Time : 3 hrs. Max. Marks: 183 Answers & Solutions...(B) dNq l e; d sckn ckã cy F vkjS ifzrjksèkd cy l rafqyr gk t k,xas (C) IysV }kjk vuHqko gvk ifzrjkèkd cy v d sl ekuikrh gS (D)

1

CODE

4

Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005Ph.: 011-47623456 Fax : 011-47623472

DATE : 21/05/2017

PAPER - 1 (Code - 4)

Answers & Solutionsfor

JEE (Advanced)-2017

Time : 3 hrs. Max. Marks: 183

INSTRUCTIONS

QUESTION PAPER FORMAT AND MARKING SCHEME :

1. The question paper has three parts : Physics, Chemistry and Mathematics.

2. Each part has three sections as detailed in the following table :

Section QuestionType

Number of Questions Full Marks

Category-wise Marks for Each Question

Partial Marks Zero Marks Negative Marks

MaximumMarksof the

Section

SingleCorrectOption

6 +3If only the bubble corresponding to the correct option

is darkened

— 0If none of the

bubbles is darkened

–1In all other

cases

183

One ormore

correctoption(s)

7 +4If only the bubble(s)

corresponding toall the correct

option(s) is(are)darkened

+1For darkening a bubblecorresponding to each

correct option, provided NO incorrect option is

darkened

0If none of the

bubbles is darkened

–2In all other

cases

281

Singledigit

Integer(0-9)

5 +3If only the bubblecorresponding to

the correct answeris darkened

— 0In all other

cases

— 152

fgUnh ekè;e

2

JEE (ADVANCED)-2017 (PAPER-1) CODE-4

PHYSICS

[ kaM-1 (vf/ dre vad : 28)

bl [ kaM esa l kr i z' u gSaA

çR;sd i z'u ds pkj mÙkj fodYi (A), (B), (C) vkSj (D) gSa ft uesa ,d ;k ,d l s vf/ d fodYi l gh gSaA

i zR; sd i z' u ds fy, vks-vkj-, l - i j l kjs l gh mÙkj (mÙkjksa) ds vuq: i cqycqys (cqycqyksa) dks dkyk djsaA

i zR; sd i z' u ds fy, vad fuEufyf[ kr i fjfLFkfr ; ksa esa l s fdl h , d ds vuql kj fn;s t k;saxs %

i w.kZ vad : +4 ; fn fl i QZ l kjs fodYi (fodYi ksa) ds vuq: i cqycqys (cqycqyksa) dks dkyk fd; k gSA

vkaf' kd vad : +1 i zR; sd l gh fodYi ds vuq: i cqycqys dks dkyk djus i j] ; fn dksbZ xyr fodYi dkykugha fd; k gSA

' kwU; vad : 0 ; fn fdl h cqycqys dks dkyk ugha fd; k gSA

½.k vad : –2 vU; l Hkh i fjfLFkfr ; ksa esaA

mnkgj.k % ; fn ,d i z' u ds l kjs l gh mÙkj fodYi (A), (C) vkSj (D) gSa] r c bu rhuksa ds vuq: i cqycqyksa dks dkys djus i j+4 vad feysaxs_ fl i QZ (A), (D) ds vuq: i cqycqyksa dks dkyk djus i j +2 vad feysaxsa_ rFkk (A) vkSj (B) ds vuq: i cqycqyksadks dkyk djus i j –2 vad feysaxs D;ksafd ,d xyr fodYi ds vuq: i cqycqys dks Hkh dkyk fd; k x; k gSA

1. , d l eku jSf[ kd ?kurkokys (uniform mass per unit length) mèokZèkj Mksj ds fupys fl js i j , d xqVdk M yVdk gqvk gSA

Mksj dk nwl jk fl jk n<+ vkèkkj (fcanq O) l s l ayXu gSA rajx&nSè; Z 0 dh vuqi zLFk r jax Li an (Li an 1, Pulse 1) fcanq O i j

mRi Uu dh xbZ gSA ; s r jax Li Un fcanq O l s fcanq A rd TOA l e; esa i gq¡prh gSA xqVds M dks fcuk fo{kksfHkr fd; s gq, fcanq A

i j fuekZ.k dh xbZ r jax&nSè; Z 0 dh vuqi zLFk r jax Li an (Li an 2, pulse 2), fcanq A l s fcanq O rd TAO l e; esa i gq¡prh gSA fuEu

esa l s dkSul k (l s) dFku l gh gS@gSa\

MA

O Pulse 1

Pulse 2

(A) Mksj ds eè; fcanq i j Li an 1(pulse 1) ,oa Li an 2(pulse 2) dk osx l eku gS

(B) Mksj ds vuqfn' k i zsf"kr fdl h Hkh Li an dk osx ml dh vkofÙk ,oa r jax&nSè; Z i j fuHkZj ugha gS

(C) Li an 1(pulse 1) dh r jax&nSè;Z fcanq A rd i gqapus esa yEch gks t k,xh

(D) l e; TAO = TOA

mÙkj (A, B, D)

gy fdl h fcUnq i j Tv ,

pw¡fd osx ,d fcUnq i j r uko rFkk i zfr bdkbZ yEckbZ nzO;eku i j fuHkZj djr k gS] bl fy, O l s A rd

rFkk A l s O rd l e; l eku gksxkA

TOA = TAO

3


pw¡fd vkofÙk l Hkh fcUnq i j fu; r jgrh gSA bl s l eku cuk; s j[ kus ds fy, v

l eku gksxk rFkk v, O(T mPpre) i j mPpre

gS bl fy, , O i j mPpre gksxkA

2. ,d xksykdkj fo| qr&jks/ h r kez r kj (insulated copper wire) dks A ,oa 2A okys nks {ks=ki Qyksa ds oy; ksa esa O; kofrZr

