wbjee sample paper 9 (wbjee2013 answers hints mathematics)

7/21/2019 WBJEE Sample Paper 9 (Wbjee2013 Answers Hints Mathematics)

1/23

@HUMDAWS - C

X. 8 X. 62 `hrry aom ihrk mh`j, far wjc`j aoly aom aptcao cs àrrm`t. Hoy wraod hoswmr wcll lmhe ta

emeu`tcao af 8/0 ihrk

8. H pacot R lcms ao tjm cr lm x7+ y79 863. Cf X 9 (=,87) hoe W 9 (87,=), tjmo tjm h dlm XRW cs

(H) 6

(G) 5

(@) 0

(E) 7

Hos > (G)

Jcots > X (=,87) W (87,=)

2 (2,2) iAX

987

=, i

AW9

=

87 i

AX.

i

AW9 8, sa XAW 9 /7 Jmo`m XRW 9 /5

7. H cr`lm phsscod tjraudj (2,2), (7,6), (6,7) ùts tjm x hxcs h tjm pacot R (2,2). Ujmo tjm lmodtj af AR, wjmrm A csarcdco, cs

(H)=

7(G)

=

7(@) = (E) 82

Hos > (@)Jcots > @cr`lm phssms tjraudj (2,2)

sa, x7+y7+7dx+7fy 9 2 (7,6) & (6,7) lcms ao ct sa, 77+67+5d+87f 9 2 (8)

77+67+87d+5f 9 2 (7)Frai (8) & (7),d 9 f 9 =/7Mqo. af `cr`lm cs x7+y7=x=y92Far y 9 2, x(x=)92 x92, x9=AR 9 =

0. Ujm laùs af tjm icepacots af tjm jares af ho mllpsm x7+5y79 5 tjht hrm erhwo fari tjm pasctcvm moe af tjm icoar hxcs,cs

(H) h cr`lm wctj `motrm8

,27 hoe rhecus 8

(G) h phrhgalh wctj faùs8

,27

hoe ecrm`trcx x 9 8

(@) ho mllcpsm wctj motrm8

2,7

, ihnar hxcs 8 hoe icoar hxcs8

7

(E) h jypmrgalh wctj motrm8

2,7

, trhosvmrsm hxcs 8 hoe aonudhtm hxcs8

7

@aem-TGNMM - 7287 (Hoswmrs & Jcots) Ihtjmihtc`s

HOPTMWP & JCOUP

far

TGNMM - 7280

P]G > IHUJMIHUC@P


2/23

TGNMM - 7280 (Hoswmrs & Jcots) Ihtjmihtc`s

Hos > Oa aptcao cs arrm`t

Jcots > Rasctcvm moe af icoar hxcs (2,8) gut ice-pt gm (j,k) x97j, y97k8 lcms ao mllcpsm

5j7+ 5(7k8)79 5

7

7

8k

j 78

885

+ 9

(2,8)

(J,k)

( ,y)Z

Jmrm ao mllcpsm af `motrm (2,8

7), ihnar hxcs 7, icoar hxcs 8

5. H pacot iavms sa tjht tjm sui af squhrms af cts ecstho`ms frai tjm pacots (8,7) hoe (7,8) cs hlwhys 6. Ujmo cts laùscs

(H) tjm strhcdjt lcom0 8

y 0 x7 7

9 +

(G) h cr`lm wctj `motrm8 0

,7 7

hoe rhecus8

7

(@) h phrhgalh wctj faùs (8,7) hoe ecrm`tcx psscod tjraudj (7,8)

(E) ho mllcpsm wctj fa`c (8,7) hoe (7,8)

Hos > (G)

Jcots > Lmt(j,k) gm a-arecohtms af tjm pacot

(j8)7+ (k7)7+ (j+7)7+ (k8)79 6

j7+k7+j 0k + 7 9 2

@cr`lm wctj motrm8 0,

77

rhecus 97

=. Far tjm vhrchglm t, tjm laùs af tjm pacots af cotmrsm`tcao af lcoms x7y 9 t hoe x+7y98

tcs

(H) tjm strhcdjt lcom x9y

(G) tjm cr lm wctj motrm ht tjm arcdco hoe rhecus 8

(@) tjm mllcpsm wctj motrm ht tjm arcdco hoe aom faùs7

,2=

(E) tjm jypmrgalh wctj motrm ht tjm arcdco hoe aom faùs=

,27

Hos > (E)

Jcots > (x7y) (x+7y) 9 8 x7- 5y79 8

7 7x y8

88

5

9


3/23


h 9 8, g 98

7m 9

=

7faùs

=,2

7

6. Lmt R 9

às sco5 5

sco às5 5

hoe Z 9

8

7

8

7

. Ujmo R0Z cs mquhl ta

(H)2

8

(G)

8

7

8

7

(@)8

2

(E)

8

7

8

7

Hos > (@)

Jcots > R7 92 8

8 2

R0 9

8 8

7 7

8 8

7 7

R0 Z 98

2

4. Ujm ouigmr af salutcaos af tjm mquhtcao x+y+z 9 82 co pasctcvm cotmdmrs x,y,z cs mquhl ta

(H) 06 (G) == (@) 47 (E) 5=Hos > (H)

Jcots > 828@08

9 3@79 06

:. Far 2 R,X7

, cf sco R + às X97, tjmo tjm vhlum af tho

R X

7

+

cs mquhl ta

(H) 8 (G)8

7(@)

8

7(E)

0

7

Hos > (H)

Jcots > R 9 7 , X 9 2

3. Cf hoe hrm tjm raats af x7 x+8 9 2, tjmo tjm vhlum af 7280+ 7280cs mquhl ta

(H) 7 (G) 7 (@) 8 (E) 8

Hos > (G)

Jcots > 9 , 7

7280 7x72809 (0)648 (0)7x6489 7


4/23


82. Ujm vhlum af tjm cotmdrhl

( )

7280

x x7

8

8

x 8ex

m x `as x m

+

+ +

cs mquhl ta

(H) 2 (G) 8m8 (@) 7 m8 (E) 7

( )

88 m

Hos > (E)

