IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ] · IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ]...

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IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ]

PART - I [PHYSICS]1. nks fi aMksa ft uesa i zR; sd dk nzO; eku M gS] ds chp dh nwj h dks 2L fLFkjj [ kk x; k gSA bu fi aMksa ds dsUnzkasa dks t ksM+us okyh j s[ kk ds e/; fcUnq l s , dm nzO; eku dk d.k yEcor ~ i z{ksfi r fd; k t kr k gSA xq: Rokd"kZ.k fu; r kadG gSA l gh i zFkdu gS ¼gSa½(A) nks fi aMksa ds xq: Rokd"kZ.k {ks=k l s i yk; u ds fy, nzO; eku m dk U; wur e

i zkj afHkd osx L

GM4 gSA

(B) nks fi aMksa ds xq: Rokd"kZ.k {ks=k l s i yk; u ds fy, nzO; eku m dk U; wur e

i zkj afHkd osx L

GM2 gSA

(C) nks fi aMksa ds xq: Rokd"kZ.k {ks=k l s i yk; u ds fy, nzO; eku m dk U; wur e

i zkj afHkd osx LGM2

gSA

(D) nzO; eku m dh Åt kZ l nSo fLFkj j gr h gSASol. B

Åt kZ l aj {k.k l s2GMm GMm GM

L L 2L =

12

mv2 + 2GM

2L

2GMmL

= 12

mv2

v = GM2L

Ei = 12

mv2 Ef = 0.

, d d.k esa fLFkfr t Åt kZ dks ugha fy; k x; k gSAnzO; eku m dh Åt kZ fu; r ugha gSA

2. ?k"kZ.kghu {kSfr t r y i j i M+h gqbZ k cy fLFkj kad dh nzO; eku j fgr fLi zaxds , d fl j s l s m nzO; eku dk d.k t qM+k gqvk gSA bl fLi zax dk nwl j k fl j kc) gSA ; g d.k vi uh l kE; koLFkk l s l e; t = 0 i j i zkj afHkd {kSfr t osxu0 l s xfr eku gks j gk gSA t c d.k dh xfr 0.5 u0 gksr h gS] ; g , d n<+nhokj l s i zR; kLFk l a?kê dj r k gSA bl l a?kê ds ckn(A) t c d.k vi uh l kE; koLFkk esa ykSVr k gS bl dh xfr u0 gksr h gSA(B) t c d.k vi uh l kE; koLFkk l s i gyh ckj xqt j r k gS og l e;

t = km

gSA

(C) t c fLi zax esa l ai hMu vf/kdr e gksr k gS og l e; km

34t

gSA

(D) t c d.k vi uh l kE; koLFkk l s nwl j h ckj xqt j r k gS og l e;

km

35t

gSA

Sol. AD

xfr ' khy ugh gS

m

MPu0

v = u /20

m

MPu0

T/4

T/4 T/6

T/6

u /20

x = A sin tv = A cos t

A2

= A cos t

t = 3

nhokj r d i gqapus esa yxk l e; = T6

e/; fcUnq r d oki l vkus esa yxk l e; = T6 +

T6 =

T3 =

2 m3 k

vf/kdr e fl dqM+us esa yxk l e; = T3 +

T4

= 7T12

= 67

mk

f} r h; l e; = T3 +

T2

= 5T6

l kE; koLFkk fLFkfr esa = 5T6 =

5 m3 k


2

3. , d R f=kT; k ds vuar yEcs [ kks[ kys pkyd csyu dh yackbZ esa , d fLFkj/kkj k I cg j gh gSA bl csyu dks 2R f=kT; k dh vuar i fj ukfydk ds vUnjl ek{k j [ kk x; k gSA bl i fj ukfydk esa n yi sVs i zfr bdkbZ yEckbZ esa gS o, d fLFkj /kkj k I gSA , d fcUnq P ds fy, t ks l ek{k l s r nwj h i j gS] dkSul k ¼l s½ i zdFku l gh gS ¼gSa½(A) 0 < r < R esa] pqacdh; {ks=k ' kwU; ugha gSA(B) R < r < 2R esa] pqqacdh; {ks=k dh fn' kk l ek{k dh fn' kk esa gSA(C) R < r < 2R esa] pqacdh; {ks=k r f=kT; k ds oÙk l s Li ' kZj s[ kh; gS ft l dkdsUnz l ek{k i j gSA(D) r > 2R esa] pqacdh; {ks=k ' kwU; ugha gSASol. AD

Bi fj ufydk

= 0 nI

B[ kks[ kys csyu

= ‘' kwU; ‘(vUnj )

R2R

= r2I0

(ckgj )

4. nks okgu] ft uesa i zR; sd dh xfr u gS] , d gh l h/kh {kSfr t l M+d i j , dnwl j s dh vksj vk j gs gSaA ok; q l M+d dh fn' kk esa w ds osx l s cg j gh gSAbuesa l s , d okgu f1 vkofÙk dh l hVh ct kr k gSA nwl j s okgu esa cSBs gq, i zs{kddks l hVh f2 vkofÙk dh l qukbZ nsr h gSA fu' py ok; q esa /ofu dh xfr V gSAl gh i zdFku gS ¼gSa½(A) ; fn ok; q i zs{kd l s L=kksr dh fn' kk esa cgr h gS f2 > f1A(B) ; fn ok; q L=kksr l s i zss{kd dh fn' kk esa cgr h gS f2 > f1 A(C) ; fn ok; q i zs{kd l s L=kksr dh fn' kk esa cgr h gS f2 < f1A(D) ; fn ok; q L=kksr l s i zs{kd dh fn' kk esa cgr h gS f2 < f1ASol. AB

çs{kd l s L=kkssr

2 1v w uf fv w u

f2 > f1

L=kkssr l s çs{kd

2 1v w uf fv w u

f2 > f1

5. O; at d 2d sin= dk mi ; ksx dj r s gq, ge dks eki dj d dk ekut kuuk pkgr s gSaA dk eku 0 o 90° ds chp esa gSA r j ax nS/; Z dk ekugesa i fj ' kq) r % Kkr gS r Fkk ds eki us esa =kqfV] ds l Hkh ekuksa ds fy, l ekugSA t Sl s dk eku 0° l s c<+r k gS r c(A) d esa fuj i s{k =kqfV fLFkj j gr h gSA(B) d esa fuj i s{k =kqfV c<+r h gSA(C) d esa fHkUukRed =kqfV fLFkj ?kVr h gSA(D) d esa fHkUukRed =kqfV ?kVr h gSA

Sol. D2dsin =

d =

sin2

dd = 2

dcoseccos

dcot.ddd ( d fu; r )

ddd

dd =

ddd

va' k

dd = fuj i s{k =kqfV

6. nks vpkyd R1 r Fkk R2 f=kT; k okys xksyksa dks Øe' k% + o -, dl ekuvk; r u vkos' k ?kuRo l s vkosf' kr fd; k x; k gSA bu xksyksa dks fp=k esa n' kkZ,vuql kj bl i zdkj t ksM+ dj j [ kk x; k gS fd os vkaf' kd : i l s vfr NkfnrgSaA vfr Nkfnr {ks=k ds i zR; sd fcanq i j

R1R2

(A) fLFkj oS| qr {ks=k ' kwU; gSA(B) fLFkj oS| qr foHko vpj gSA(C) fLFkj oS| qr {ks=k dk i fj ek.k vpj gSA(D) fLFkj oS| qr {ks=k dh fn' kk , dl eku gSASol. CD

net 1 20

E r r3

O O'

r1 r2

= 0

OO'3

= fu; r

7. fp=k esa fdl h Bksl dh fof' k"V Å"ek /kkfj r k (C) dk r ki eku (T) i jfuHkZj r k dks n' kkZ; k x; k gSA r ki eku esa 0 l s 500 K r d l eku nj l s l ar rof) gksr h gSaA eku dj fd vk; r u esa i fj or Zu mi s{kuh; gS] fuEu i zdFkuesa dkSu l k ¼l s½ r dZl axr l fUudV l gh gS ¼gSa½\