fd; k x; k gSA rkjksa ds vfrØe.k fcanq fo| qr jks/ h jgr s gS (t Sl k fp=k esa n' kkZ; k x; k gS)A l ai w.kZ oy; dkxt + ds r y

esa fLFkr gSA dkxt ds r y ds vfHkyEcor fLFkj rFkk ,dl eku pqEcdh; {ks=k B l oZ=k mi fLFkr gSA oy; vi us

l keqnkf; d O; kl ksa l s cus v{k ds i fjr% l e; t = 0 l s dks.kh; osx (angular velocity) l s ?kweuk ' kq: djr k gSA

fuEu esa l s dkSul k(l s) dFku l gh gS@gSa\

× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×× × × × × × × ×

area 2A

area A

B

× × × × × × × ×

(A) nksuksa oy; ksa l s mRi Uu dqy çsfjr fo| qr okgd cy (emf induced) cos t ds l ekuqi krh gS

(B) t c oy; ksa dk r y dkxt + ds r y l s vfHkyac fn' kk esa gksrk gS r c vfHkokg ds i fjor Zu ds nj vf/ dregksrh gS

(C) nksuksa oy; ksa l s mRi Uu vf/ dr e dqy çsfjr fo| qr okgd cy (net emf) dk vk; ke] NksVs oy; esa mRi Uuvfèkdre çsf"kr fo| qr okgd cy ds vk; ke ds cjkcj gksxk

(D) çsfjr fo| qr okgd cy (emf induced) oy; ksa ds {ks=ki Qyksa ds ; ksx ds l ekuqi kfrd gS

mÙkj (B, C)

gy = BA cost

e = B sint

ywi 1 ds fy,

1 = BA cost : |e1| = AB sint

2 = 2BA sint |e2| = 2BA sint

e1 o e2 , d nwl js ds foi jhr gSa] bl fy, usV çsfjr fo-ok-c- dk vk; ke = 2BA – BA = BA

e1 rFkk e2, t = /2 ; k = 90° i j f' k[ kj gksaxsA

4


3. oÙkkdkj pki okys ,d xqVds dk nzO; eku M gSA ; s xqVdk ,d ?k"kZ.k jfgr est i j fLFkr gSA est ds l ki s{; (in a coordinate

system fixed to the table) xqVds dk nkfguk dksj (right edge) x = 0 i j fLFkr gSA nzO; eku m okys ,d fcanq d.k (point

mass) dks oÙkkdkj pki ds mPpre fcanq l s fojkekoLFkk l s NksM+k t krk (released from rest) gSA ; s fcanq d.k oÙkkdkj i Fk i juhps dh vkSj l jdrk gSA t c fcanq d.k xqVds l s l ai dZ foghu gks t krk gS] rc ml dh r kR{kf.kd fLFkfr x vkSj xfr gSA fuEuesa l s dkSul k (l s) dFku l gh gS@gSa\

R

Rm

M

y

x

x = 0

(A) fcanq d.k (m) dk osx 2

1

gRmM

gS

(B) xqVds (M) ds l agfr dsUnz ds foLFkki u dk X ?kVd (X co-ordinate) mR

M m

gS

(C) fcanq d.k (m) dk LFkku 2 mRxM m

gS

(D) xqVds (M) dk osx 2mV gRM

gSS

mÙkj (A, B)

gy v = M dk osx

u = m dk osx

mu = –MV ...(i)

2 21 12 2

mgR mu MV ...(ii)

2 – 2

1 1

gR m gRu vm mMM M

rFkkMx = m(R – x) t gk¡ x = CykWd M dk foLFkki u

x(M + m) = mR mRx

m M

ck;ha vksj

;k – mRxm M

4. , d l i kV IysV (flat plate) vYi ncko ds xSl (gas at low pressure) esa] vi us r y dh vfHkyac fn' kk esa] ckã cy F dsi zHkko esa vxzl fjr gSA IysV dh xfr v,xSl v.kqvksa ds vkSl r xfr u l s cgqr de gSA fuEu esa l s dkSul k (l s) dFku l gh gS@gSa?

5


(A) i zfr xkeh ,oa vuqxkeh i "B ds ncko dk var j uvds l ekuqi krh gS

(B) dqN l e; ds ckn ckã cy F vkSj i zfr jksèkd cy l arqfyr gks t k,axs

(C) IysV }kjk vuqHko gqvk i zfr jksèkd cy vds l ekuqi krh gS

(D) IysV l oZnk ' kqU;sr j fLFkj Roj.k (constant non-zero acceleration) l s pyrh jgsxh

mÙkj (A, B, C)

gy n = i zfr bdkbZ vk; ru v.kqvksa dh l a[ ; k

u = xSl v.kqvksa dh vkSl r pky

t c IysV v pky l s xfr ' khy gS] r c eq[ ; cy ds l ki s{k v.kqvksa dh l ki sf{kd pky = v + u

l Eeq[ k VDdj esa i zfr VDdj IysV dks LFkkukUr fjr l aosx = 2m(u + v)

l e; t esa VDdj dh l a[ ; k 1 ( )2

u v n tA

t gk¡ A = i "Bh; {ks=ki Qy

bl fy, vkxs dh l rg l s l e; t esa LFkkukUr fjr l aosx = m(u + v)2nAt

i hNs dh l rg l s l e; t esa LFkkukUrfjr l aosx = m(u – v)2nAt

usV cy = mnA[(v + u)2 – (u – v)2]

= mnA[4vu]