Jcots > ( )

7280

x 7

xcsaee

m x `asx+

C 9 ( )8 8

x 8

x8 2

8ex 7 m ex 7 8 m

m

9 9

88. Lmt

f(x) 9 7822x+8,

d(x) 9 0822

x+8.Ujmo tjm smt af rmhl ouigmrs x su`j tjht f(d(x)) 9 x cs

(H) mipty (G) h scodlmtao (@) h fcoctm sm wctj iarm tjho aom mlmimot

(E) cofcoctm

Hos > (G)

Jcots > f(x) 9 7822x+8 ; d(x) 9 0822x+8

f (d(x)) 9 x x 9( )822

822

8 7

6 8

+

87. Ujm lcict af x sco(m8/x) hs x 2

(H) cs mquhl ta 2 (G) cs mquhl ta 8 (@) cs mquhl ta m/7 (E) eams oat mxcst

Hos > (H)

Jcots > 8scom8/x8, -xxsco(m8/ ) x

lcix 2 xsco(m8/x)9 lcix 2 x 9 2,

80. Lmt C 9

8 2 2

2 8 2

2 2 8

hoe R 9

8 2 2

2 8 2

2 2 7

. Ujmo tjm ihtrcx R0+7R7cs mquhl ta

(H) R (G) CR (@) 7C+R (E) 7CR

Hos > (@)

Jcots > R C 2 9 , jhrh`tmrcstc`s mquhtcao af R cs R0+7R7R7C 9 2

R0+7R79R+7C


5/23


85. Cf ,hrm tjm raats af tjm quherhtc` mquhtcao x7+hx+g92, (g2); tjmo tjm quherhtc` mquhtcao wjasm raats hrm

8 8,

cs

(H) hx7+h(g8)x+(h8)792 (G) gx7+h(g8)x+(g8)792 (@) x7+hx+g 9 2 (E) hgx7+gx+h 9 2

Hos > (G)

Jcots > +9 h, 9 g

9 8

,8

9

+ 9( ) ( )

7h g 8 g 8

g g

9

Mquhtcao cs x7(+) x +9 2

gx7+ h(g8)x + (g8)79 2

8=. Ujm vhlum af

8 8 8 8

8222 ......8x7 7x0 0x5 333x8222

+ + + + cs mquhl ta

(H) 8222 (G) 333 (@) 8228 (E) 8/333

Hos > (G)

Jcots > 82228

87

8 8

7 0+

8

333 +

8

8222

9 82228

88222

9 333

86. Ujm vhlum af tjm emtmricohot

7 7

7 7

7 7

8 h g 7hg 7g

7hg 8 h g 7h

7g 7h 8 h g

+ +

cs mquhl ta

(H) 2 (G) (8+h7+g7) (@) (8+h7+g7)7 (E) (8+h7+g7)0

Hos > (E)

Jcots > (8+h7+g7)0

@8 @

8 g@

0, @

7h@

0+ @

7

( ) ( )

7 7

7 7

7 7 7 7 7 7

8 h g 2 7g

2 8 h g 7h

g 8 h g h 8 h g 8 h g

+ + + +

+ + + +

9 (8+h7+g7)7 7 7

8 2 7g

2 8 7h

g h 8 h g

9 (8+h7+g7)0


6/23


84. Cf tjm ecstho`m gmtwmmo tjm fa`c af ho mllcpsm cs mquhl ta tjm lmodtj af tjm lhtus rm`tui, tjmo cts m``motrc`cty cs

(H) ( )8

= 85

(G) ( )8

= 87

+ (@) ( )8

= 87

(E) ( )8

= 85

+

Hos > (@)

Jcots > 7hm 977g

h

m 97

7

g

h9 8 m7

m7+ m8 9 2

m 9= 8

7

8:. Far tjm urvm x7+5xy+:y7965 tjm thodmots hrm phrhllml ta tjm x-hxcs aoly ht tjm pacots

(H) (2,7 7 ) hoe (2,7 7 )

(G) (:,5) hoe (:,5)

(@) ( ): 7, 7 7 hoe ( ): 7,7 7(E) (:,2) hoe (:,2)

Hos > (G)

Jcots > x7+5xy+:y79 65

7x+5xy+5y+86yy92 (5x+86y)y9 (7x+5y) 7x + 5y 9 2

83. Ujm vhlum af C 9 ( ) ( )o 8 o 85 7

2 2

8tho x ex tho x / 7 ex

7+

+ cs mquhl ta

(H)8

o(G)

o 7

7o 8

++

(@)7o 8

o

(E)

7o 0

0o 7

Hos > (H)

Jcots > C9o 8 o 8

5 /7

2 2

xtho x.ex tho ex

7 7+

+

Far sm`aoe cotmdrhl sugstctutmx

y7

9

C 9 ( )o 8 o 8/5

2tho x tho x ex+

+o

o 8 7/5

/52

2

tho x 8tho x.sm` x ex

o o

9 9

72. Lmt f() 9 (8+sco7)(7sco7). Ujmo far hll vhlums af

(H) f() ?3

5(G) f()


7/23


Hos > (E)

Jcots > f() 9 (8+sco7) (7sco7)f()9(8+sco7)(8+`as7)97+sco7`as7

97+8

5sco7

7f() 35

78. Lmt f(x) 9

0

0 7

x 0x 7, x 7

x 6x 3x 7, x 7

+ (@)

Jcots >x 7lci

+f(x) 9 5

x 7lci

f(x) 9 5

f(x) 97

7

0x 0, x 7

0x 87x 3, x 7

(@)

Jcots >7 0 5 8222

xlci ( x x x x .... ........ x )

+ + +

9( )8222

x

( x) 8lci ( x ).

x 8

9

8228

x

x xlci

x 8

+ 9 +

70. Cf f(x) 9 mx(x 7)7tjmo

(H) f cs co`rmhscod co ( ),2 hoe ( )7, hoe em`rmhscod co (2, 7)

(G) f cs co rmhscod co ( ),2 hoe em`rmhscod co ( )2,

(@) f cs co`rmhscod co ( )7, hoe em`rmhscod co ( ),2

(E) f cs co rmhscod co (2, 7) hoe em`rmhscod co ( ),2 hoe ( )7,Hos > (H)