100 200 300 400 500

C


3

(A) 0-100 K ds chp vo' kksf"kr Å"ek dh nj r ki eku i j j Sf[ kdvkfJr r k fn[ kk, xhA(B) 0-100 K r d r ki eku dks c<+kus i j vo' kksf"kr Å"ek 400-500 Kr ki eku ds chp c<+kus dh Å"ek dh r qyuk esa de gSA(C) vo' kksf"kr Å"ek dh nj 400-500 K r ki eku ds chp vi fj ofr Zr gSA(D) Å"ek vo' kks"k.k dh nj 200-300 K r ki eku ds chp c<+ j gh gSASol. ABCD

dTdt = fu; r

d = mC.dT A fodYi(B) C, 0 – 100 ds e/; de gS 400 – 500(C) 400 – 500 = C fu; r gSA

(D) dQdt =

MCdTdt C

dQdt

8. , d gkbMªkst u&l eku i j ek.kq ds bysDVªkWu d{k dh f=kT; k 4.5 a0 gS

t gk¡ a0 cksj f=kT; k gSA bl bysDVªkWu dk d{kh; dks.kh; l aosx 2h3

gSA fn; k

gS fd h Iykad fu; r kad o R fj McxZ fu; r kad gSA i j ek.kq ds O; qÙksft r gksusi j mRl ft Zr fofdj .k ds r j axnS/; Z dh l aHkkouk, ¡ gS

(A) R329

(B) R169

(C) R59

(D) R34

Sol. AC

L = nh2

n = 3

R = a0

2nZ

4.5 a0 = a0

9Z

Z = 2.

hc

= Rhc × 2 21 2

1 1n n

(l aØe.k 1 3)

1

1 = R

119

=

32R9 1 =

932R

(l aØe.k 1 2)

2

1 = R

114

× 4 2 =

13 R

(l aØe.k 2 3)

3

1 = R

1 14 9

× 4 2 = 13 R

3

1 =

20R36 =

95R

Paragraph for Question 9 and 10, d : {k i Fk ds mPpr e fcUnq l s , d 1 kg nzO; eku ds xqVds dksfoj kekoLFkk l s NksM+k x; k gSA ; g i Fk 40 m f=kT; k dk oÙkh; pki gSA xqVdkvi us i Fk i j fcuk yq<+ds gq, l j dr k gSA bl xqVds i j , d ?k"kZ.k cyr kR{kf.kd osx dh foi j hr fn' kk esa yxr k gSA fp=k esa n' kkZ; s vuql kj ] fcUnqQ r d vkus ds fy, ?k"kZ.k dks vfr Øe dj us ds fy, 150 J dk; Z dj uki M+r k gSA ¼xq: Roh; Roj .k g dk eku = 10 ms-2 yhft , )

1 kg300

R P

O x

R

y

9. t c xqVdk fcUnq Q i j i gqapr k gS] bl dh xfr gS

(A) 5 ms-1 (B) 10 ms-1 (C) 310 ms-1 (D) 20 ms-1

Sol. B

1kg

30º

R = 40m P

R = 40m

mg

60º

mg cos60º

R sin 30º = 20

30º

y

x

xq: Roh; fLFkfr t Åt kZ = 0

Q

wf = Ef – Ei (Ei = 0)

– 150 = 2v200

2

v = 10 m/s

10. fcUnq Q i j xqVds i j yxus okys vfHkyac cy dk i fj ek.k gS(A) 7.5 N (B) 8.6 N (C) 11.5 N (D) 22.5 NSol. A

mg cos 60º + 2mv

R = 10 ×

12

+ 10040 = 7.5 N


4

Paragraph for Question 11 and 12, d r ki h; fo| qr l a; a=k 600 kW dh ' kfDr 4000 V i j mRi kfnr dj r kgS] t ks 20 km dh nwj h i j mi HkksDr vksa ds mi ; ksx ds fy, ys t k; h t kr hgSA bl dks ; k r ks mPp /kkj k ogu&{ker k okys dsfcy l s Hkst k t k l dr k gS; k nksuksa fl j ksa i j mPpk; h o vi pk; h VªkUl QkWeZj dk i z; ksxdj fd; k t kl dr k gSA i zR; {k i zs"k.k dk nks"k ; g gS fd bl esa Åt kZ dk {k; cgqr vf/kdgksr k gS t cfd VªkUl QkWeZj ds mi ; ksx ds r j hds esa {k; cgqr de gksr k gSAbl r j hds esa , d mPpk; h VªkUl QkWeZj l a; a=k dh vksj yxk; k t kr k gS ft l l s/kkj k dk eku de gks t k, A mi HkksDr k ds fl j s esa vi pk; h VªkUl QkWeZj dki z; ksx fd; k t kr k gS ft l l s mi HkksDr vksa dks , d fo' ks"k de oksYV i j fo| qr' kfDr nh t k l dsA ; g ekuk t k l dr k gS fd dsfcy ' kq) i zfr j ksf/kr gS r FkkVªkUl QkWeZj vkn' kZ gSa] o mudk ' kfDr xq.kkad , d gSA mfYyf[ kr l eLr /kkj kvksa o oksYVr kvksa dk eki rms eku esa gSA

11. ; fn , sl s dsfcy dk mi ; ksx fd; k t k, ft l dk i zfr j ks/k 4.0 km-1

gS r c i zR; {k i zs"k.k dh fLFkfr esa ' kfDr {k; (% esa) gS(A) 20 (B) 30 (C) 40 (D) 50Sol. B

P = vii = 150 AR = 0.4 /km = 0.4 × 20 = 8 gkfuH = I2 R= 150 × 150 × 8= 180000 J

% ' kfDr {k; = 1000600180000

× 100 = 30 %

12. VªkUl QkWeZj ds i z; ksx dj us okyh fof/k esa] ; g ekusa fd mPpk; hVªkUl QkWeZj ds i zkFkfed o f} r h; d esa yi sVksa dh l a[ ; k dk vuqi kr 1 : 10gSA ; fn fo| qr ' kfDr ] mi HkksDr kvksa dks 200 V i j nh t kr h gS r ks vi pk; hVªkUl QkWeZj esa i zkFkfed o f} r h; d ds y i sVksa dh l a[ ; k dk vuqi kr gS(A) 200 : 1 (B) 150 : 1(C) 100 : 1 (D) 50 : 1Sol. A

mPpk; h

110

4000v

v = 40000 oksYV

vi Ppk; h

200200

40000

200 : 1

Paragraph for Questions 13 and 14x-y r y esa R f=kT; k dh oÙkh; d{kk esa , d Q fcUnq vkos' k dks.kh; xfr

l s i fj Øek dj j gk gSA bl s ywi esa cgr h

2Q

vi fj or hZ /kkj k ds r qY; ekuk

t k l dr k gSaA vc , d , dl eku pqacdh; {ks=k dks /kukRed z-fn' kk esa pkywdj r s gSa ft l dk eku 0 l s B r d , d l Sfd.M esa , dl eku nj l s c<+r kgSA ; g ekfu; s fd bl nkSj ku d{k dh f=kT; k fLFkj j gr h gSA paqcdh; {ks=kds yxkus l s d{k esa , d emf i zsfj r gksr k gSA , d i zsfj r fo| qr {ks=k } kj k bdkbZ/ku vkos' k dks l aoÙk ywi ds pkj ksa vksj ?kqekus esa fd; s x; s dk; Z dh ek=kk dksi zsfj r fo| qr okgd cy (emf) dgk t kr k gSA ; g Kkr gS fd t c , d vkos' k, d d{k esa i fjHkze.k dj r k gS r c ml dk pqacdh; f} /kwzo vk?kw.kZ ml ds dks.kh;l aosx ds vkuqi kfr d gksr k gS ft l dk vkuqi kfr d fLFkj kad gSA13. pqacdh; {ks=k esa i fj or Zu ds nkSj ku d{k esa fdl h fo' ks"k {k.k i j i zsfj rfo| qr {ks=k dk eku gS

(A) 4

BR(B)

2BR

(C) BR (D) 2BR

Sol. B

dE.d l B.dAdt

E (2R) = ddt (B) (R2)

E (2R) = R2 dBdt

E = R dB2 dt =

BR2

14. ft l l e; vUr j ky esa pqacdh; {ks=k esa i fj or Zu gks j gk gS] ml vUr j kyds vUr esa] vkos' k ds d{k l s l acaf/kr pqacdh; f} /kqzo vk?kw.kZ esa i fj or Zu gS