F v

Pvxz – P

i ' p = mn[u + v]2 – mn[u – v]2

= mn[4uv] = 4mnuv

P uv

5. fp=k esa fn[ kk; s x, i fj i Fk esa L = 1H, C = 1F, R = 1kgSA ,d i fjorhZ oksYVrk (V = V0sint) l zksr l s Jzs.kh l acaèk gSAfuEu esa l s dkSu l k (l s) dFku l gh gS@gSa\

V t0 sin

L = 1 H C = 1 F R = 1 k

(A) t c = 104 rad.s–1 gksxh rc fo| qr èkkjk (electric current) oksYVrk dh l edyk esa gksxh

(B) t c >>106 rad. s–1, i fji Fk l aèkkfj=k (capacitor) dh r jg O;ogkj djrk gS

(C) t c fo| qr èkkjk oksYVrk dh l edyk esa gksxh rks og vkofÙkZ R i j fuHkZj ugha djsxh

(D) t c ~ 0 gksxh rc i fji Fk esa cgrh èkkjk ' kwU; ds fudV gksxh

mÙkj (C, D)

gy >> 106 i j

XL = L

= = 106 ds fy, 106 × 10–6

6


= 1 >> 106 ds fy, XL >> 1

XC = 6 –61 1 1

10 10C

= 106 i j >> 106 ds fy, Xc << 1

R = 1 k

R LC –6 –6

1 1

10 10 = 106 rad.s–1

0 Xc i j / kjk 0

6. , d l ef}ckgq fi zTe dk fi zTe dks.k A gS (isosceles prism of angle A)A bl fi zTe dk vi orZukad gSA bl fi zTe dk

U;wure fopyu dks.k (angle of minimum deviation) m = A gSA fuEu esa l s dkSu l k (l s) dFku l gh gS@gSa\

(A) U;wure fopyu esa vki fr r dks.k i1 , oa i zFke vi orZd ry ds vi orZd dks.k r1 = (i1/2) }kjk l acafèkr gS

(B) fi zTe dk vi orZukad ,oa fi zTe dks.k (A), 11cos2 2

A

}kjk l acafèkr gS

(C) Tkc fi zTe dk vki ru dks.k i1 = A gS rc fi zTe ds Hkhr j i zdk' k fdj.k fi zTe ds vkèkkj ds l ekukUr j gksxhA

(D) t c i gys r y i j vki ru dks.k 21 sin 1 sin 4cos 1 cos

2Ai A A

gS] r c bl fi zTe ds fy, f}rh; ry l s fuxZr

fdj.k fi zTe ds i "B l s Li ' khZ; gksxh (tangential to the emergent surface)

mÙkj (A, C, D)

gy m = (2i) – A

2A = 2i i = A ds fy, r dh x.kuk

i = A rFkk r = A/2 (nk; ha vksj ds gy dks nsf[ k, )

sin2

sin2

A A

A

A

i1c

A

r A1 = /2

A Ar A2 = /2

2sin cos2 2

sin2

A A

A 1sinA = sinr

2cos2A

sinA = 2cos .sin2A r

1sini1 = × sin(A – C)

2sin cos2 2sin

2cos2

A A

rA

= sin2A

= 2cos sin cos – cos sin2 C CA A A

2Ar

7


= 2 12cos sin 1– sin – cos2 CA A A

= 21 cos2cos sin 1– –

2 2cos2

A AAA

= 2

1 cos2cos sin 1– –2 4cos 2cos

2 2

A AAA A

i1 = –1 2sin sin 4cos – 1 – cos2AA A

U;wure fopyu ds fy, ] 1 2Ar

–1cos2 2A

–12cos2

A

7. ekuoh; i "Bh; {ks=ki Qy yxHkx 1 m2 gksrk gSA ekuo ' kjhj dk r ki eku i fjos' k ds r ki eku l s 10 K vfèkd gksrk gSA i fjos' k

rki eku T0 = 300 K gS] bl i fjos' k r ki eku ds fy, 4 20 460 WmT gSA t gk¡ LVhi Qku&cksYV~t eku fu; rkad (Stefan-

Boltzmann constant) gSA fuEu esa l s dkSu l k (l s) dFku l gh gS@gSa\

(A) i fjos' k r ki eku vxj T0 l s ?kVr k gS (T0 << T0) r c ekuo ds ' kjhj dks r ki eku dk vuqj{k.k djus ds fy,30 04W T T vfèkd Åt kZ fofdfjr djuh i M+rh gS

(B) i "Bh; {ks=ki Qy ?kVkus (t Sl s% fl dqM+us l s) l s ekuo vi us ' kjhj l s fofdfjr Åt kZ ?kVkrs gSa ,oa vi us ' kjhj dk r ki eku vuqjf{krdjr s gSa

(C) ekuoh; ' kjhj ds r ki eku esa vxj l kFkZd of¼ gks rc i zdk' k pqEcdh; fodj.k Li SDVªe dh f' k[ kj r jax&nSè; Z (peak in the

electromagnetic spectrum) nh?kZ r jax&nSè;Z dh vksj foLFkkfi r gksrh gS

(D) ekuoh; ' kjhj l s 1 l sadM esa fudVre fofdfjr Åt kZ 60 t wy (60 Joules) gS

mÙkj (A, B, D)

gy ' kjhj dk mi ki p; ra=k r ki dks cuk;s j[ kus ds fy, vfr fjDr Åt kZ mRi Uu djsxk

ekuk vkUr fjd mi ki p; ds dkj.k] mRi Uu ' kfDr I gS]

t c ' kjhj dk r ki fu; r gS] Q usV

= 0

I = A [T4 – 40T ]

dI = 4A 30T (dT0) = 4A 3

0T .T0 (fodYi A)

= 4× 460 × 10300 60 J/s (fodYi D)