Jcots > f(x) 9 mx[(x 7)7+ 7(x 7)Y9 mx[x7 7xY 9 mx.x(x 7)

scdo s`jmim af f(x) wcll gm+

2 7

+

sa f cs co`rmhscod co (, 2) hoe (7, ) hoe em`rmhscod co (2, 7)


8/23


75. Lmt f > gm su`j tjht f cs conm`tcvm hoe f(x)f(y) 9 f(x +y) far hll x, y . Cf f(x), f(y), f(z) hrm co D.R., tjmo x, y,z hrm co

(H) H.R. hlwhys

(G) D.R. hlwhys

(@) H.R. empmoecod ao tjm vhlums af x, y, z

(E) D.R. empmoecod ao tjm vhlums af x, y, z

Hos > (H)

Jcots > f (x + y) 9 f(x).f(y), sa f(x) 9 hkx

hkx, hky, hkzhrm co D.R

h7ky9 hk(x + z)

7y 9 x + z, sa x, y, z hrm co H.R7=. Ujm ouigmr af salutcaos af tjm mquhtcao

730

8 x 8lad lad (x =) 8

7 x =

++ + 9 +

cs

(H) 2 (G) 8 (@) 7 (E) cofcoctm

Hos > (G)

Jcots > 0 0x 8

lad lad (x =) 8x =

++ + 9 +

(x + 8) 9 0, x 9 7 sa aoly aom salutcao

76. Ujm hrmh af tjm rmdcao gauoeme gy tjm phrhgalh y 9 x75x + = hoe tjm strhcdjt lcom y 9 x + 8 cs

(H) 8/7 (G) 7 (@) 0 (E) 3/7

Hos > (E)

Jcots >

y9x

+8

8 5

y 8 9 (x 7)7

Wmqucrme Hrmh 9 ( )5

7

8

(x 8) (x 5x =) ex+ +

93

sq.uoct7

74. Ujm vhlum af tjm cotmdrhl7

xm

8

x 8m lad x ex

x

+ + cs

(H) m7(8 + ladm7) (G) m7 m (@) m7(8 + lad

m7) m (E) m7m(8 + lad

m7)

Hos > (@)

Jcots >

7x

m

8

8m lad x 8 ex

x

+ + 97 7

x xm

8 8

8m .ex m lad x ex

x

+ +

9 (m7 m8) +7x

m8

m lad x

9 m7 m + m7ladm7

9 m7(8 + ladm7) m


9/23


7:. Lmt R 9 78 8

8 ........7 7 0 7

+ + +

hoe X 98 8 8

........8 7 0 5 = 6

+ + +

Ujmo

(H) R 9 X (G) 7R 9 X (@) R 9 7X (E) R 9 5X

Hos > (@)

Jcots >

7 08 8

R 8 7 7.........ta

7 7 7 0

9 + + +

9 ladm

88

7

sa R 9 7 ladm7

X 98 8 8 8 8 8

........ta8 7 0 5 = 6

+ + +

9 ladm(8 + 8) 9 lad

m7

R 9 7X

73. Lmt f(x) 9 sco x + 7 as7x,0

x5 5

. Ujmo f htthcos cts

(H) icociui ht x 95

(G) ihxciui ht x 9

7

(@) icociui ht x 97

(E) ihxciui ht x 9

8 8sco5

Hos > (@)

Jcots > f(x) 9 scox + 7`as7x

9 7sco7x + scox + 7

9 7 (sco7x scox) + 7

9 7

78 8

scox 75 :

+ +

9

784 8

7 sco x: 5

; oemr tjm dcvmo eaihco

f(x) wcll gm icociui wjmo

78

scox5

cs ihxciui wjc`j cs ht x 9 7

02. Mh`j af h hoe g ho thkm vhlums 8 ar 7 wctj mquhl praghgclcty. Ujm praghgclcty tjht tjm mquhtcao hx7+ gx + 8 9 2 jhsrmhl raats, cs mquhl ta

(H)

8

7 (G)

8

5 (@)

8

: (E)

8

86

Hos > (G)

Jcots > hx7+gx + 8 9 2 jhs rmhl raats far g75h 2

Pa h jhs ta gm 8 hoe g jhs ta gm 7

sa praghgclcty cs 98 8 8

7 7 5 9


10/23


08. Ujmrm hrm twa acos, aom uogchsme wctj praghgclcty8

7af dmttcod jmhes hoe tjm atjmr aom cs gchsme wctj praghgclcty

0

5

af dmttcod jmhes. H aco cs smlm`tme ht rhoeai hoe tassme. Ct sjaws jmhes up. Ujmo tjm praghgclcty tjht tjm uogchsmeàco whs smlm`tme cs

(H)7

0(G)

0

=(@)

8

7(E)

7

=

Hos > (E)Jcots > J Mvmot af jmhe sjawcod up

G Mvmot af gchsme àco `jasmo

]G Mvmot af uogchsme aco jasmo

JR(]G).R

]G ]GR

J JJR(]G).R R(G).R

]G G

9 +

8 877 7

8 8 8 0 =7 7 7 5

9

+

07. Far tjm vhrchglm t, tjm laùs af tjm pacot af cotmrsm`tcao af tjm lcom 0tx 7y + 6t 92 hoe 0x + 7ty 6 9 2 cs

(H) tjm mllcpsm7 7x y

85 3

+ 9 (G) tjm mllcpsm7 7x y

83 5

+ 9

(@) tjm jypmrgalh7 7x y

85 3

9 (E) tjm jypmrgalh7 7x y

83 5

9

Hos > (H)

Jcots > Ujm pacot af cotmrsm`tcao af 0tx 7y + 6t 9 2 hoe 0x + 7ty 6 9 2 cs

x 97

7

7(8 t )

(8 t )

