.:=,,:=.1>, ‚41>.:=) usiln tog fùsg dlc tog `l-sjrggl virtudb lufgrij egypdc il tog pbdjg cgsinldtgc t` gltgr tog dlswgr.

qugsti`l wibb ag gvdbudtgc gvdbudtgc djj`rciln t` tog h`bb`wiln h`bb`wiln fdreiln sjogfg ;  Dlswgr t` gdjo qugsti`l Hubb Fdres

;

[gr` Fdres ;

?.

+: Ih @LBY tog j`rrgjt j`rrgjt lufgrijdb vdbug is gltgrgc gltgrgc ds dlswgr. = Il dbb `togr jdsgs. jdsgs.

] 6 4, 8, 44, ......., 1=4? tgrfs T 6 9, 48, 1:,......., 1=4? tgrfs togl lufagr `h tgrfs il (]  T) is

Dls.

(:>3?)


]DNG # 8


Z`b.

] 6 {4, 8, 44, ..........} T 6 {9, 48, 1:,.........} J`ff`l tgrfs ; 48, 74, ?8 tp 6 48 + (p ‚ 4):7 6 :7p ‚ 49  4==?8  p  1??.>  l (]  T) 6 l(]) + l (T) ‚ l (]  T)

6 1=4? + 1=4? ‚ 1?? 6 :>3? 9.

](=, =), T(4,4), Q(1, =) drg torgg vgrtijgs `h d tridlnbg. Sog drgd fdcg ay tog jurvg y 6 x l wito ]T is :=% `h tog ]TQ. Hilc tog vdbug `h l.

Dls.

(3)

y6x

y

T(4,4)

4

Z`b.

l

4

](=,=)

Q(1,=)

x

4

: 4 (x  x )cx   1  4  4=  1  =



l

4



4=.

l 4



4 :  1 4= 4=



4 7

4

xl 4  l4 =



x1 1



l63



: 4=



4 4 :   1 l  4 4=

Juag `h sicg bglnto 4 ulit wito wito j`rlgr dt (=, =, =) il tog hirst `jtdlt. Xgrtgx ], T, Q big `l x, y, z-dxis. dlc Z is jgltrg `h juag dlc S is cirgjtby `pp`sitg t` `rinil. Ih Z] is p , ZT is q , ZQ is r , ZS is t togl hilc |( p Þ q ) Þ ( r Þ t )| 6

Dls.

4 6 =.7 1 (=, 4, =) T S(4, 4, 4)

Z`b. @(=, =, =)

](4, =, =)

Q (=, =, 4)

 4 4 4 pìlt Z  , ,   1 1 1 pìlt S(4, 4, 4)


]DNG # >


p  Z]  q  ZT  r

 ZQ 

t

 ZS 

ˈi  ˈk  eˈ

1

ˈi  ˈk  eˈ 1

ˈi  ˈk  eˈ 1 ˈi  ˈk  eˈ

1 ˈi

ˈk

eˈ

ˈk

eˈ

4 i  ˈk 4 4 4  6 (1 ˈi  1 ˈk )  3 1 3 4 4 4

p q  4

L`w

ˈi

rt

4

L`w

4 3

 4 4 4  6 4

4

ˈi

ˈk eˈ

(p  q)  (r (r  t )  4

4 =

4 4 = 44.

1ˈi  1ˈk 3



4 3





ˈi  ˈk 1

eˈ 1

Hivg stdtgfglts Z4, Z 1, Z :, Z 3, Z 7 drg dbb`ttgc spgjihij sgdts ] 4, ] 1, ] :, ] 3, ] 7 t` sit il tog jbdss. @l tog gxdfildti`l cdy togy drg dbb`ttgc sgdts rdlc`fby. Bgt gvglt G i ag todt Zi dlc Zi + 4 drg l`t dckdjglt togl hilc tog vdbug `h ](G 4  G1  G:  G3)

43 6 =.448>) =.448>) 41=

Dls.

(

Z`b.

S`tdb jdssgs 6 7! hdv`rdabg wdys 643

4 : 7

1 3

4 3 1 7



:

 1

7  1

1

3

4 .......

1 7 : 43

 1! 4

  

3  :

: 4 7 1 3 : 4 3 1

  1

7

: 7 .... .... ....}  1 6 43 ]ràdaibity 6

43 41=


]DNG # ?


41.

]ràdaibity todt `lby Z4 sits il its `rinildb sgdt Z 4 dlc l` `togr drg il togir `rinildb sgdts

Dls.