8


[ kaM - 2 (vf/ dre vad % 15)

bl [ kaM esa i kap ç'u gSaA

çR; sd ç'u dk mÙkj 0 l s 9 rd (nksuksa ' kkfey) ds chp dk ,d , dy vadh; i w.kk±d gSA

çR; sd ç'u ds fy, vks- vkj- , l - i j l gh i w.kk±d ds vuq: i cqycqys dks dkyk djsaA

çR; sd ç' u ds fy, vad fuEufyf[ kr i fjfLFkfr ; ksa esa l s fdl ,d ds vuql kj fn; s t k; saxs%

i w.kZ vad : +3 ; fn fl i QZ l gh mÙkj ds vuq: i cqycqys dks dkyk fd; k gSA

' kwU; vad : 0 vU; l Hkh i fjfLFkfr ; ksa esa

8. i "B&ruko (surface tension) –10.1Nm4

S

ds æo ds ,d cwan dh f=kT; k R = 10–2 m gS] ft l s K l e: i cwanksa esa

foHkkft r fd; k x; k gSA i "B&Åt kZ dk cnyko U = 10–3 Joules gSA ; fn K = 10 gS r c dk eku gksxkmÙkj (6)gy ekuk NksVh cw¡n dh f=kT; k = r

3 34 43 3

R K r

13R K r ...(i)

S(K4r2 – 4R2) = 10–3

22 3

23

0.14 104

Rk R

K

12 23 1 10R K 13 1 100K 13 101K

310 101

a 6

9. ,do.khZ çdk' k (monochromatic light) vi orZukad n = 1.6 okys ekè; e esa çxkeh gSA ; g çdk' k dk¡p dh phrh(stack of glass layers) i j fupys l r g l s = 30° dks.k i j vki fr r gksrk gS (t Sl k fd fp=k esa n' kkZ; k x; k gS)Ad k¡pksa d s Lr j i j Li j l ekar j gSA d k¡p d s phr h d s v i or Zukad , d fn"V nm = n – mn, Øe l s ?kV jgs gSaA ; gk¡m Lr j dk vi orZukad nm gS vkSj n = 0.1 gSA çdk' k fdj.k (m – 1) ,oa m Lr j ds i "Bry l s l ekar j fn' kk esankbZa vksj l s ckgj fudyrk gSA rc m dk eku gksxk

m n m n – n m n – ( – 1)m – 1

321

n n – 3n n – 2n n – n

9


mÙkj (8)

gy i zFke i jr rFkk m oha i jr ds eè; Lusy fu; e yxkus i j

sin sin90º n n m n

11.6 1.6 0.1 12

m

0.8 80.1

m

10. vk; ksMhu dk l eLFkkfud (isotope)1311, ft l dh v/ Z&vk; q 8 fnu gS] -{k; ds dkj.k t suksau (xenon) ds l eLFkkfudesa {kf; r gksrk gSA vYi ek=k dk 1311 fpfëòr (labelled) l hje (serum) ekuo ' kjhj esa vUr%f{kIr (inject) fd; k x; k]ft l ek=kk dh vWfDVork (activity) 2.4 × 105 csdsjsy (Becquerel) gSA ; g l hje : f/ j / kjk esa vk/ s ?kaVs esa ,dl ekufor fjr gksr k gSA vxj 11.5 ?kaVs ckn 2.5 ml jDr 115 csdsjsy dh vWfDVork n' kkZrk gS] r c ekuo ' kjhj esa jDrvk; ru (yhVj esa) gS (vki ex 1 + x for |x| << 1 ,oa ln 2 0.7 dk mi ; ksx dj l dr s gSaA)

mÙkj (5)

gy ekuk jDr dk dqy vk; ru V ml gS

2.5 ml jDr l fØ; rk = 115 Bq

1 ml jDr l fØ; rk = 1152.5

l Ei w.kZ jDr dh l fØ; rk = 115 V Bq2.5

5115 2.4 102.5

tV e

ln2 11.55 8 242.4 10 e

0.7 11.55 8 242.4 10 e

15 242.4 10 e

5 12.4 10 124

5 23 2.5V 2.4 1024 115

5000 ml

jDr dk dqy vk; ru = 5 L

11. ,d gkbMªkst u i jek.kq dk ,d bysDVªkWu ni DokaVe l a[ ; k (quantum number) okys d{k l s nf DokaVe l a[ ; k (quantum

number) ds d{k esa ços' k djr k gSA Vi rFkk Vf çkFkfed ,oa vafre fLFkfr t Åt kZ,a gSaA ; fn 6.25,i

f

vv

r c nf

dh U; wure l EHkkoh l a[ ; k (smallest possible nf) gS

10


mÙkj (5)

gy2

2PE 27.2 Zn

H ds fy, Z = 1

227.2PEn

1 227.2PEi

i

Vn

2 2

27.2PE ff

vn

22

2 6.25i f f

f ii

V n nV nn

nf = 2.5 ni

52

f

i

nn

ni feuV = 2

nf feuV = 5

nf = 5

12. ,d fLFkj L=kksr vkofÙk f0 = 492 Hz dh èofu mRl ft Zr djr k gSA 2 ms–1 ds xfr ds vi xeuh dkj l s ; g èofui jkofrZr gksrh gSA èofu L=kksr i jkofrZr l adsr dks çkIr dj ds ewy l adsr i j vè; kjksfi r (superpose) djr k gSArc i fj.kkeh fl Xuy dh foLi an&vkofÙk (beat frequency) gS