+ , y 9 7

6t

(8 t )+

@aoscemrcod t 9 tho , x 9 7às 7, y 9 0.sco 7

sa laùs af pacot af cotmrsm`tcao cs tjm mllcpsm

7 7x y8

5 3+ 9

00. @hres hrm erhwo aom-gy-aom wctjaut rmplh`mimot frai h wmll sjufflme ph`k af =7 hres. Ujmo tjm praghgclcty tjht hfh`m hre (Nh`k, Xummo ar Kcod) wcll hppmhr far tjm fcrst tcim ao tjm tjcre turo cs mquhl ta

(H)022

7834(G)

06

:=(@)

87

:=(E)

5

=8

Hos > (@)

Jcots > R (fh`m hre ao tjcre turo) 9 R (oa fh`m hre co fcrst turo) R (oa fh`m hre co 7oe turo) R (fh`m hre co 0returo)

952 03 87 87

=7 =8 =2 := 9


11/23


05. Lcoms x + y 9 8 hoe 0y 9 x + 0 cotmrsm`t tjm mllcpsm x7+ 3y79 3 ht tjm pacots R,X,W. Ujm hrm af tjm trchodlm RXW cs

(H)06

=(G)

8:

=(@)

3

=(E)

8

=

Hos > (G)

Jcots > W ( 3=

5=

, )

X

R

0y 9 x + 0 (0, 2)

x + y 9 8

(2, 8)

Hrmh af RXW 9

32 0

=88 2 5 / =

78 8 8

98:

=sq. uocts

0=. Ujm ouigmr af aota fuo tcaos frai tjm smt {8, 7, ..........., 88} ta smt {8, 7, .... 82} cs

(H) = 88 (G) 82 (@)88

7(E) 82 88

Hos > (E)

Jcots > Oa. af aota fuo`tcao 9 88 82@ 82 82

9 82 88

06. Ujm lcict af

x

7 x x

8 (7280) 8

x m 8 m 8

+

hs x 2

(H) hpprah`jms + (G) hpprah`jms (@) cs mquhl ta ladm(7280) (E) eams oat mxcst

Hos > (H)

Jcots >

x

7 x xx 2

8 (7280) 8lci

x m 8 m 8

+

9

x

7 xx 2 x 2

8 (7280) 8 xlci lci

xx m 8

+

9 mlad (7280) +

9 +04. Lmt z

89 7 + 0c hoe z

79 0 + 5c gm twa pacots ao tjm àiplmx plhom. Ujmo tjm smt af àiplmx ouigmrs z shtcsfycod

|z z8|7+ |z z

7|79 |z

8 z

7|7rmprmsmots

(H) h strhcdjt lcom (G) h pacot (@) h cr`lm (E) h phcr af strhcdjt lcoms

Hos > (@)

Jcots > ^8

^ (0, 5)7

(7, 0)

^

@lmhrly tjm laùs af cs h `cr`lm wctj 8&

7hs moe pacot

af echimtmr.

0:. Lmt p(x) gm h quherhtc` palyoaichl wctj aosthot tmri 8. Puppasm p(x) wjmo ecvceme gy x 8 lmhvms rmihcoemr 7 hoewjmo ecvceme gy x + 8 lmhvms rmihcoemr 5. Ujmo tjm sui af tjm raats af p(x) 9 2 cs

(H) 8 (G) 8 (@)8

7 (E)

8

7


12/23


Hos > (E)

Jcots > R(x) 9 hx7+ gx + 8

R(8) 9 h + g + 8 9 7

R(8) 9 h g + 8 9 5

sa g 9 8, h 9 7

sui af raats af R(x) csg

h

9

8

7

03. Mlmvmo hpplms hrm ecstrcgutme hiaod h dcrl hoe h gay. Ujmo wjc`j aom af tjm fallawcod sthtmimots cs trum 1

(H) Ht lmhst aom af tjmi wcll rm`mcvm 4 hpplms

(G) Ujm dcrl rm`mcvms ht lmhst 5 hpplms ar tjm gay rm`mcvms ht lmhst 3 hpplms

(@) Ujm dcrl rm`mcvms ht lmhst = hpplms ar tjm gay rm`mcvms ht lmhst : hpplms

(E) Ujm dcrl rm`mcvms ht lmhst 5 hpplms ar tjm gay rm`mcvms ht lmhst : hpplms

Hos > ()

Jcots >

52. Fcvm ouigmrs hrm co J.R. Ujm iceelm tmri cs 8 hoe tjm rhtca af tjm sm`aoe hoe tjm faurtj tmris cs 7>8. Ujmo tjm suiaf tjm fcrst tjrmm tmris cs

(H) 88/7 (G) = (@) 7 (E) 85/0

Hos > (H)

Jcots > Lmt h, g, 8, `, e hrm J.R

sa8

h,

8

g, 8,

8

`,

8

ehrm H.R, g 9 7@.

8 87

g `+ 9 sa g 9

0

7, ` 9

0

5, h 9 0.

sa, sui af tjm fcrst tjrmm tmris 90 88

0 87 7

+ + 9

58. Ujm lcict af 78 8

8 x 8x x

+ + hs x 2

(H) eams oat mxcst (G) cs mquhl ta 8/7 (@) cs mquhl ta 2 (E) cs mquhl ta 8

Hos > (H)

Jcots > W.J.L.7

x 2

8 x x 8lci

x x+ +

97 7

7x 2

8 x x 8 8 x x 8lci

x 8 x x 8+ + + + +

+ + +

9 ( )

7

x 2 7

8 x 8 xlci

x 8 x 8 x

+

+ + +

9 ( )x 2 7x(8 x)

lci x 8 x 8 x

+ + + 9 87. L.J.L. 9

7

x 28 x x 8lci

x+ + + 9

W.J.L. L.J.L.57. Ujm ihxciui hoe icociui vhlums af as6+ sco6hrm rmspm`tcvmly

(H) 8 hoe 8/5 (G) 8 hoe 2 (@) 7 hoe 2 (E) 8 hoe 8/7

Hos > (H)

Jcots > sco6+ as69 8 0sco7.`as7

9 80

5sco77


13/23


50. Cf h, g, hrm co H.R., tjmo tjm strhcdjt lcom hx + 7gy + ` 9 2 wcll hlwhys phss tjraudj h fcxme pacot wjasm à-arecohtmshrm(H) (8, 8) (G) (8, 8) (@) (8, 7) (E) (7, 8)