(A) -BQR2 (B) 2

BQR2

(C) 2

BQR2

(D) BQR2

Sol. B or C

E = BR2

F = QE = QBR

2

= FRI

=

2

RQBR2 MR

= QB2M

= T = QB2M

(T)

fn; k gSM = L. bl fy; s M = L

= (I) ()


5

= 2MR QB T

2M

bl fy, M = 2R QBT2

T = 1 j [ kus i j

M = ± 2R QB

T

i z' u 15 vkSj 16 ds fy, vuqPNsn

, d ukfHkd AZ X dk nzO; eku (A–Z) U; wVkªWuksa , oa Z i zkWVksuksa ds

nzO; ekuksa ds ; ksx l s de gksr k gSA nzO; ekuksa dh deh ds l er qY; Åt kZ dksca/ku Åt kZ dgr s gS A , d nzO; eku M dk Hkkjh ukfHkd m1 , oa m2 æO; ekuksads nks gYds ukfHkdksa esa fo?kfVr gks l dr k gS] ; fn (m1+m2) < M A r Fkkm3 , oa m4 �æO; ekuksa�ds�nks�gYds�ukfHkd�i w.kZ�l ay; u�dj ds�, d�M'æO; eku dk Hkkj h ukfHkd cuk l dr s gSa] ; fn (m3+m4) > M' A dqNi j ek.kqvksa ds æO; eku uhps Vscy es fn; s x; s gS %

11H 1.007825 u 2

1H31H 3.016050 u 4.002603 u2.014102 u 4

2He

6.015123 u 69.925325 u 81.916709 u7.016004 u73Li 70

30Zn 8234Se

151.919803u 208.980388u 209.982876u205.974455u20682 Pb 209

83 Bi 21084 Po

63 Li

15264 Gd

15. l gh i zdFku gSA

(A) ukfHkd 63Li , d , sYQk d.k mRl ft Zr dj l dr k gSA

(B) ukfHkd 21084 Po , d i zksVkWu mRl ft Zr dj l dr k gSA

(C) M~; wwVkWu vkSj , sYQk d.k i w.kZ l ay; u dj l dr s gSA

(D) ukfHkd 7030Zn , oa ukfHkd 82

34Se i w.kZ l ay; u dj l dr s gSASol. C

36Li 6.015123 u

24He 4.002603 u

12H 2.014102 u

36Li 1 2

2 4H + HenzO; eku

denzO; ekuvf/kd

( )l aHko ugh

84210Po 1 83

1 209H + BinzO; eku

denzO; ekuvf/kd

( )l aHko ugh

1 22 4H + He 3

6Li nzO; eku

denzO; ekuvf/kd

( )l aHko

30 3470 82Zn + Se 64

152GdnzO; eku

denzO; ekuvf/kd

( )l aHko ugh

16. t c foj kekoLFkk esa ukfHkd 21084 Po , sYQk&{k; dj r k gS] r c , sYQk d.k

dh xfr t Åt kZ (keV esa) gksr h gSA(A) 5319 (B) 5422(C) 5707 (D) 5818Sol. D

84210Po 82 2

206 4Pb + He M = 0.005818 u( M)c2 = 5.419467 MeV ~ 5420 KeV

K() = 206210 (5420)

= 5316 KeV ~ 5319 KeV

17. , d µ1 vi or Zukad ds l edks.k fi zTe dks µ2 vi or Zukad ds vk; r kdkjCykd esa j [ kk x; k gSA i w.kZ O; oLFkk µ3, vi or Zukad ds ek/; e l s fp=k esan' kkZ; s vuql kj f?kj h gqbZ gSA i zdk' k dh fdj .k ‘e’ vk; r kdkj CykWd i jvfHkycaor vki fr r gksr h gSA µ1,µ2, vkSj µ3, ds ekuksa i j fuHkZj gksr h gqbZi zdk' k dh fdj .k pkj l aHko i Fkksa ‘ef’, ‘eg’, ‘eh’ ; k ‘ei’ eas l s , d ysr h gSA

l wph I esa fn; s x; s i Fkksa dks l wph II dh vi or Zukad dh ' kr ksZ l sl qesfyr dhft , vkSj l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx dj ds mÙkjpqfu; s % l wph I l wph II

(P) e f 1. µ1> 2 µ2

(Q) e g 2. µ2>µ1 , oa µ2>µ3(R) e h 3. µ1=µ2

(S) e i 4. µ2<µ1< 2 µ2 , oa µ2>µ3

Codes :P Q R S

(A) 2 3 1 4(B) 1 2 4 3(C) 4 1 2 3(D) 2 3 4 1


6

Sol. D(P) e f : i w.kZ vkUr fj d i j kor Zu ugh gks j gk gS r Fkkfdj .k 1 o 2 l r g ds vfHkyEcor ~ t k j gh gS vkSj fdj .k l r g2 o 3 vfHkyEcor ~ l ss nwj t k j gh gSA2 > 1

(Q) e g : dksbZ >qdko ugh gS 1 = 2

(R) e h : 1 o 2 l r g dh l r g ds vfHkyEcor ~ l ss nwji w.kZ vkUr fj d i j kor Zu ugh gks j gk gS i qu% 2 o 3 ds vfHkyEcor ~

l s nwj t k j gh gSA 2 > 3 vkSj 2 2 > 1 gSA

(S) e i : i w.kZ vkUr fj d i j kor Zu gks j gk gS

sin 45º > 2

1

2 < 1

2

1 > 2 2.

18. l wph I dks l wph II l s l qesfyr dhft , vkSj l wfp; ksa ds uhps fn; sx; s dksM dk i z; ksx dj ds l gh mÙkj pqfu; s %

l wph I l wph II(P) cksYV~l eku fu; r kad 1. [ML2 T–1](Q) ' ; kur k xq.kkad 2. [ML-1 T–1](R) Iykad fu; r kad 3. [MLT-3 K–1](S) Å"ek pkydr k 4. [ML2T–2 K–1]

Codes :P Q R S

(A) 3 1 2 4(B) 3 2 1 4(C) 4 2 1 3(D) 4 1 2 3Sol. C

32

KT = E , k = [ML2 T–2 K–1]

F = 6rv, = [ML–1 T–1]E = h h = [ML2 T–1]

Q = 1 2KA T T t

L

K = [ML T–3 K–1]

19. , d , d&i j ek.kqd vkn' kZ xSl ds , d eksy dks fp=k esa n' kkZ; s PVvkj s[ k ds vuql kj nks pØh; i zØeksa EFGE o EFHE esa yst k; k t kr k gSA l acf) r i zØe ' kq) r % l evk; r fud] l enkch l er ki h; ; k: nks"e gSA

Vl wph I esa fn; s x; s i Fkksa dks l wph II esa fd; s x; s dk; Z ds i fj ek.k

ds l kFk l qesfyr dhft , vkSj l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx l ghmÙkj pqfu; sA l wph I l wph II(P) GE 1. 160 P0V0In2(Q) GH 2. 36 P0V0(R) FH 3. 24 P0V0(S) FG 4. 31 P0V0Codes :

P Q R S(A) 4 3 2 1(B) 4 3 1 2(C) 3 1 2 4(D) 1 3 2 4Sol. A

(32T )0

(32P V )0 0

T0

(8T , 8V )0 0

(P , 32V )0 0

V0 V

P0

32P0

P

G – E l enkch;W = Pv = P0(32 V0 – V0) = 31P0V0 ..........(4)

G – X l enkch;P0(32 V0 – 8V0) = P0 × 24V0

= 24 P0V0 ..........(3)

F – X r1

nR

(Tf – Ti) = WÅ/kks"e

)T8T32(TVP

23

000

00 00

00 T24TVP

23

36 P0V0 ..........(2)


7

F – G nR (32 T0) ln 32 0

0

vv

nR 32 T0 ln 25

0 00

0

P V5 32T ln2

T

= 160P0V0 ln2 ..........(1)

20. l wph I esa dqN ukfHkdh; i zfØ; k, ¡ nh xbZ gSSA l wph II esa bu i zfØ; kvksads t ud ukfHkd o , d vafr e ukfHkdh; [ kaM fn, x, gSA l wfp; ksa ds uhpsfn; s x, dksM dk i z; ksx dj ds l gh mÙkj pqfu, % l wph I l wph II

(P) , sYQk&{k; 1. 15 158 7O N .....