(èofu dh xfr 330 ms–1 gSA dkj èofu dks ml dh çkIr gqbZ vkofÙk i j i jkofrZr djrh gSA)

mÙkj (6)

gy fLFkj L=kksr }kjk mRl ft Zr èofu dh vkofÙk = f0 = 492 Hz

xfr ' khy dkj }kjk i jkofrZr èofu dh vkofÙk

0'

c

c

c vf f

c v

t gk¡ c = ok;q esa èofu dh pky = 330 m/s

vc = dkj dh pky = 2 m/s

gy djus i j

332' 492 498 Hz328

f

i fj.kkeh fl Xuy dh foLi Un vkofÙk = f ' – f0

= (498 – 492) Hz

= 6 Hz

11


[ kaM-3 (vf/ dre vad : 18)

bl [ kaM esa l qesy i zdkj ds Ng i z'u gSaA bl [ k.M esa nks Vscy gSa (i zR;sd Vscy esa 3 dkWye vkSj 4 i afDr ; ka gSa) i zR; sd Vscy i j vk/ kfjr rhu i z'u gSaA çR;sd i z' u ds pkj mÙkj fodYi (A), (B), (C) vkSj (D) gSa ft uesa fl iQZ ,d fodYi l gh gSA i zR; sd i z' u ds fy, vks-vkj-, l - i j l gh mÙkj ds vuq: i cqycqys dks dkyk djsaA i zR; sd i z' u ds fy, vad fuEufyf[ kr i fjfLFkfr ; ksa esa l s fdl h , d ds vuql kj fn;s t k;saxs %

i w.kZ vad : +3 ; fn fl i QZ l gh fodYi ds vuq: i cqycqys dks dkyk fd; k gSA' kwU; vad : 0 ; fn fdl h cqycqys dks dkyk ugha fd; k gSA½.k vad : –1 vU; l Hkh i fjfLFkfr ; ksa esaA

uhps nh x;h Vscy ds rhu dkyeksa esa mi yC/ l wpuk dk mi ;qDr <ax l s l qesy dj i z'uksa Q.13, Q.14 vkSj Q.15 dsmÙkj nhft ;sA

, d pkt Z; qDr d.k (bysDVªkWu ; k i zksVksu) vkjafHkd xfr l s ewy fcUnq (x = 0, y = 0, z = 0) i j i zLrqr (introduced) gksrk

gSA fLFkj rFkk ,dl eku fo| qr~ {ks=k E , oa pqEcdh; {ks=k

B l oZ=k mi fLFkr gSA d.k dh xfr

, fo| qr {ks=k

E rFkk pqEcdh;

{ks=k B fuEu dkWyeksa 1, 2 , oa 3 esa Øe' k% n'kkZ;s x; s gSaA E0, B0 ds eku / ukRed gSaA

(I) bysDVªkWu (i) (P)

(II) bysDVªkWu (ii) (Q)

(IV) i zksVksu (iv) (S)

(III) i zksVksu (iii) (R)

0

0

2E

xB

0

0

Ey

B

0

0

0

2E

xB

0E E z

0 E E y

0 E E x

0E E x

0 B B x

0B B x

0B B y

0B B z

l s

l s

l s

l s

13. fdl fLFkfr esa d.k vpy xfr l s l h/ h js[ kk esa pyu djr k gS\

(A) (IV) (i) (S) (B) (III) (ii) (R)(C) (II) (iii) (S) (D) (III) (iii) (P)

mÙkj (C)

gy bysDVªkWu fu; r osx l s l jy js[ kk esa xfr djsxk ; fn

0

0

Ev y

B , 0

E E x ,

0B B z

14. fdl fLFkfr esa d.k +z-v{k vuqfn' k dqaMfyuh i Fk (helical path along positive z-axis) dk vuql j.k djsxk\(A) (IV) (i) (S) (B) (II) (ii) (R)

(C) (III) (iii) (P) (D) (IV) (ii) (R)

mÙkj (A)

gy i zksVkWu dq.Myhnkj i Fk esa xfr djsxk rFkk v{k / ukRed z-fn' kk ds vuqfn'k gSA

; fn 0

0

2 Ev x

B,

0E E z rFkk

0B B z

12


15. fdl fLFkfr esa d.k l h/ h js[ kk esa ½.kkRed y-v{k (negative y-axis) dh fn' kk esa pysxk\

(A) (III) (ii) (R) (B) (IV) (ii) (S)

(C) (III) (ii) (P) (D) (II) (iii) (Q)

mÙkj (A)

gy i zksVkWu ½.kkRed y-fn' kk ds vuqfn' k l jy js[ kk esa xfr djsxk t c

0v ,

0 E E y rFkk

0B B y

uhps nh x;h Vscy ds rhu dkyeksa esa mi yC/ l wpuk dk mi ;qDr <ax l s l qesy dj i z'uksa Q.16, Q.17 vkSj Q.18

ds mÙkj nhft ;sA

P

V

1 2

P

V

1

2

P

V

1 2

P

V

1

2

(i)

l erki h;

l evk; r fud (Isochoric)

l enkch; (Isobaric)

: ¼ks"e (adiabatic)

1 2 2 2 1 11 ––1

W P V PV

1 2 2 1–W PV PV

1 2 0W

21 2

1–

VW nRT ln

V

(I)

,d vkn'kZ xSl fofHkUu pØh; Å"eki kfrd i zØeksa l s xqt jrk gSA ; g fuEu dkWye esa vkjs[ k }kjk n' kkZ; k x; k gSA dsoy fLFkfr l s fLFkfr t kusokys i Fk dh vksj è; ku nsaA bl i Fk i j fudk; i j gqvk dk; Z gS A ; gk¡ fu; r nkc ,oa fu; r vk; ru Å"ek&/ kfjrkvksa dk vuqi kr gS A xSl ds eksyksa dh l a[ ; k gSA

(ideal gas) 3 — 1 2 (work done on the system)

g (ratio of the heat capacities) (moles)

P V W

n

(P)

(ii)(II) (Q)