Hos > (H)

Jcots > h(x + y) + `( y + 8) 9 2x 9 8, y 9 8

55. Cf aom moe af h echimtmr af tjm `cr`lm 0x7+ 0y7 3x + 6y + = 9 2 cs (8, 7) tjmo tjm atjmr moe cs(H) (7, 8) (G) (7, 5) (@) (7, 5) (E) (5, 7)

Hos > (@)

Jcots >@motmr0

, 87

Lmt tjm atjmr pacot gm (j, k)j 8 0

j 77 7

+9 9

k 78 k 5

7

+9 9

5=. Ujm vhlum af as74=2+ as75=2+ as78=2 as7022 as7622cs

(H) 2 (G) 8 (@) 8/7 (E) 8/5

Hos > (@)

Jcots > às8=9 sco4=

às74= + às75= + às78= às702 às762

9 às74= + sco74= + às75= às702 às762

98 0 8

87 5 5

+

98

7

56. Puppasm z9 x + cy wjmrm x hoe y hrm rmhl ouigmrs hoe c 89 . Ujm pacots (x, y) far wjc`jz 8

z c

cs rmhl, lcm ao

(H) ho mllcpsm (G) h cr`lm (@) h phrhgalh (E) h strhcdjt lcom

Hos > (E)

Jcots >(x 8) cy

kx c(y 8)

+9

+

(x 8) cy x c(y 8)k

x c(y 8) x c(y 8)

+ 9

+

Cihdcohry phrt 9 2 x + y 9 854. Ujm mquhtcao 7x7+ =xy 87y7 9 2 rmprmsmots h

(H) `cr`lm

(G) phcr af oao-pmrpmoec ulhr cotmrsm`tcod strhcdjt lcoms

(@) phcr af pmrpmoec ulhr strhcdjt lcoms

(E) jypmrgalh

Hos > (G)

Jcots > 7x7+ =xy 87y79 2

(x + 5y) (7x 0y) 9 2


14/23


5:. Ujm lcom y 9 x cotmrsm ts tjm jypmrgalh7 7x y

83 7=

9 ht tjm pacots R hoe X. Ujm m``motrc`cty af mllcpsm wctj RX hs ihnar

hxcs hoe icoar hxcs af lmodtj=

7cs

(H)=

0

(G)=

0

(@)=

3

(E)7=

3Hos > ()

Jcots > Far y 9 x,7 8 8x 8

3 7=

9

7 7

7 = 0 8=x5 5

9 9

8= 7 8=h

5 7 7 9 9 ,

=g

7 79

7 8 :m 83 3

9 9

7 7m0

9

53. Ujm mquhtcao af tjm cr lm phsscod tjraudj tjm pacot (8, 8) hoe tjm pacots af cotmrsm`tcao af x7+ y7 6x : 9 2 hoe x7

+ y7 6 9 2 cs

(H) x7+ y7+ 0x = 9 2 (G) x7+ y7 5x + 7 9 2 (@) x7+ y7+ 6x 5 9 2 (E) x7+ y7 5y 7 9 2

Hos > (H)

Jcots > @cr`lm phsscod tjraudj pacot af cotmrsm`tcao af cr`lms cs x7+ y7 6x : + (x7+ y7 6) 9 2Ct phssms tjraudj (8, 8) sa, 9 0@cr`lm cs x7+ y7+ 0x = 9 2

=2. Pcx pasctcvm ouigmrs hrm co D.R., su`j tjht tjm r praeu`t cs 8222. Cf tjm faurtj tmri cs 8, tjmo tjm lhst tmri cs

(H) 8222 (G) 822 (@) 8/822 (E) 8/8222Hos > (@)

Jcots > = 0h h h

, , ,rr r

hr, hr0, hr=

h69 8222 h79 82

dcvmo hr 9 8, h7r79 8, 78

r82

9

hr=98

822

=8. Co tjm smt af hll 00 rmhl ihtrc`ms h rmlhtcao cs emfcome hs fallaws. H ihtrcx H cs rmlhtme ta h ihtrcx G cf hoe aoly cf tjmrm

cs h oao-scodulhr 00 ihtrcx R su`j tjht G 9 R8HR. Ujcs rmlhtcao cs(H) Wmflmxcvm, Pyiimtrc` gut oat Urhosctcvm (G) Wmflmxcvm, Urhosctcvm gut oat Pyiimtrc`

(@) Pyiimtrc`, Urhosctcvm gut oat Wmflmxcvm (E) ho Mqucvhlmo`m rmlhtcao

Hos > (E)

Jcots > W 9 {(H, G) | G 9 R8HR}

H 9 C8HC (H, H) W W cs rmflmxcvmLmt (H, G) W , G 9 R8HRRG 9 HR RGR89 H H 9 (R8)8G(R8)


15/23


(G, H)W, W cs syiimtrc`Lmt (H, G) W, (G, @) WH 9 R8GR hoe G 9 X8@X

H 9 R8X8@ XR 9 (XR)8@(XR)(H, @)W=7. Ujm ouigmr af lcoms wjc`j phss tjraudj tjm pacot (7, 0) hoe hrm ht tjm ecstho`m : frai tjm pacot (8, 7) cs

(H) cofcoctm (G) 5 (@) 7 (E) 2

Hos > (E)

Jcots > Ujm ihxciui ecstho`m af tjm lcom phsscod tjraudj (7, 0) frai (8, 7) cs 05 . Pa tjmrm cs oa passcglm lcom=0. Cf , hrm tjm raats af tjm quherhtc` mquhtcao hx7+ gx + ` 9 2 hoe 0g79 86h` tjmo

(H) 9 5 ar 9 5 (G) 9 5 ar 9 5 (@) 9 0 ar 9 0 (E) 9 0 ar 9 0Hos > (@)