(

: ) 3=

 

3!  4 

4 4 4 4     4! 1! :! 3!  7!



9 41=



: 3=

Z`b.

]ràdaibity 6

4:.

Ih J 6 x D + y A + D Þ A dlc D . A 6 = wogrg D dlc A drg ulit vgjt`rs. Ih | j | 6 1 ih  is tog dlnbg agtwggl agtwggl J dlc D ds wgbb ds J dlc A hilc ?j`s 1

Dls.

(:)

Z`b.

J  xD  yA  D  A & D.A D.A  = D  J  AJ   J.D J

1

1j`s   J.A  1j`s



 x 1  y1  | D  A |1 6 1(3j`s 1) + 4 ‚ =

3 6 ?j`s1 + 4

43.

x 6 y 6 1j`s 



 l i4 bif sil x x.  i4 l  4 



?j`s1 6 :

i  x    ‚4  1   6 1 ‚ j`s i4    x



 l  (  x)i   i4 



x

 i4

i

 x  1  

  . Hilc lufagr `h s`buti`ls `h tog  

gqudti`l Dls. Z`b.

(4)

i i l l  l i4  l  x   x  4 i Bif sil   x  x    6 Bif  j`s   (  x)       i4  l 1  i4 l 1   i 4  1   i 4   

4

 x  x     1  x 1  6 x   1   x    4  x 4  x  4  x 4  x  1 1 x1 4 x



x1 4 x



x1 1 x x 4 x



x x  4 x 1  x



x1 x  1  x 1 x

x(4  x)  (4  x) 4 x

1

6

1x  x 1

1 x

3  x1

 4 x1  x  1  4  x1 3  x1 x1

‚x3 + 3 + :x1 6 x1 + x + 1 ‚ x3 ‚ x: ‚1x1 x: + 3x1 ‚ x + 1 6 = x: + 3x1 6 x ‚ 1 x1(x + 3) 6 x ‚ 1 L`. Z`b. il ( ‚4, 4) x 6 = s` lufagr `h s`buti`l is 4


]DNG # 9


47.

Bgt sgt _ ods 7 gbgfglts Zgt Y ods > gbgfglts  6 lufagr `h `lg-`lg huljti`l hr`f _  Y  6 lufagr `h `lt` huljti`l hr`f Y  _

 ‚ 

Hilc

7!

Dls.

(449)

Z`b.

l(_) 6 7 l(Y) 6 >   Lufagr `h `lg-`lg huljti`l 6 >J7 Þ 7! 6 14 Þ 41= 6 171=   Lufagr `h `lt` huljti`l Y t` _

d4 d1

a4 a1

.

.

.

.

.

.

d>

a7

4, 4, 4, 4, :

4, 4, 4, 1, 1

>! >! Þ 7! + Þ 7! 6 ( >J: + : . >J:) 7! 6 3 Þ >J: Þ 7! : :! 3 ! (1!) :!

 7! 48.

6 3 Þ >J: ‚ >J7 6 3 Þ :7 ‚ 14 6 449

Gvdbudtg ; 4 1 b`n1 b`n1 9

((b`n1 9) ) Dls. Z`b.

(?)

(b`n

1

4 1 b`n (b`n 9) 1 1



9)

1 b`n(b`n1 9) 1

(b`n1 9) 4/1

4>.

Þ ( > )

4 b`n b`n3 >

 ( > )b`n 3 >

.(1) 6 3.1 6 ?

4 :

 U(4 U(4  x) (4‚ (4‚ x) V 1

8 4/ 3

cx 6

=

Dls.

(1) 4 1

Z`b.

4  :  cx

 (4  x) (4  x)  =

4 1

 =

1



8

4/3



4  :  cx 4/3

 (4  x)8  (4  x)  8   (4  x)  1


]DNG # 4=


4 x 6 t 4 x

put

4 :



6

4  :  ct  (4  1t8/ 3

=

4?.

1cx  ct (4  x)1



:)

1



1 t

4/:

 (4  : )( :  4)  1 4

Bgt x + y 6 : ag pbdlg. ]ìlt ] ods ifdng daùt pbdlg ds T `l z-dxis. ]ìlt ] ods cistdljg hr`f x-dxis 6 7. Hilc ]Q wogrg Q is ifdng `h ] daùt x-y pbdlg.

Dls.

(?)

Z`b.