(iii)(III) (R)

(iv)(IV) (S)

13


16. fuEu fn, fodYi ksa esa dkSul k l a; kst u U = Q – P V i zfØ; k dk vdsys l gh i zfr fuf/ Ro djr k gS\

(A) (III) (iii) (P) (B) (II) (iii) (S)

(C) (II) (iii) (P) (D) (II) (iv) (R)

mÙkj (A)

gy fodYi (A) l gh i zn' kZu gSA

W1 2 = –PV2 + PV1 l enkch;

17. fuEu fodYi ksa esa dkSu l k l a; kst u l gh gS\

(A) (II) (iv) (P) (B) (IV) (ii) (S)

(C) (II) (iv) (R) (D) (III) (ii) (S)

mÙkj (D)

gy fodYi (D) l gh l a; kst u gSA

W1 2 = 0 l evk; r fud i zØe

18. fuEu fodYi ksa esa l s dkSu l k l a; kst u vkn' kZ xSl esa èofu dh xfr dh eki ds l a' kks/ u esa i z; qDr Å"ekxfrd i zfØ; k dks l ghn' kkZrk gS\

(A) (III) (iv) (R) (B) (I) (ii) (Q)

(C) (IV) (ii) (R) (D) (I) (iv) (Q)

mÙkj (D)

gy : ¼ks"e i zØe dks ,d vkn' kZ xSl esa èofu dh pky ds fu/ kZj.k esa l a' kks/ u ds : i esa i z; qDr fd; k t krk gSA

l gh fodYi (D) gS

:¼ks"e i zØe esa fudk; i j fd; k x; k dk; Z = 2 2 1 1– –

– 1P V PV

END OF PHYSICS

14


CHEMISTRY

[kaM[kaM[kaM[kaM[kaM-1 (vf/dre vad (vf/dre vad (vf/dre vad (vf/dre vad (vf/dre vad : 28)))))

• bl [kaM esa lkrlkrlkrlkrlkr iz'u gSaA

• çR;sd iz'u ds pkjpkjpkjpkjpkj mÙkj fodYi (A), (B), (C) vkSj (D) gSa ftuesa ,d ;k ,d ls vf/d,d ;k ,d ls vf/d,d ;k ,d ls vf/d,d ;k ,d ls vf/d,d ;k ,d ls vf/d fodYi lgh gSaA

• izR;sd iz'u ds fy, vks-vkj-,l- ij lkjs lgh mÙkj (mÙkjksa) ds vuq:i cqycqys (cqycqyksa) dks dkyk djsaA

• izR;sd iz'u ds fy, vad fuEufyf[kr ifjfLFkfr;ksa esa ls fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %

iw.kZ vad : +4 ;fn fliQZ lkjs lgh fodYi (fodYiksa) ds vuq:i cqycqys (cqycqyksa) dks dkyk fd;k gSA

vkaf'kd vad : +1 izR;sd lgh fodYilgh fodYilgh fodYilgh fodYilgh fodYi ds vuq:i cqycqys dks dkyk djus ij] ;fn dksbZ xyr fodYi dkykugha fd;k gSA

'kwU; vad : 0 ;fn fdlh cqycqys dks dkyk ughaughaughaughaugha fd;k gSA

½.k vad : –2 vU; lHkh ifjfLFkfr;ksa esaA

• mnkgj.k % ;fn ,d iz'u ds lkjs lgh mÙkj fodYi (A), (C) vkSj (D) gSa] rc bu rhuksa ds vuq:i cqycqyksa dks dkyk djus ij+4 vad feysaxs_ fliQZ (A), (D) ds vuq:i cqycqyksa dks dkyk djus ij +2 vad feysaxsa_ rFkk (A) vkSj (B) ds vuq:i cqycqyksadks dkyk djus ij –2 vad feysaxs D;ksafd ,d xyr fodYi ds vuq:i cqycqys dks Hkh dkyk fd;k x;k gSA

19. ,d vkn'kZ xSl dks (p1, v

1, T

1) ls (p

2, v

2, T

2) rd fofHkÂ voLFkkvksa ds v/hu iQSyk;k x;k gSA fuEufyf[kr fodYiksa esa lgh

dFku gS (gSa)

(A) tc v1 ls v

2 rd :¼ks"e voLFkk ds v/hu bldk mRØe.kh; (reversible) iQSyko fd;k tk; rks xSl }kjk fd;k x;k dk;Z

v1 ls v

2 rd lerkih (isothermal) voLFkkvksa ds v/hu mRØe.kh; iQSyko esa fd;s x, dk;Z dh rqyuk esa de gS

(B) xSl dh vkarfjd mQtkZ esa cnyko (i) 'kwU; gS ;fn bls T1 = T

2 ds lkFk iQSyko mRØe.kh; (reversible) rjhds ls fd;k

tk,] vkSj (ii) /ukRed gS ;fn bls T1 ≠ T

2 ds lkFk :¼ks"e (adiabatic) ifjfLFkfr;ksa ds v/hu mRØe.kh; (reversible)

iQSyko fd;k tk;

(C) ;fn iQSyko eqDr :i ls fd;k tk; rks ;g lkFk&lkFk nksuksa lerkih (isothermal) ,oa :¼ks"e (adiabatic) gS

(D) tc bls vuqRØe.kh; rjhds ls (irreversibly) (p2, v

2) ls (p

1, v

1) rd fLFkj nkc p

1 ds fo:¼ nck;k tkrk gS rks xSl ds

mQij fd;k x;k dk;Z vf/dre gksrk gS

mÙkjmÙkjmÙkjmÙkjmÙkj (A, C, D)

gygygygygy (A)

V1

V2

V

P

W > Wlerkih; :¼ks"elerkih;