Jcots > 0g79 86h`

7g `

0 86h h

9

( )7

0 86 + 9 , 7 70 0 82 + 9

0 0 82

+ 9

, Lmt y

9

0y7 82y + 0 9 2,(0y 8) (y 0) 9 2

8y

09 ar y 9 0

09 ar 9 0=5. Far hoy twa rmhl ouigmrs h hoe g, wm emfcom h W g cf hoe aoly cf sco7h + às7g 9 8. Ujm rmlhtcao W cs

(H) Wmflmxcvm gut oat Pyiimtrc` (G) Pyiimtrc` gut oat trhosctcvm

(@) Urhosctcvm gut oat Wmflmxcvm (E) ho Mqucvhlmo`m rmlhtcao

Hos > (E)

Jcots >sco7h + às7g 9 8

Wmflmxcvm > sco7h + as7h 9 8

hWhsco7h + às7g 9 8, 8 as7h + 8 sco7g 9 8sco7g + às7h 9 8

gWhJmo`m syiimtrc` Lmt hWg gW

sco7h + às7g 9 8 ............. (8)

sco7g + às7` 9 8 ............... (7)

(8) + (7)

sco7h + às7` 9 8

Jmo`m trhosctcvm tjmrmfarm mqucvhlmo`m rmlhtcao.

==. Lmt o gm h pasctcvm mvmo cotmdmr. Ujm rhtca af tjm lhrdmst amffc cmot hoe tjm 7oelhrdmst amffc`cmot co tjm mxphoscao af(8 + x)ocs 88>82. Ujm tjm ouigmr af tmris co tjm mxphoscao af (8 + x)ocs

(H) 72 (G) 78 (@) 82 (E) 88

Hos > (G)

Jcots > Lmt o 9 7i

7ii

7ii 8

@ 88

82@ 9

i 9 82, o 9 72Uathl Oa. af tmri 9 78


16/23


=6. Lmt mxp(x) emoatm mxpaomotchl fuo`tcao mx. Cf f(x) 9

8

xmxp x ,

x ? 2 tjmo tjm icociui vhlum af fco tjm cotmrvhl [7, =Y cs

(H)8

mmxp m

(G)8

7mxp 7

(@)8

=mxp =

(E)8

0mxp 0

Hos > (@)

Jcots >8/xxf(x) m9

d(x) 9 lad f(x) 98

xx

d(x) co`rmhsms cs (2, m) & em`rmhsms co (m, ) ct wcll gm icociui ht mctjmr 7 ar =

88

=77 =? icociui vhlum af f(x) 98

==m

=4. Ujm sui af tjm smrcms

7= 7= 7= 7=

2 8 7 7=

8 8 8 8@ @ @ ......... @

8 7 7 0 0 5 76 74+ + + +

(H)747 8

76 74

(G)

747 7:76 74

(@)

76

8 7 87 76 74

+

(E)767 8=7

Hos > (G)

Jcots > Ao cotmdrhtm (8 + x)7=twc`m 8st uoemr tjm lcict 2 ta x & tjmo 2 ta 8 wm dmt sui 9747 7:

76 74

=:. Fcvm ouigmrs hrm co H.R. wctj aiiao ecffmrmo`m 2 Cf tjm 8st, 0rehoe 5tjtmris hrm co D.R., tjmo

(H) tjm =tjtmri cs hlwhys 2 (G) tjm 8sttmri cs hlwhys 2

(@) tjm iceelm tmri cs hlwhys 2 (E) tjm iceelm tmri cs hlwhys 7

Hos > (H)

Jcots > Lmt h, h + e, h + 7e, h + 0e, h + 5e hrm fcvm ouigmr co H.R.

Dcvmoh 7e h 0e

h h 7e

+ +9

+

h + 5e 9 2=3. Ujm icociui vhlum af tjm fuo`tcao f(x) 9 7|x 8| + |x 7| cs

(H) 2 (G) 8 (@) 7 (E) 0

Hos > (G)

Jcots > f(x) wcll gm icociui ht x 9 8

62. Cf R, X, W hrm hodlms af ho csas`mlms trchodlm hoe R ,7

9 tjmo tjm vhlum af

0R R

às csco0 0

+ (àsX + cscoX)

(àsW cscoW) + (àsR cscoR) (àsX cscoX) (àsW cscoW) cs mquhl ta(H) c (G) c (@) 8 (E) 8

Hos > (G)

Jcots > R7

9 , X W

5

9 9

( )0

R Ràs csco àsX cscoX (àsW cscoW) (àsR cscoR)(àsX cscoX)(àsW cscoW)

0 0

+ + +


17/23


9 cp cX cW cR cX cWm m .m m m + +

9 c /7 c (X W) c(R X W)m m m + ++ +

9 c /7 2 cm m m + +

9 ( )às csco 8 às csco7 7

+ +

9 c + 8 8 2 9 c

@HUMDAWS - CC

X. 68 X. 4= `hrry twa ihrks mh`j, far wjc`j aoly aptcao cs àrrm`t. Hoy wraod hoswmr wcll lmhe taemeu`tcao af 7/0 ihrk.

68. H lcom phsscod tjraudj tjm pacot af cotmrsm`tcao af x + y 9 5 hoe x y 9 7 ihkms ho hodlm tho8(0/5)wctj tjm x-hxcs. Ctcotmrsm`ts tjm phrhgalh y79 5(x0) ht pacots (x

8, y

8) hoe (x

7, y

7) rmspm`tcvmly. Ujmo |x

8x

7| cs mquhl ta

(H)86

3(G)

07

3(@)

52

3(E)

:2

3

Hos > (G)

Jcots > H(0, 8)

( )0 0 3

y 8 x 0 y x 85 5 5

9 9 + ar, ( )70 =

y x ,y 5 x 05 5

9 9

( )7

0x =5 x 0

5

9

ar, 73x 02x 7= 65x 65 0 + 9

73x 35x 784 2 + 9

8 735

x x3

+ 9

8 7

784x x

39

( ) ( )77 7

8 7 8 7 8 735 784x x x x 5x x 53 3

9 + 9

( )7

7

35 5.784.3

3

9

07

39

67. Lmt [hY emoatm tjm drmhtmst cotmdmr wjc`j cs lmss tjho ar mquhl ta h. Ujmo tjm vhlum af tjm cotmdrhl

7

7

[scoxàsxYex

cs

(H)7

(G) (@) (E) / 7

Hos > (E)