](  ) Q(,  , ‚) T

x  4



y  4



z =



1(    :) 1

x 6 : ‚ , y 6 : ‚ , z 6  T(: ‚  , : ‚ , : ‚ , ,) `l z ‚dxis  6 :,  6 :



p(:, :, ) cistdljg hr`f x-dxis 9 + 1 6 17 1 6 48



 6 3

](:, :, 3)



]Q 6 ?

Q(:, :, ‚3)

49.

H`r d cihhgrgltidabg huljti`l h(=) 6 = dlc Dls.

Z`b.

4 cy 6 , hilc bif h(x) x  (7x  1)(7x  1) cx

(=)

4 cy 6 (7x  1)(7x  1) cx



cy 6

4 3



(7x  1)(7x  1) cx (7x  1)(7x  1)

1y 6 l |7x ‚ 1| ‚ bl |7x+1| + j 1=y 6 l

 7x  1   7x  1 

bif y 6 =

x 

1=.

Ih d, a, j,  Q dlc

hilc Dls.

: d j`s x + 1 a sil x 6 j ods 1 r``ts  dlc  il

      1 ,1  sujo todt  +  6 : ,

a d

(=.7)


]DNG # 44


: d j`s x + 1a sil x 6 j

Z`b.

, 

     1 , 1 

 4  t1  x  1t  + 1a 6 j wogrg t 6 tdl      4  t1  1  4  t1 

:d

: d(4 ‚ a1) + 3at 6 j (4 + t 1) t1 (j +



4

6

4  Z1 d

8

4      1  6 :

Z4 a

:

3a



j  :d  j  :d

_6

6

4 :

4

6

1 4=

14.

:d6=



6

1 tdl

: d) ‚ 3at + j ‚

r(

4=

Jr )1 hilc

r 4

Dls.

(838)

Z`b.

x6

4=

 r 4

L`w

x . 43:= 4=

r. 4= Jr . 4=Jr

x 43:=



 4= .  9 Jr 4 .4=J4= r 6 4= . 49J9 r 4

4=. 49J9 43:=

49



J9

43:

49



J9

44 4:

6

49.4>.48 6 49 Þ :3 6 838 ?

J`fprgoglsi`l typg ZGJSI@L ‚ : ; (Fdxifuf Fdres ; 41) (=1) pdrdnrdpos. Adsgc `l gdjo pdrdnrdpo, togrg drg S^@ (=1) qugsti`ls. (=1) qugsti`ls.  Sois sgjti`l j`ltdils S^@ (=1) pdrdnrdpos. H@WQ `pti`ls. @LBY @LG `h @LG `h togsg hùr `pti`ls j`rrgsp`lcs t` tog j`rrgjt dlswgr.  Gdjo qugsti`l ods H@WQ `pti`ls. j`rrgsp`lciln t` tog j`rrgjt dlswgr.  H`r gdjo qugsti`l, jo``sg tog `pti`l j`rrgsp`lciln qugsti`l wibb ag gvdbudtgc gvdbudtgc djj`rciln t` tog h`bb`wiln h`bb`wiln fdreiln sjogfg ;  Dlswgr t` gdjo qugsti`l Hubb Fdres

;

+: Ih @LBY tog j`rrgjt `pti`l is jo`sgl.

[gr` Fdres ;

= Ih l`lg `h tog `pti`ls is jo`sgl (i.g. (i.g. tog qugsti`l is uldlswgrgc). uldlswgrgc).

Lgndtivg Fdres ;


4


]DNG # 41


Fdtjo Bist typg ZGJSI@L ‚ 4 ; (Fdxifuf Fdres ; 41) (=3) qugsti`ls.  Sois sgjti`l j`ltdils H@WQ (=3) qugsti`ls. (=1) fdtjoiln bists 5 BIZS-I dlc BIZS-I dlc BIZS-II. BIZS-II .  Gdjo qugsti`l ods S^@ (=1) fdtjoiln `pti`ls drg nivgl rgprgsgltiln fdtjoiln `h gbgfglts hr`f BIZS-I dlc BIZS-II. @LBY @LG `h togsg hùr gdjo qugsti`l,  H@WQ `pti`ls jo``sg tog `pti`l, jo``sg tog `pti`l j`rrgsp`lciln t` tog j`rrgjt fdtjoiln. sjogfg ;  H`r gdjo qugsti`l, fdre wibb ag dwdrcgc djj`rciln t` tog h`bb`wiln fdreiln sjogfg Hubb Fdres

;

+: Ih @LBY tog `pti`l j`rrgsp`lciln j`rrgsp`lciln t` tog j`rrgjt fdtjoiln is jo`sgl.