(Q) +{k; 2. 238 23492 90U Th .....

(R) fo[ kaMu 3. 185 18483 82Bi Pb .....

(S) i zksVkWu mRl t Zu 4. 239 14094 57Pu La .....

Codes :P Q R S

(A) 4 2 1 3(B) 1 3 2 4(C) 2 1 4 3(D) 4 3 2 1

Sol. C(P) , YQk {k;

42

23490

23892 HeThu (2)

(Q) B+ {k;

01

157

158 NO

(1)

(R) fo[ k.Mu

140239 995794 37Pu La (4)

n 1.5 unstablep

vfLFkj

(S) çksVksu mRl t Zu

PbBi18482

18583 + çksVksu (3)


8

PART - II [CHEMISTRY]

SECTION – ASINGLE CORRECT

21. Ksp (Ag2CrO4) dk eku 298 K i j 1.1 × 10–10 gSA0.1 M AgNO3 ds foy; u esa Ag2CrO4 dh foys; r k eksy@yhVjesa gSA(A) 1.1 × 10–11 (B) 1.1 × 10–10

(C) 1.1 × 10–12 (D) 1.1 × 10–9

Sol. BAg2CrO4 = 2Ag+

(aq.) + CrO4––

(aq.)—— 0.1 +2s s

0.1 Mksp = [Ag+]2 [CrO4

––]1.1 × 10–12 = 0.12 × (s)s = 1.1 × 10–10 M

22. fuEufyf[ kr vfHkfØ; k ds mRi kn@mRi knksa dks cr k, ¡

OH

CH3

CHCl3OH–

OH

CH3

OHC CHO

P Q

O

H C3 CHCl2

R

H C3 CHCl2

OH OH

CH3

S

CHO

(A) P(eq[ ; ) (B) Q(xkS.k)(C) R(xkS.k) (D) S(eq[ ; )

Sol. B DO–H O

O

CH3 CH3

CH3CH3

CH3

OH CCl2

OC Cl2

H

CCHOCl

Cl-2Cl

OH

OH

Major(S)

O

CH3

CCl2 HOH

O O

C Cl2 CHCl2CH3 CH3

Minor(Q)

23. fuEufyf[ kr vfHkfØ; k dk eq[ ; mRi kn gS

OH

SO H3

t yh; r qY; eku Br (3.0, )2

OH

SO H3

Br Br

P Q R

OH OH

SO H3

S

Br

Br BrOH

BrBr

BrBr Br Br

Br

Sol. B

SO3

SO3

SO3

Br Br

SO3

SO H3

O

O

O

O OH

OH

Br – Br

H

Br

O

BrBr

Br

Br Br

Br Br

Br Br

Br

Br2

– HBr

Br – Br

O

BrO=S=O

– SO 3

O

HOH


9

Hint:

SO H3

OH

Br

OHBrBr

aq. Br (3.0 mole)2

Bromo desulphonation

24. j kl k; fud vfHkfØ; kvksa (I vkSj II) ds i wj s gksus ds ckn j kl k; fudfeJ.k esa dkcZfud ; kSfxd ¼; kSfxdksa½ dks cr k, ¡A

vfHkfØ; k I :

O

H C3 CH3

Br2(1.0 mol)t yh; NaOH

vfHkfØ; k II :

O

H C3 CH3

Br2(1.0 mol)

CH COOH3

(1.0 mol)

(1.0 mol)

O

H C3 CH Br2

P Q R

O

H C3 CBr3

O

BrH C2 CH Br2

O

BrH C2 CH Br2

S T

O

H C3 ONa

U

CHBr3

(A) vfHkfØ; k I : P and vfHkfØ; k II : P(B) vfHkfØ; k I : U, , l hVksu vkSj vfHkfØ; k II : Q , l hVksu(C) vfHkfØ; k I : T, U, , l hVksu vkSj vfHkfØ; k II : P(D) vfHkfØ; k I : R, , l hVksu vkSj vfHkfØ; k II : S, , l hVksu

Sol. A

Reaction-1

CH –C–CH23

O HOH CH –C–CH23

O

–Br Br—Br (1 me)

CH –C–CH23 –Br

O

Reaction-2

CH –C–CH33

CH –C–OH3

O O

CH –C=CH23

–Br

–Br

Br—Br

–HBr



O

O

OH

Enol is Nu

H

25. O3 ds l anHkZ esa l gh oDr O; o gS (gS)(A) O-O vkca/kksa dh yackbZ cj kcj gSA(B) O3 dk r ki h; fo; kst u Å"ek' kks"kh gSA(C) O3 i zfr paqcdh; gSA(D) O3 dh l aj puk cafdr gksr h gSA

Sol. A,C,D

O OO OO O

sp2

vuqukn ds dkj .k l eku cU/k yEckbZ eqM+h gqbZ vkdkj dh r Fkk

i zfr pqEcdh; bl dk fo| Vu m"ek{ksi h gksr k gSA2O3 3O2 + Heat

26. fuEufyf[ kr ukfHkdh; r Rokar j .k8944Be X Be Y

esa (X, Y) gS@gSa(A) (, n) (B) (p, D)(C) (n, D) (D) (, p)

Sol. A,B

(A) 9 0 9 14 0 4 0Be Be r (T)

(B) DBepBe 20

84

11

94 (T)

(C) DBenBe 21

84

10

94 (F)

(D) 9 0 8 14 0 4 1Be Be P (F)


10

27. /kkr qvksa ds fu"d"kZ.k esa dkcZu vk/kkfj r vi p; d fof/k dk i z; ksx fduv; Ldksda esa ugha gksr k gS\(A) SnO2 l s fVu(B) Fe2 O3 l s vk; j u(C) Al2O3 l s , syqfefu; e(D) MgCO3. CaCO3 l s eSXuhf' k; e

Sol. C D(A) SnO2 + C Sn + COdkcZu vi p; u fof/k(B)Fe2O3 + CO Fe + CO2dkcZu vi p; u fof/k ysfdu eq[ ; vi pk; d CO. gS

(C) Al2O3 l s , Y; wfefu; e gkWy & gsj ksYV fof/k ds } kj k

vi pf; r gksr k gSA

(D) MgCO3.CaCO3 Hkt Zu MgO.CaO

2(MgO.CaO) +FeSi 1150º 2Mg + Ca2SiO4

+ Fe; g fl fydk r ki h; i zØe dgykr k gSa

28. CaCO3 (Bksl ) ds Å"eh; fo?kVu dh l kE; koLFkk dk v/; ; ufofHkUu voLFkkvksa esa fd; k x; kACaCO3 (s) CaO(s) + CO2(g)bl l kE; koLFkk ds fy; s] l gh i zdFku gS (gSa)(A) H r ki eku i j fuHkZj dj r k gSA(B) l k: E; koLFkk fLFkj kad (K) CaCO3 ds i zkj fEHkd i fj .kke i jfuHkZj ugha dj r k gSA(C) K fu; r r ki eku i j CO2 ds nkc i j fuHkZj dj r k gSA(D) H mRi zsj (vxj gks) ds i zHkko i j fuHkZj ugha dj r k gSA

Sol. ABD(A) H T i j fuHkZj dj r k gSA

2

112

T

TprTT dTCHH (T)

(B) l kE; oLFkk fLFkj kad l kUnzr k i j fuHkZj ugh gSA(T)(C) K fu; r r ki eku i j CO2 ds nkc i j fuHkZj dj r k gSA(F)(D) H mRi zsj d (vxj gks) ds i zHkko i j fuHkZj ugh dj r k gsA(T)