Jcots >

7

7

8C Pco7x ex Rut7x ar,7ex e

7

9 9 9


18/23


( )2

2

8 8 8Pco e 8 ex 2ex

7 7 7

9 9 + 2

( )( ) ( )28 8

x 2 27 7 7

9 + 9 + 9

60. Cf

7 7 5

R 8 0 5

8 7 0

9

tjmo R=mquhls

(H) R (G) 7R (@) R (E) 7R

Hos > (H)

Jcots >7

7 7 5 7 7 5

R 8 0 5 8 0 5

8 7 0 8 7 0

9

9

7 7 5

8 0 5

8 7 0

R79 p ; p59 p; p=9p79p

65. Cf 7sco 0`as 7 + 9 , tjmo 0 0`as sm` + cs(H) 8 (G) 5 (@) 3 (E) 8:

Hos > (E)

Jcots > 7@as 0@as 8 2 + 9 ar,8

@as 0@as

+ 9

,

00

8 8 8@ 0.@. @ 74

@ @@

+ + + 9

ar, 0 0@as sm` 0.0 74 + + 9

0 0@as sm` 8: + 9

6=.8 8 8

x 8 ........7 8 5 7 : 0

9 + + + + hoe

7 5 6x x xy 8 ......

8 7 09 + + + + . Ujmo tjm vhlum af lad

my cs

(H) m (G) m7 (@) 8 (E) 8/m

Hos > (H)

Jcots > 78

x 7y m , x m9 9

lo y 9 x79 m


19/23


66. Ujm vhlum af tjm cofcoctm smrcms

7 7 7 7 7 7 7 7 78 7 8 7 0 8 7 0 5..........

0 5 =

+ + + + + ++ + cs

(H) m (G) =m (@) =m 8

6 7 (E)

=m

6

Hos > (@)

Jcots >( )77 7 7o

r 8

8 7 0 ....... r 8

r 79

+ + + ++ 9 r 8

8 7r 0 8 7 0

6 r 6 r 8 r9

+ 9 + 9 ( )8 = 87m 0 m 8 m

6 6 7 + 9

64. Ujm vhlum af tjm cotmdrhl( )

( )

0

6

sco x x `as xex

x x scox

+ cs mquhl ta

(H)( )

m

7 0lad

7 0 0

+

+ (G) ( )

m

0lad

7 7 0 0

+ +

(@) ( )m7 0 0

lad7 0

+ +

(E)( )

m

7 7 0 0lad

0

+ +

Hos > (H)

Jcots >( ) ( )

( )

x Pcox x 8 @asxC ex

x x Pcox

+ +9

+ 98 8 @asx

exx x Pcox

+ +

90 8

lo lo lo lo0 6 0 7 6 7

+ +

7 6lo

7 0 0

+ 9 +

6:. Lmt8 8 8

f(x) x ,x 8x 8 x x 8

9 + + ? + Ujmo

(H) f(x) 8 (G) 8 < f(x)7 (@) 7 < f(x)0 (E) f(x)?0

Hos > (E)

Jcots >8 8 8

f(x) xx 8 x 8 x

9 + + +

9 7x 8 x 8

x 8x 8

+ + +

7

7

7x8

x 89 +

7

78

8

8 x

9 +


20/23


63. Lmt F(x) 9 ( )

x

72

àstet,2 x 7

8 t

+ . Ujmo

(H) F cs co`rmhscod co0

,7 7

hoe em`rmhscod co 2,7

hoe0

,77

(G) F cs co`rmhscod co ( )2, hoe em`rmhscod co ( ),7

(@) F cs co`rmhscod co ( ),7 hoe em`rmhscod co ( )2,

(E) F cs co`rmhscod co 2,7

hoe0

,77

hoe em`rmhscod co0

,7 7

Hos > (E)

Jcots > ( ) 7@asx

F x8 x

9+

0@as x 2 x 2, ,7

7 7

?

0@asx 2 x ,

7 7

<

42. Lmt f(x) 9 x7/0, x 2. Ujmo tjm hrmh af tjm rmdcao mo`lasme gy tjm rvm y 9 f(x) hoe tjm tjrmm lcoms y 9 x, x 9 8 hoex 9 : cs

(H)60

7(G)

30

=(@)

82=

4(E)

873

82

Hos > (E)

Jcots >

y9x

y9x7/0

x98 x9:

( ) ( )7:

: :7 =/00

8 88

8 0H x x ex x x

7 =

9 9

( ) ( )8 0 87365 8 07 87 = 82

9 9

48. Lmt R gm h pacot ao tjm phrhgalh y79 5hx wctj faùs F. Lmt X emoatm tjm faat af tjm pmrpmoecùlhr frai R aota tjm

ecrm`trcx. Ujmotho RXF

tho RFX

cs

(H) 8 (G) 8/7 (@) 7 (E) 8/5

Hos > (H)

Jcots >

X(-h, 7h)

R(h, 7h)

F(q,2)2

RX 9 RF sa 9


21/23


47. Ho agnm`tcvm typm tmst phpmr jhs = qumstcaos. Aut af tjmsm = qumstcaos, 0 qumstcaos jhvm faur aptcaos mh`j (H, G, @,E) wctj aom aptcao gmcod tjm arrm`t hoswmr. Ujm atjmr 7 qumstcaos jhvm twa aptcaos mh`j, ohimly Urum hoe Fhlsm. H`hoecehtm rhoeaily tc`ks tjm aptcaos. Ujmo tjm praghgclcty tjht jm/sjm wcll tc`k tjm àrrm`t aptcao co ht lmhst faurqumstcaos, cs

(H)=

07(G)

0

87:(@)

0

7=6(E)

0

65

Hos > (E)

Jcots > o(P) 9 50

. 77

, o(m) 9 (0

@8.0 +7

@8. 8) + 8

0 7

0.0 7 8 87 0R

5.5.5.5 655 .7

+ +9 9 9

40. H fhicly af urvms cs su`j tjht tjm lmodtj cotmr`mptme ao tjm y-hxcs gmtwmmo tjm arcdco hoe tjm thodmot ht h pacot cstjrmm tcims tjm arecohtm af tjm pacot af aoth`t. Ujm fhicly af urvms cs