:¼ks"e

(C) eqDr çlkj esa Pex

= 0 ⇒ w = 0

,d vkn'kZ xSl ds lerkih; eqDr çlkj ds fy,,

ΔU = 0 ⇒ q = 0 ∴ :¼ks"e Hkh gS

(D) vuqRØe.kh; lEihM+u esa P o V vkjs[k ds uhps dk {ks=kiQy mRØe.kh; lEihM+u esa P o V vkjs[k ds uhps ds {ks=kiQy lsvfèkd gS

15


P1

P2

P

(P , V )1 2

(P , V )2 2

V1

V2

V

(P , V )1 1

vuqRØe.kh;mRØe.kh;

20. fuEufyf[kr ;ksfxd dk(ds) vkbZ- ;w- ih- ,s- lh- (IUPAC) uke gS(gaS)

ClH C3

(A) 4-eSfFkyDyksjks csathu (B) 4-Dyksjks Vksyqbu

(C) 1-Dyksjks-4-eSfFky csathu (D) 1-eSfFky-4-Dyksjkscsathu

mÙkjmÙkjmÙkjmÙkjmÙkj (B, C)

gygygygygy IUPAC uke

(B)

CH3

Cl

4-DyksjksVksyqbu

1

2

3

4

(C)

Cl

CH3

1- -4-Dyksjks esfFkycsathu

1

2

3

4

21. L vkSj M nzoksa ds feJ.k }kjk cuk;s ,d foy;u esa nzo M ds xzke&v.kqd fHkÂ (mole fraction) ds fo:¼ nzo L ds okLinkc dks fp=k esa fn[kk;k x;k gSA ;gk¡ x

L vkSj x

M, L vkSj M ds Øe'k% xzke&v.kqd fHkÂksa dks fu:fir djrs gSaA bl fudk; dk

(ds) mi;qDr lgh dFku gS (gSa)

PL

Z

xM1 0

(A) fcanq Z 'kq¼ nzo M ds ok"i nkc dks fu:fir djrk gS vkSj xL = 0 ls x

L = 1 rd jkmYV dk fu;e (Raoult's law) dk

ikyu gksrk gS

(B) 'kq¼ nzo L esa L-L ds chp esa vkSj 'kq¼ nzo M esa M-M ds chp esa varjk&v.kqd fØ;k,a L-M ds chp esa varjk&v.kqdfØ;kvksa ls izcy gSa tc mUgsa foy;u esa fefJr fd;k tkrk gS

(C) fcanq Z 'kq¼ nzo M ds ok"i nkc dks fu:fir djrk gS vkSj tc xL → 0 rks jkmYV dk fu;e (Raoult's law) dk ikyu gksrk gS

(D) fcanq Z 'kq¼ nzo L ds ok"i nkc dks fu:fir djrk gS vkSj tc xL → 1 rks jkmYV dk fu;e (Raoult's law) dk ikyu

gksrk gS

mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)

gygygygygy Z

x = 0M

x = 1L

'kq¼ L

xM

→x = 1M

M'kq¼

pL

16


fcUnq Z 'kq¼ nzo L ds ok"i nkc dks iznf'kZr djrk gS

xL

→ 1 ij foy;u cgqr ruq gS] L foyk;d cu tkrk gSA L esa m dk cgqr ruq foy;u yxHkx vkn'kZ gS rFkk jkÅV fu;e

(pL = x

Lp

L

°) dk ikyu djrk gSA

vkSj fcUnqfdr js[kk (vkn'kZ foy;u ds fy, visf{kr) ds Åij xzkiQ }kjk fufnZ"V gS fd blesa /ukRed fopyu gS

∴ L-M ikjLifjd fØ;k < L - L ,oa M - M ikjLifjd fØ;k

22. fuEufyf[kr ladyu vfHkfØ;kvksa (addition reactions) ds fy, lgh dFku gS (gSa)

(i)

H C3

H

H

CH3

Br /CHCl2 3

M N vkSj

(ii)H C

3

H H

CH3 Br /CHCl

2 3

O P vkSj

(A) (M vkSj O) vkSj (N vkSj P) ,uUVhvksesjks (enantiomers) ds nks ;qxy gSa

(B) nksuksa vfHkfØ;kvksa esa czksfefudj.k Vªkal ladyu }kjk c<+rk gS

(C) O vkSj P le:i v.kq gSa

(D) (M vkSj O) vkSj (N vkSj P) MkbZLVhfjvksesjksa (diastereomers) ds nks ;qxy gSa


gygygygygy C = CH

CH3

H C3

H

foi{k

Br /CHCl2 3

CH3

Br

Br

CH3

H

H+

CH3

H

H

CH3

Br

Br

eslks ,oa (M N)

CH3

Br

H

CH3

H

Br+

CH3

H

Br

CH3

Br

H

çfrfcEc leko;oh ;qXeO P

(i)

C = CCH

3

H

CH3

H

Br2

(ii)

lei{k izfr jslhfedlei{k izfr jslhfedlei{k izfr jslhfedlei{k izfr jslhfedlei{k izfr jslhfed

M rFkk N eslks gS (le:i)