[ k.M-2 : (vuqPNsn i zdkj )bl [ k.M esa fl ) kar ksa] i z; ksxksa vkSj vk¡dMksa vkfn dks n' kkZus okys 4vuqPNsn gSA pkj ksa vuqPNsnksa l s l acaf/kr vkB i z' u gSa] ft uesa l s gjvuqPNsn i j nks i z' u gSA fdl h HKh vuqPNsn esa gj i z' u ds pkj fodYi(A), (B), (C) vkSj (D) gSA ft uesa l s dsoy , d gh l gh gSA

i z' u 29 , oa 30 ds fy; s vuqPNsnnks vdkcZfud yo.kksa ds , d feJ.k dk t yh; foy; u r uq HCI vEy} kj k vi p; u dj , d vo{ksi (P) vkSj , d fQYVªsV (Q) nsr k gSAvo{ksi P xeZ t y esa ?kqyu' khy gSA fQYVªssV (Q) r uq [ kfut vEyh;ek/; e esa H2S } kj k foospu dj vi fj ofr Zr j gr k gSA fdUr q , eksfudyek/; e esa H2S ds l kFk vi {ksi (R) nsr k gSA vo{ksi R ds l kFk t yh;NaOH ek/; e r Fkk H3O2 dh vfHkfØ; k j axhu foy; u (S) nsr h gSA

29. vo{ksi P esa mi fLFkr gSA(A) Pb2+ (B) Hg2

2+

(C) Ag+ (D) Hg2+

Sol. APb2+ + Cr3+ dil. HCl PbCl2 + CrCl3 (Q)

' osr vo{ksi xeZ t y esa

foys; gSa t cfd B.Ms

t y esa vfoys;

30. j axhu foy; u S esa mi fLFkr gSA(A) Fe2(SO4)3 (B) CuSO4(C) ZnSO4 (D) Na2CrO4d

Sol. DCrCl3 + H2S + dil HCl No ppt.CrCl3 + H2S + NH4OH Cr(OH)3(s)

Green (R)

H S 2

NH OH4

2H + OH+

S2– vk; u l kUnzr k c<+r h gSa t ks fd vk; u vo{ksi dkj .k gSA

2H + S+ 2–

NH + OH4+

H O2

Cr(OH)3(s) + NaOH + H O2 2

vkWDl hdkj d Na2CrO4

i hyk foy; u (R)

i z' u 31 , oa 32 ds fy; s vuqPNsn

P vkSj Q , d MkbdkcksZfDl fyd vEy C2H4O4 ds nks l eko; oh gSAnksuksa Br2/H2Odks j axghu dj r s gSA xeZ duus i j P pØh; , ugkbMªbMcukr k gSAr uq {kkj h; KMnO4 } kj k P vkSj Q vyx&vyx vfHfØ; k dj , dvFkok , d l s vf/kd ; kSfxd S, T vkSj U.

COOH COOH COOH

COOH COOH COOH

H H H

H HO H

OH OH OH

OH H OH

S T U


11

31. P vkSj Q } kj k cus ; kSfxd Øe' k% gS(A) /kzqo.k ; kSfxd S , d /kzqo.k fuf"Ø; ; qXe (T, U)(B) /kzqo.k fuf"Ø; S , oa /kqzo.k fuf"Ø; ; qXe (T, U)(C) /kzqo.k ?kw.kZd ; qXe (T, U) , oa /kzqo.k ?kw.kZd S(D) /kzqo.k fuf"Ø; ; qXe (T, U) , oa /kqzo.k fuf"Ø; S

Sol. B

H C–OH

HHO–C

O

O

+

H C–OH

H C–O–H

O

O

Br /H O2 2

j axghu gks x; k(Q) (P)

KMnO4Rkuq

fl l 0 gkbMªksDl hyhd j .k KMnO4Rkuq

fl l 0 gkbMªksDl hyhd j .k

COOH COOH COOH

COOH COOH COOH

H H H

H HO H

OH OH OH

OH H OH

S T U

(T) (U) (S) ehl ks ; ksfxd

(i zdkf' kd fu"Øh; ); qXe j sl sfed feJ.k gSA

vr % i zdkf' kd fu"Øh;

32. fuEufyf[ kr vfHkfØ; k vuqØeksa esa V vkSj W

2H /NlQ

+ VAlCl (andhydrous)3

2 4

1.Zn Hg /HCl2.H PO

(A)

O

O

O vkSj

OWV

(B) W

CH OH2

CH OH2V

vkSj

(C)

O

O

O

WV

vkSj

(D) CH OH2

HOH C2

CH OH2V W

vkSj

Sol. A

C – OH

O

H

H

C

OHO

Q

H /Ni2 O

O

O(V)

Fumeric acid

+ AlCl (anhydrous)3

Fridel craft acylation

O

CH2

CH2

CH2

CH2

C – OH

O

O

O

O1.Zn, Hg/HClClemmenson

reduction

COOH

H PO3 4

O(W)

(V)

i z' u 33 , oa 34 ds fy; s vuqPNsn, d xSl ds fuf' pr nzO; eku 'm' dh voLFkk i fj or Zu K l s L M l sN r Fkk oki l K esa fp=k } kj k fn[ kkbZ gS

Pressure

Volume

K L

N M

nkc

vk; r u


12

33. Øfed i fj pkyu t ks bu voLFkk i fj or Zuksa esa l gk; d gS] og gS(A) xeZ] BaMk] xeZ] BaMk dj us i j(B) BaMk] xeZ] BaMk] xeZ dj us i j(C) xeZ] BaMk] BaMk] xeZ dj us i j(D) BaMk] xeZ] xeZ] BaMk dj us i j

Sol. C

P

V

N M

LK

K L ; xeZ

L M ; B.Mk

M N ; B.Mk

N K ; xeZ

34. voLFkk i fj or Zuksa dh fLFkfr ; ksa esa l evk; fud çØe ; qXe gS(A) K l s L vkSj L l s M (B) L l s M vkSj N l s K(C) L l s M vkSj M l s N (D) M l s N vkSj N l s K

Sol. BL M & N K : nksuks l evk; r uhd i z+de

i z' u 35 , oa 36 ds fy; s vuqPNsnCl2 xSl r uq vkSj l kanz NaOH ds t yh; foy; u } kj k Øe' k% BaMsvkSj xeZ voLFkk esa vfHkfØ; k dj nks ¼fHkUu½ Dyksjhu ds vkWDl h&vEyds l ksfM; e yo.k] P vkSj Q nsr s gSA Cl2(g) pkj dksy dh mi fLFkfresa SO2 l s vfHkfØ; k dj mRi kn R nsr k gSA R l Qsn QkLQksj l } kj kvfHkfØ; k dj ; kSfxd S nsr k gSA S dh t y&vi ?kVu fØ; k QkLQksj ldk , d vkDl ksvEy T nsr h gSA

35. P and Q, respectively, are the sodium salts of(A) gkbi ksDyksj l vkSj Dyksfj d vEy(B) gkbi ksDyksj l vkSj Dykj l vEy(C) Dyksfj d vkSj i j Dyksfj d vEy(D) Dyksfj d vkSj gkbi ksDyksj l vEy

Sol. ACl2 + NaOH NaOCl + NaCl

(cold & dil.) (P)Cl2 + NaOH NaClO3 + NaCl

(hot & conc.) (Q)NaOCl l ksfM; e gkbi ksDyksj kbV

NaClO3 l ksfM; e Dyksj sV

vr % l EcfU/kr vEy HOCl r Fkk HClO3 gSA

36. R, S vkSj T, Øe' k% gS(A) SO2Cl2, PCl5 vkSj H3PO4

(B) SO2Cl2, PCl3 vkSj H3PO4

(C) SOCl2 PCl3 vkSj H3PO2

(D) SOCl2 PCl5 vkSj H3PO4

Sol. ACl2 + SO2 charcoal SO2Cl2

(R)SO2Cl2 + P4 PCl5 + SO2

(S)PCl5 + H2O H3PO4 (T)

(Oxy acid)

37. l wph I esa fyf[ kr vl ar qfyr vfHkfØ; kvksa esa vi znf' kZr vfHkdeZd @voLFkk, ¡ l wph II eas nh xbZ gSA l wph I dks l wph II l s l qesfyr dhft ,r Fkk l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx dj ds l gh mÙkj pqfu; s&l wph I l wph II

P. PbO2+H2SO4 ? PbSO4+O2 vU; mRi kn 1. NO

Q. Na2S2O3+H2O? NaHSO4+ vU; mRi kn 2. I2

R. N2H4 ? N2 + vU; mRi kn 3. xeZ

S. XeF2 ? Xe + vU; mRi kn 4. Cl2

dksM :P Q R S

(A) 4 2 3 1(B) 3 2 1 4(C) 1 4 2 3(D) 3 4 2 1

Sol. D(S)XeF2 + NO NOF + Xe(Q)Na2S2O3 + Cl2 + H2O NaHSO4+NaCl

+ H2O, sUVhDyksj (Cl2 gVkus i j )

(P)PbO2 + H2SO4 warm PbSO4 + O2 + H2O(R)N2H4 + I2 I– + N2 + H3O

+

bl vfHkfØ; k dk mi ; ksx vk; r uhd fo' ys"k.k esa gkbMªst hu ds

ek=kkRed fo' ys"k.k esa fd; k t kr k gSA38. l wph I eas fn; s x; s j kl k; fud Li kUr j .kksa dks l wph II eas fn, x,

mi ; qDr vfHkdeZdksa ds l kFk l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn,x; s dksM dk i z; ksx dj ds l gh mr j pqfu; s %l wph I

ClP.