(H) xy 9 , cs h aosthot (G) xy79 `, ` cs h àosthot

(@) x7y 9 , cs h aosthot (E) x7y79 `, ` cs h àosthot

Hos > (@)

Jcots > yxey

ex9 0y ar, x

ey

ex9 7y

ar,ey ex

7y x9 (Cotmdrhtm) ar, lo y 9 7 lo x + lo ` ar, lo y + lo x79 lo ar, y x79 `

45. Ujm salutcao af tjm ecffmrmotchl mquhtcao ( )7 eyy 7x yex

+ 9 shtcsfcms x 9 8, y 9 8. Ujmo tjm salutcao cs

(H) x 9 y7(8 + ladmy) (G) y 9 x7(8 + lad

mx) (@) x 9 y7(8 lad

my) (E) y 9 x7(8 lad

mx)

Hos > (H)

Jcots >

7ex y 7x

ey y

+9 ar,ex 7

.x yey y

9 ar,7

eyloyy

7

8CF m m

y

9 9 9 ar, 7 7

8 8x. y. ey `

y y9 +

ar, 7x

loy `y

9 + y(8) 9 8 ar, 8 9 2 + `

x 9 y7(lo y + 8)

4=. Ujm salutcao af tjm ecffmrmotchl mquhtca y sco (x/y) ex9 (x sco(x/y)y) ey shtcsfycod y ( )/ 5 98 cs

(H) mx 8

às lad yy 7

9 + (G) mx 8

sco lad yy 7

9 + (@) mx 8

sco lad xy 7

9 (E) mx 8

às lad xy 7

9

Hos > ()

Jcots >

x xsco 8

ex y yxey scoy

9

Rutx

y9 ar, x 9 y . tjmo,

ex ey

ey ey

9 +

ar,e Pco 8

yey Pco

+ 9

ar,e Pco 8 8

yey Pco Pco

9 9

ar,ey

sco ey

9

ar, loy às ` y 85

9 9

8

`7

9

ar,x 8

loy àsy 7

9


22/23


@HUMDAWS - 0

X. 46 X. :2 `hrry twa ihrks mh`j, far wjc`j aom ar iarm tjho aom aptcaos ihy gm àrrm`t. Ihrkcod afàrrm`t aptcaos wcll lmhe ta h ihxciui ihrk af twa ao pra rhth ghscs. Ujmrm wcll gm oa omdhtcvm ihrkcod

far tjmsm qumstcaos. Jawmvmr, hoy ihrkcod af wraod aptcao wcll lmhe ta hwhre af zmra ihrk hdhcost tjmrmspm`tcvm qumstcao crrmspm`tcvm af tjm ouigmr af arrm`t aptcaos ihrkme

46. Ujm hrmh af tjm rmdcao mo`lasme gmtwmmo phrhgalh y79 x hoe tjm lcom y 9 ix cs8

5:. Ujmo tjm vhlum af i cs

(H) 7 (G) 8 (@) 8 (E) 7Hos > (H, E)

Jcots > .

8

i7

2

yH y ey

i

9 9

( ) ( )8

8/i7 0 i

2 2

8 8y y

7i 0

0 0

8 8 8

5: 7i 0i9 ar, 0

8 8

5: 6i9

(8)0 8i .5: 8.: :ar,i 7

69 9 9 9

(7)0 8

i .5: 8.:ar,i 769 9 9 44. @aoscemr tjm systmi af mquhtcaos>

x + y + z 9 2

x + y +z 9 2 7x + 7y + 7z 9 2

Ujmo tjm systmi af mquhtcaos jhs

(H) H uocqum salutcao far hll vhlums , ,(G) Cofcoctm ouigmr af salutcaos cf hoy twa af , hrm mquhl(@) H uocqum salutcao cf , ,hrm ecstco`t(E) Iarm tjho aom, gut fcoctm ouigmr af salutcaos empmoecod ao vhlums af , ,Hos > (G, @)

Jcots >( ) ( )( )

7 7 7

8 8 8

9

4:. Ujm mquhtcaos af tjm cr lms wjc`j tau`j gatj tjm hxms hoe tjm lcom 5x + 0y 9 87 hoe jhvm motrms co tjm fcrst quherhot,hrm

(H) x7+ y7 x y + 8 92 (G) x7+ y7 7x 7y + 8 92

(@) x7+ y7 87x 87y + 0692 (E) x7+ y7 6x 6y + 06 92

Hos > (G, @)

Jcots >5j 0j 87

j=

+ 9 ar, |4j87| 9 =j

(c) 4j 87 9 =j ar, 7j 9 87 ar, j 9 6 @motrm (6, 6)

x7+ y7 87x 87y + 0692

(cc) 4j 87 9 =j ar, j 9 8 (8, 8) ar, r 9 8

43. Tjc`j af tjm fallawcod rmhl vhlume fuo`tcaos cs/hrm oat mvmo fuo`tcaos1

(H) 0f(x) x scox9

(G) f(x) 9 x7às x

(@) f(x) 9 mxx0 sco x

(E) f(x) 9 x[xY, wjmrm [xY emoatms tjm drmhtmst cotmdmr lmss tjho ar mquhl ta x


23/23


Hos > (@, E)

Jcots > (H) f( x) f(x) 9 mvmo (G) f( x) f(x) 9 mvmo

(@) f( x) f(x) oat mvmo (E) f( x) f(x) oat mvmo

:2. Lmt sco , `as gm tjm raats af tjm mquhtcao x7gx + 9 2. Ujmo wjc`j af tjm fallawcod sthtmimots cs/hrm arrm`t1

(H)8

`7

(G) g 7 (@)8

`7

? (E) g 7?

Hos > (H, G)

Jcots > Pco @as g,Pco .@as ` + 9 9

8g 7, ` sco7

7 9

8`

7

wbjee sample paper 9 (wbjee2013 answers hints mathematics)

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