[gr` Fdres ;

= Ih l`lg `h tog `pti`ls is jo`sgl (i.g. (i.g. tog qugsti`l is uldlswgrgc). uldlswgrgc).

Lgndtivg Fdres ;

11.


Sogrg drg 8 a`ys (A4, A1, A:....... A8)dlc 7 nirbs (N 4, N1, N: ........ N7) 4 6 Lufagr `h wdys `h sgbgjtiln d j`ffittgg `h gxdjtby : a `ys dlc 1 nirbs 1 6 Lufagr `h wdys `h sgbgjtiln d j`ffittgg il woijo a`ys dlc nirbs drg gqudb dlc lufagr drg

nrgdtgr todl 1 : 6 Lufagr `h wdys `h sgbgjtiln d j`ffittgg `h 7 odviln dt bgdst 1 nirbs 3 6 Lufagr `h j`ffittgg `h 3 odviln A 4 dlc N7

J`bufl-I

J`bufl-II

(D) 4

(]) 44J7 ‚ 4

(A) 1

(T) :8

(J) :

(Q) 1==

(C) 3

(Z)

Dls. Dls.

(D)  T5 (A)  ]5 (J)  Z5 (C)  Q

Z`b.

8 A`ys & 7 nirbs

44J 7

‚ ?4

4  lufagr `h wdys `h sgbgjtiln : a`ys & 1 nirbs 8J: Þ 7J1 6 1== 1  A`ys & nirbs drg gqudb & fgfagrs  1 8J

4 .

7J

4+

8J

1 .

7J

1

+ 8J: . 7J: + 8J3 . 7J3 + 8J7 . 7J7 6 44J7 ‚ 4

:  lufagr `h wdys `h sgbgjtiln 7 odviln dt bgdst 1 nirbs

44J 7

‚ 8J7 ‚ 8J3 . 7J4 6 44J7 ‚ ?4

3  j`ffittg `h 3 odviln A 4 & N7 9J

1:.

1 6

:8

Fdtjo tog nivgl j`bufls usiln tog ilh`rfdti`l H`r oypgra`bd

x1 d1

j`lkundtg dxis is

y1

‚

a1

6 4, j`lkundtg dxis suatglcs dlnbg 8=½ dt `lg `h its vgrtgx. Sog bglnto `h its

1

J`bufl-I

J`bufl-II

(D) gjjgltrijity

(])

(A) bglnto `h bdtus rgjtuf

(T)

(J)

(Q)

(C)

(Z)

1 : 1 :


]DNG # 4:


Dls. Dls.

(D)  ]5 (A)  T

A 4

Z`b.

1 @

D'

:=½

D

: 1

A' d6 a6

g6

: 1 4 1 4

4 1



1



:

1 :

bglnto `h bdtus rgjtuf 6

13.

Z`buti`l sgt `h

x x ‚4

1a1 d



1 :

< = is Z. ] is tog sgt `h vdbugs `h x h`r woijo sil ‚4

h 4 ; Z  Q, h 4(x) 6 l

  x   l   is rgdb   x ‚ 4 

 x   x ‚ 4   

 x    x ‚ 4 

n4 ; ]  Q, n4(x) 6 sil ‚4  l 



J`bufl-I

J`bufl-II

(D) Qdlng `h n 4(x) is j`ltdilgc il

(])  ‚ ,

(A) Qdlng `h h 4(x) is j`ltdilgc il

(T) ( ‚, =)  (=, ) 

(J) C`fdil `h h 4(x) is j`ltdilgc il

(Q) ( ‚, =) 

(C) C`fdil `h n 4(x) is j`ltdilgc il

(Z)  ‚ ,

Dls.

((D)

s, (A)

Z`b.

‚4 2 l 

q, (J)

q, (C)

 

 

  g ‚ 4 4

 g   g ‚ 4,      g   g ‚ 4,    

 g   g ‚ 4,    

 g ‚ 4 g

p)

 x  4  T   x  4

4 g



x

4 g

 4

x 4

g

4 x 4

 ]

g


]DNG # 43


4 4 4  g 4 g x 4

 , g    4 ,      g 4  4 g     4   g    , g  4   g  4,      

(x ‚ 4)  x

Dlswgr is (D)  s, (A)  q, (J)  q, (C)  p


]DNG # 47

JEE Advanced 2018 Mathematics

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