ONaQ. OEt

R.OH

S.

OH


13

l wph II1. (i) Hg(OAc)2; (ii) NaBH22. NaOEt3. Et-Br4. (i) BH3; (ii) H2O2/NaOH

dksM :P Q R S

(A) 2 3 1 4(B) 3 2 1 4(C) 2 3 4 1(D) 3 2 4 1

Sol. A

(P) NaOEtCl + EtOH + NaCl

(Q) O Na + Et–Br O – Et

(R) Hg(OAc)2NaBH4

OH (Oxymercuration)

demurcuration

(S) (i) BH3

(ii) H O /NaOH2 2

OH (Hydroboration

oxidation)

39. X ds t yh; foy; u esa Øe' k% Y dk t yh; foy; u /khj s & /khj s Mkykt kr k gS t Sl s l wph I es fn[ kk; k x; k gSA bu vfHkfØ; kvksa l s mRi Uupkydr k dh fHkUur k l wph II esa nh xbZ gSA l wph I dks l wph II l sl wesfyr dhft , r Fkk l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx dj dsl gh mr j pqfu; s &l wph IP. (C2H5)3N + CH3COOH

X YQ.KI (0.1M) + AgNO3(0.01M)

X YR. CH3COOH + KOH

X YS. NaOH + HI

X Yl wph II1. pkydr k ?kVr h gS vkSj r Ri ' pkr ~ c<r h gSA2. pkydr k ?kVr h gS vkSj r Ri ' pkr ~ vf/kd i fj ofr Zr ugha gksr h gSA3. pkydr k c<r h gS vksj r Ri ' pkr ~ vf/kd i fj ofr Zr ugha gksr hA4. pkydr k vf/kd i fj ofr Zr ugah gksr h vkSj i Ri ' pkr ~! c<r h gSA

Sol. A(P) (C2H5)3N + CH3COOH salt WB WA

Eq. Point

G

Vol. of WB added

Ans = 3

(Q) KI + AgNO3 AgI + KNO3

Eq. Point

G

Vol. of KI added

Ans = 4

(R) CH3COOH + KOH CH3COOk + H2O WA SB

Eq. Point

G

Vol. of WA added

Ans = 2

(S) NaOH + HI NaI + H2O SB SA

Eq. Point

G

Vol. of SB added

Ans = 1

40. ekud vi pk; d foHko 25ºC i j fuEufyf[ kr gSAEº (Fe3+, Fe2+) = + 0.77 V;Eº (Fe2+, Fe) = 0.44 VEº (Cu2+, Cu) = +0.34 VEº (Cu+, Cu) = +0.52 VEº [O2(g) + 2H2O+ 4e– 2H2O] = + 1.23 V;Eº [O2(g) + 2H2O+ 4e– 4OH–] = + 0.40 VEº (Cr3+, Cr) = –0.74 V;Eº (Cr2–, Cr) = 0.91 Vl wph I esa fn; s x; s j sMkDl ; qXeksa dks l wph II esa fn; s x; s vi pk; dfoHko l s l wesfyr dhft , r Fkk l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksxdj ds l gh mÙkj pqfu; s &


14

l wph IP. Eº (Fe3+, Fe)Q. Eº (4H2O 4H+ + 4OH–)R. Eº (Cu2+ + Cu 2Cu+)S. Eº (Cr3+, Cr2–)l wph II1. –0.18 V2. –0.4 V3. –0.04 V4. –0.83 V

Sol. D

(P) (1) e– + Fe+3 Fe+2, 01E = + 0.77 V

(2) 2e– + Fe+2 Fe, 02E = – 0.44 V

__________________________ (3) 3e– + Fe+3 Fe(s) __________________________(3) = (1) + (2)

02

01

03 GGG

)FE2()FE1(FE3 02

01

03

3)44.0(277.0

3E2E

E02

010

3

= – 0.037= – 0.04VP = 3

(Q) (1) O2(g) + 2H2O + e– OH– V40.0E0

(2) 2H2O O2 + 4H+ + 4e– –1.23 V __________________________ (3) 4H2O 4H+ + 4OH–

__________________________(3) = (1) + (2)

02

01

03 GGG

0 0 03 1 24FE ( 4FE ) ( 4FE )

0 0 03 1 2E E E

= 0.40 + (–1.23)= – 0.83 VQ = 4

(R) (1) Cu+2 + 2e– Cu 01E = + 0.34 V

(2) Cu+ + e– Cu 02E = + 0.52 V

(3) Cu+2 + e– Cu+ 03E

(3) = (1) – (2)0 0 03 1 2G G G

0 0 03 1 21FE 2FE ( 1FE )

0 0 03 1 2E 2E E

= 2 × 0.34 – (0.52)= 0.16 V

(1) Cu+2 + e– Cu+ 01E = + 0.16 V

(2) Cu Cu+ + e– 02E = – 0.52 V

(3) Cu+2 + Cu 2Cu+ 03E = ?

(3) = (1) + (2)0 0 03 1 2G G G

0 0 03 1 21FE ( 1FE ) ( 1FE )

0 0 03 1 2E E E = + 0.16 + (–0.52) = – 0.36 V

(S) (1) Cr+3 + 3e– Cr 01E 0.74V

(2) Cr+2 + 2e– Cr 02E 0.91V

__________________________

(3) Cr+3 + e– Cr+2 03E ?

__________________________(3) = (1) – (2)

0 0 03 1 2G G G

)FE2()FE3(FE1 02

01

03

)91.0(2)74.0(3E2E3E 02

01

03

= – 0.4 VS = 2


15

PART - III [MATHEMATICS]

SECTION – ASingle Correct

41. a R (l Hkh okLr fod l a[ ; kvksa dk l eqPp; ), a – 1, ds fy,

nlim

)]nna(...)2na()1na[()1n()n...21(

1a

aaa

= 601

r c a =

(A) 5 (B) 7 (C) 215

(D) 217

Sol. B,D

nlim

a 1 a a 11 2

a 1 2

1 .n a n a n .....1(a 1)

1 601n(n 1) .n a

2

nlim

1 22

a

1 a a .... 1a 1 n n1 6011 n1 a

n 2

11a 1

1 60a2

(a + 1) (2a + 1) = 1202a2 + 3a – 119 = 02a2 + 17a – 14a – 119 = 0

(a – 7) (2a + 17) = 0

a = 7, 172

42. x – v{k dks ewyfcUnq l s 3 nwj h i j Li ' kZ dj us okyk (okys) r Fkk

y – v{k i j 72 var %[ k.M cukus okyk (okys) oÙk gS (gSa)(A) x2 + y2 – 6x + 8y + 9 = 0(B) x2 + y2 – 6x + 7y + 9 = 0(C) x2 + y2 – 6x – 8y + 9 = 0(D) x2 + y2 – 6x – 7y + 9 = 0

Sol. A,C

2 Cf2 = 72f2 – C = 7 .....(i)r Fkk oÙk (3, 0) l s

; k (–3, 0) .....(ii)r Fkk f=kT; k = 4

43. nks j s[ kk, ¡ L1 : x = 5, 2z

3y

r Fkk L2 : x =

,

2

z1

y l er yh; gSaA r c dk eku gks l dr k gS

(A) 1 (B) 2 (C) 3 (D) 4Sol. A,D

05x

= 3

y =

z2

0x

= 1

y

= 2

z

005210

230

= 0

a = 4, 1

44. f=kHkqt PQR, esa P cgÙke dks.k gS r Fkk cos P = 31

A bl ds

vfr fjDr f=kHkqt dk var %oÙk Hkqt kvksa PQ, QR r Fkk RP dks Øe' k%N, L r Fkk M i j bl r j g Li ' kZ dj r k gS fd PN, QL r Fkk RMdh yEckb; k¡ Øekxr l e i w.kZl a[ ; k, ¡ gSA r c f=kHkqt dh Hkqt k(Hkqt kvksa) dh l EHkkfor yEckbZ (yEckb; k¡) gS (gSa)(A) 16 (B) 18 (C) 24 (D) 22

Sol. B,D

cos P = PR.PQ.2QRPRPQ 222

31

= 1m3m2

1m2

2

P 2m (2m+2) R

(2m+2)

(2m+4)

Q

(2m+4)

2m

m = 4 r Fkk m = –1bl fy, Hkqt kvksa dh yEckbZ = 10, 20, 22


16

45. ekuk fd w =2

i3 r Fkk P = {wn : n = 1, 2, 3, .....}.

bl d s v fr fj Dr H1 =

21zRe:Cz r Fkk H2

=

21zRe:Cz , t gk¡ C l fEeJ l a[ ; kvksa dk l eqPp;

gSA ; fn z1 P H1, z2 P H2 r Fkk O ewyfcUnq i znf' kZrdj r k gS, r c z1 Oz2 =

(A)2

(B)6

(C)32

(D)65

Sol. C,D

½/6

/3

–½

z1z1z '2

z "2

z "'2

H1

H2

O

P = , 2, 3 ...............

1 2

1 2

z o z ' 2 /3z oz " 5 /6

46. ; fn 3x = 4x – 1, r c x =

(A) 12log22log2

3

3

(B) 3log22

2

(C) 3log11

4 (D) 13log23log2

2

2

Sol. A,B,C3x = 4x–1

x = (x – 1) log34 x (1 – log34) = –log34

x = 12log22log2

3

3 = 3log2

2

2 = 3log11

4

47. bdkbZ dk , d l fEeJ ?kuewy yhft ; s, t gk¡ 1 r Fkk

P = [pij] , d n × n vkO; wg yhft ; s, t gk¡ pij = i+jA r c

P2 0, t c n =(A) 57 (B) 55 (C) 58 (D) 56

Sol. B,C,DLi "Vr % n, 3 dk xq.kt gksuk pkfg,

48. Qyu f(x) = 2 |x| + |x + 2| – ||x + 2 |– 2|x|| dk

, d LFkkuh; U; wur e ; k , d LFkkuh; vf/kdr e ft u x ds eku i j

gS] og gSa

(A) – 2 (B) 32

(C) 2 (D) 32

Sol. A,B

f(x) = (2|x| + |x + 2|) – |x|2|2x|

f(x) =

|x|2|2x|if|2x|2|x|2|2x|if|x|4

y=2|x|

y=|x+2|

2–2 O32

x

y

So f(x) =

2x,2x2x0,x2

0x32,x2

3/2x2,2x2x,2x

f'(x) =

2x12x02

0x3/223/2x21

2x1

Li "Vr % U; wur e r Fkk vf/kdr e x = –2, 32

, 0 i zkIr gksr k gS


17

i z' u 49 r Fkk 50 ds fy; s vuqPNsn

ekuk fd f : [0, 1] R (l Hkh okLr fod l a[ ; kvksa dk l eqPp; ), d Qyu gSA eku yhft ; s Qyu f nks ckj vodyuh; gS,f (0) = f (1) = 0 r Fkk f" (x) – 2f' (x) + f(x) ex,x [0, 1] dks l ar q"V dj r k gSA

49. fuEu esa l s dkSu 0 < x < 1 ds fy, l R; gS ?

(A) 0 < f(x) < (B) 21)x(f

21

(C) 1)x(f41

(D) – < f (x) < 0

Sol. Df(x) = f(x) e–x

g''(x) 1g(1) = g(0) = 0

dsoy l aHko vkys[ k gks l dr k gS

1/4 1

g'(x) < 0 f(x) < 0 – < f(x) < 0

50. ; fn Qyu e–x f(x) var j ky [0, 1] esa vi uk U; wur e eku

x =41

i j ysr k gS, r c fuEu esa l s dkSu l R; gS ?

(A) f '(x) < f(x), 41

< x < 43

(B) f '(x) > f(x), 0 < x < 41

(C) f ' (x) < f (x), 0 < x < 41

(D) f ' (x) < f (x), 43

< x < 1

Sol. C

f(x),

41,0 esa âkl eku gS

g(x) = e–x f(x)g'(x) = e–x f'(x) – e–x f(x) < 0

x

41,0

e–x (f'(x) – f(x)) < 0

x

41,0

i z' u 51 , oa 52 ds fy; s vuqPNsn

ekuk fd PQ i j oy; y2 = 4ax dh , d ukHkh; t hok gSA fcUnqvksa

P r Fkk Q i j i j oy; dh Li ' kZ j s[ kk, ¡ , d fcUnq i j feyr h gSa t ks

fd j s[ kk y = 2x + a, a > 0 i j fLFkr gSA

51. t hok PQ dh yEckbZ gS :(A) 7a (B) 5a (C) 2a (D) 3a

Sol. B

(–a,–a)

P(at ,2at)2

(a,0)

y =4ax2

Q

O

yEckbZ = 4a cosec2 y (–a) = 2a (x – a)COC: y = –2x + 2atan = –2

2

1

5 sin = 52

cosec = 25

yEckbZ = 45

× 4a

52. ; fn t hok PQ, y2 = 4ax ds ' kh"kZ i j dks.k var fj r dj r h gS

r c tan =

(A) 732

(B) 732

(C) 532

(D) 532

Sol. D

(–a,–a)

P(at ,2at )1 12

(a,0)

y =4ax2

Q

O

12

1 ta2,

ta


18

MOP = 1t2

MOQ = –2t1

– 2 = aat

at22

1

1

t1 = 2

15

tan = 41

t2t2

11

tan = –32

(t1 + 1/t1)

t1 dk eku j [ kus i j (D)

i z' u 53 , oa 54 ds fy; s vuqPNsnekuk fd S = S1 S2 S3, t gk¡S1 = {z C : |z| < 4},

S2 =

0

i31i31zIm:Cz r Fkk

S3 = {z C : Re z > 0}.

53. S dk {ks=kQy =

(A) 10

3 (B) 20

3

(C) 16

3 (D) 32

3Sol. B

S1 = {z C: |z| < 4}

S2 =

0

i31i31zlm:Cz

S3 = {z C: Re z > 0}

(4,0)

x+y = 03

Nk; kafdr Hkkx dk {ks=kQy = 203

oxZ bdkbZ

54.Z S

min |1–3i – z| =

(A) 2 – 3

2(B)

2 32

(C) 3 – 3

2(D)

3 32

Sol. C

z Smin

|1 – 3i – z| =

(1, –3) l s j s[ kk 0yx3 dh yEcor ~ nwj h 2

33

bdkbZ

i z' u 55 , oa 56 ds fy; s vuqPNsn, d i sVh B1 esa 1 l Qsn xsan, 3 yky xsans r Fkk 2 dkyh xsans gSaA

, d nwl j h i sVh B2 esa 2 l Qsn xsans, 3 yky xsans r Fkk 4 dkyh xsansa gSaA , dr hl j h i sVh B3 esa 3 l Qsn xasnsa, 4 yky xsansa r Fkk 5 dkyh xsansa gSaA

55. ; fn i sfV; ksa B1, B2 r Fkk B3 esa i zR; sd l s 1 xsan fudkyh t kr h gS]r c fudkyh xbZ l Hkh 3 xsanksa ds , d gh j ax ds gksus dh i zkf; dr k gS :

(A) 82648 (B)

90648

(C) 558648 (D)

566648

Sol. A; fn B1, B2 r Fkk B3 esa l s , d ckWy fudkyh t kr h gS .......

B1

1 W3 R2 B

B2

2 W3 R4 B

B3

3 W4 R5 B

WWW BBB RRR

= 1 2 3 3 3 46 9 12 6 9 12 +

2 4 56 9 12

6 36 406 8 12

= 82

6 9 12

56. ; fn 2 xsansa , d ; knfPNd p; fur i sVh l s (fcuk i zfr LFkki u ds)fudkyh t kr h gSa r Fkk , d xsan l Qsn o nwl j h xsan yky fudyr h gS, r cbu 2 xsanksa dh i sVh B2 l s fudyus dh i zkf; dr k gS :

(A) 116181 (B)

126181

(C) 65

181 (D) 55181


19

Sol. D

212

29

26

29

C43

C32

C31

C3x2

112

61

51

61

=

18155

57. l wph I dks l wph II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn,x, dksM dk i z; ksx dj ds l gh mÙkj pqfu; s :

l wph I l wph II

P.

1 /22–1 –14

2 –1 –1

1 cos(tan y) y sin(tan y) yy cot (sin y) tan(sin y)

1.

1 52 3

dk eku gS

Q. ; fn cos x+cos y+cos z = 0 = sin x 2. 2

+ sin y + sin z r c cos 2

yx dk l EHkkfor

eku gS

R. ; fn cos

– x4 cos 2x + sin x 3.

12

sin 2x sec x=cos x sin 2x sec x+

cos

x4 cos 2x dk l EHkkfor eku gS

S. ; fn cot –1 2 –1sin 1– x sin tan x 6 , 4. 1

x 0. r c x dk l EHkkfor eku gSCodes :

P Q R S(A) 4 3 1 2(B) 4 3 2 1(C) 3 4 2 1(D) 3 4 1 2

Sol. B(P)

2/1

42

11

11

2 y)ytan(sin)ycot(sin)ysin(tany)ycos(tan

y1

tan–1 y = y = tan i zfr LFkkfi r dj us i j

1 /222

2 42

y 1 y1 1 y y1y

1 /2

2 4 42

1 y (1 y ) yy

= 1

(Q)oxZ dj ds ; ksx dj us i j3 + 2 (cos (x – y) + cos (y – z) + cos (z – x))=0cos (x – y) + cos (y – z) + cos (z – x) = –3/2

cos 2

yx cos

2zx

cos 2

xz =

81

....(i)

; fn cos 2

)yx( = 1

x = y

cos2 2

zy =

81

(vl aHko)

bl fy, cos 2

)yx( =

21

(R) cos 2x 2 sin 4

sin x = sec x sin 2x (cos x – sin x)

cos x sin x = 2

x = 4

sec x = 2

(S) 1

58. , d j s[ kk L:y = mx + 3, y–v{k ds fcUnq E(0, 3) r Fkki j oy; ds pki y2 = 16x, 0 y 6 ds fcUnq F(x0,y0) i j feyr h gSAi j oy; dh fcUnq F(x0,y0) i j Li ' khZ y-v{k dks fcUnq G(0,y1) i j dkVr hgSA j s[ kk L dh i zo.kr k m , sl h pquh t kr h gS fd f=kHkqt EFG ds {ks=kQy dk, d LFkkuh; vf/kdr e gSAl wph I dks l wph II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn, dksM dki z; ksx dj ds l gh mÙkj pqfu; s :

l wph I l wph II

P. m = 1. 12

Q. EFG dk egÙke {ks=kQy gS 2. 4R. y0= 3. 2S. y1= 4. 1

P Q R S(A) 4 1 2 3(B) 3 4 1 2(C) 1 3 2 4(D) 1 3 4 2


20

Sol. A

(0,3)E

(0,y )1

F(x ,40 x0 ) (x , y )0 0

Tf = yy0 = 8(x + x0)Tf, G l s xqt j r h gSy1 y0 = 8 (0 + x0)y1 y0 = 8 x0

x0 = 8yy 01

vc, 16y 2

0 = 8yy 01

2yy 0

1

= 21

(3 – y1) . x0

= 21

(3 – y1). 8yy 01

= 161

(3 – y1). y1. (2y1)

= 18

(3 – y1). y12

dyd

= 18

(6y1 – 3y12) = 0

y1 0 y1 = 2y1 = 2 & y0 = 4

vc m = 0x3y

0

0

= 1

m = 1

EFG = 21

. (3 – 2). 1 = 21

D = 21

P – 4 (m = 1)

Q – 1 (A = 21

)

R – 2 (y0 = 4)S – 3 (y1 = 2)

59. l wph-I dks l wph-II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn,x, dksM dk i z; ksx dj ds l gh mÙkj pqfu; s :

l wph-I l wph-II

P. l fn' kksa

a,b r Fkk c } kj k fu/kkZfj r l ekUr j 1. 100

"kV~Qyd dk vk; r u 2 gSA r c l fn' kksa

2(a b),3(b c) r Fkk

(c a) } kj k fu/kkZfj r

l ekUr j "kV~Qyd dk vk; r u gS

Q. l fn' kksa

a,b r Fkk c } kj k fu/kkZfj r l ekUr j 2. 30

"kV~Qyd dk vk; r u 5 gSA r c l fn' kksa

3(a b),(b c) r Fkk

2(c a) } kj k fu/kkZfj r

l ekUr j "kV~Qyd dk vk; r u gSR. , d f=kHkqt dk {ks=kQy] ft l dh l ayXu 3. 24

Hkqt k, ¡ l fn' kksa a r Fkk b } kj k fu/kkZfj r

gS] 20 gSA r c l fn' kksa

(2a 3b) r Fkk

(a– b)

} kj k fu/kkZfj r l ayXu Hkqt kvksa okys f=kHkqt dk {ks=kQy gSS. , d l ekUr j pr qHkqZt dk {ks=kQy ft l dh l ayXu 4. 60

Hkqt k, ¡ l fn' kksa a r Fkk b } kj k fu/kkZfj r gS, 30 gSA

r c l fn' kksa

(a b) r Fkk a } kj k fu/kkZfj r l ayXu

Hkqt kvksa okys l ekUr j pr qHkqZt dk {ks=kQy gSCodes :

P Q R S(A) 4 2 3 1(B) 2 3 1 4(C) 3 4 1 2(D) 1 4 3 2

Sol. C

(P) ]accb3ba2[

= 6 ]accbba[

= 6 ]cba[ 2 = 24

P 3

(Q) [3(a b) b c 2c a]

= 6 [a b b c c a]

= 6 × 2 ]cba[

= 60Q 4

(R) 21

ba

= 20, ba

= 40

21

)ba()b3a2(

= 25

|ba|

= 100

R 1

(S) a)ba(

= ab

= 30

S 2


21

60. j s[ kk, ¡ 1x –1 y z 3L :

2 –1 1

,

2x – 4 y 3 z 3L :

1 1 2

r Fkk l er y P1:7x + y + 2z = 3,

P2 : 3x + 5y – 6z = 4 yhft ; sA ekuk fd ax + by + cz = d,j s[ kkvska L1 r Fkk L2 ds i zfr PNsn fcUnq l s xqt j us okyk r Fkk l er y P1 o P2

ds yEcor , l er y dk l ehdj .k gSAl wph-I dks l wph-II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn, x, dksMdk i z; ksx dj ds l gh mÙkj pqfu; s :

l wph-I l wph-IIP. a = 1. 13Q. b = 2. – 3R. c = 3. 1S. d = 4. – 2

Codes :P Q R S

(A) 3 2 4 1(B) 1 3 4 2(C) 3 2 1 4(D) 2 4 1 3

Sol. AL1 i j dksbZ fcUnq ( 2 + 1, –, – 3)L2 i j dksbZ fcUnq ( + 4, – 3, 2 – 3)bl fy, = 1 r Fkk = 2bl fy, i zfr PNsn fcUnq (5, –2, –1) gS

P1 r Fkk P2 ds nksuksa ds yEcor ~ l fn' k = 653217kji

= k32j48i16 , vr % r y dk l ehdj .k

–16 (x – 5) + 48 (y + 2) + 32 (7 + 1) = 0x – 3y – 2z = 13a = 1b = –3c = –2d = 13

IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ] · IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ]...

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Transcript of IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ] · IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ]...