O rFkk P çfrfcEc leko;oh ;qXe gS

(B) czksehuhdj.k çfr&;ksx }kjk lEiUu gksrh gS

(D) (M rFkk O) rFkk (N rFkk P) vçfrfcEc f=kfoe leko;fo;ksa ds nks ;qXe gSA

23. ,d xqykch jax okys MCl2·6H

2O(X) vkSj NH

4Cl ds tyh; foy;u esa vf/D; tyh; veksfu;k ds feykus ij] ok;q dh mifLFkfr

esa ,d v"ViQydh; ladj (octahedral complex) Y nsrk gSA tyh; foy;u esa ladj Y 1:3 fo|qr vi?kV~; (electrolyte) dhrjg O;ogkj djrk gSA lkekU; rki ij vf/D; HCl ds lkFk X dh vfHkfØ;k ds ifj.kkeLo:i ,d uhys jax dk ladj Z curkgSA X vkSj Z dk ifjdfyr izpdj.k ek=k pqEcdh; vk?kw.kZ (spin only magnetic moment) 3.87 B.M. gS] tcfd ;g ladjY ds fy, 'kwU; gSA fuEu esa ls dkSulk (ls) fodYi lgh gS (gSa)\

(A) Y esa dsUnzh; /krq vk;u dk ladj.k (hybridization) d2sp3 gS

(B) Y esa flYoj ukbVªsV feykus ij flYoj DyksjkbM ds dsoy nks lerqY; feyrs gSa

(C) tc 0°C ij X vkSj Z lkE;koLFkk esa gSa rks foy;u dk jax xqykch gS

(D) Z ,d prq'iQYdh; (tetrahedral) ladj gS

17



gygygygygy (X) = CoCl2⋅6H

2O, ;k [Co(H

2O)

6]Cl

2 (xqykch)

Co =2+

[Ar]

3d

H2O nqcZy {ks=k yhxs.M gSA vr% bysDVªkWuksa dk ;qXeu ugha gksxk

∴ v;qfXer bysDVªkWuksa dh la[;k n = 3

⇒s

n(n 2)B.M.μ = +

= 15 B.M.= 3.87 B.M.

2 22 6 3(aq) 3 6 2[Co(H O) ] 6NH [Co(NH ) ] 6H O+ ++ +��⇀

↽��

O2 ; [Co(NH

3)6]2+ dks [Co(NH

3)6]3+, esa vkWDlhÑr djsxhA vr% vxz fn'kk esa LFkkukUrfjr gksxk

∴ Y = [Co(NH3)6]3+Cl

3[1 : 3 ladqy]

Co(III) = [Ar]

3d

NH3 çcy {ks=k yhxs.M gS

∴ bysDVªkWuksa dk ;qXeu gksxk

∴ n = 0

μ = 0 B.M.

[Co(NH ) ] = [Ar]3 6

3+

3d 4s 4p

d sp2 3

vkSj, [Co(H O) ] + 4Cl2 6

2+ –[CoCl ] + 6H O; H = +ve4 (aq) 2

2– Δ( )

( )

X

xqykch jax( )Z

uhyk

0°C ij lkE; i'p fn'kk esa LFkkukUrfjr gksxkA vr% xqykch jax

24. HClO4 vkSj HClO ds ckjs esa lgh dFku gS(gSa)

(A) HClO4 vkSj HClO nksuksa esa dsUnzh; ijek.kq sp3 ladfjr gS

(B) Cl2 dh H

2O ds lkFk vfHkfØ;k gksus ij HClO

4 curk gS

(C) HClO4 dk la;qXeh {kkj (conjugate base) H

2O ls nqcZy {kkj gS

(D) ½.kk;u ds vuqukn fLFkjhdj.k (resonance stabilization) ds iQyLo:i HClO4, HClO ls vf/d vEyh; gS


gygygygygy (A) HCIO4 rFkk HClO nksuksa esa dsfUnz; ijek.kq sp3 ladfjr gS

(C) HClO4 > H

2O vEyh; y{k.k

ClO4

– < OH– la;qXeh {kkj

çcy vEyksa ds la;qXeh {kkj nqcZy gksrs gSaA

(D) HClO4 > HClO vEyh; lkeF;Z

+

4 4HClO H + ClO

−⎯⎯→

18


+

HClO H + ClO−⎯⎯→

CI

O–

O

OO

CI

O

O

O–

O

CI

O

OO

O–

CI

O

O–

OO

4ClO

− esa ½.kkos'k pkj vkWDlhtu ij iSQyk gqvk gSA vr% mÙke vuquknh LFkkf;Ro gS

25. lewg 17 ds rRoksa ds X2 v.kqvksa dk jax buds oxZ esa uhps tkus ij ihys jax ls /hjs&/hjs cSaxuh jax esa cnyrk gSA ;g fuEu esa

ls fdlds iQyLo:i gS\

(A) oxZ esa uhps tkus ij π*-o* dk varj ?kVrk gS

(B) oxZ esa uhps tkus ij vk;uu mQtkZ ?kVrh gS

(C) lkekU; rki ij oxZ esa uhps tkus ij X2 dh HkkSfrd voLFkk xSl ls Bksl esa cnyrh gS

(D) oxZ esa uhps tkus ij HOMO-LUMO dk varj ?kVrk gS

mÙkjmÙkjmÙkjmÙkjmÙkj (A, D)

gygygygygy jax çdk'k ds vo'kks"k.k ls bysDVªkWu ds vkè;koLFkk ls mPp voLFkk esa mÙksftr gksus ds dkj.k mRiUu gksrk gSA oxZ esa uhps

pyus ij ÅtkZ Lrj lehi vk tkrs gSa rFkk HOMO-LUMO ds eè; vUrjky de gks tkrk gS

HOMO, π* gS

LUMO, σ* gS

[kaM[kaM[kaM[kaM[kaM 2 (vf/dre vad(vf/dre vad(vf/dre vad(vf/dre vad(vf/dre vad % % % % % 15)))))

• bl [kaM esa ik¡p iz'u gSaA

• izR;sd iz'u dk mÙkj 0 ls 9 rd (nksuksa 'kkfey) ds chp dk ,d ,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad gSA

• izR;sd iz'u ds fy, vks-vkj-,l- ij lgh iw.kkZad ds vuq:i cqycqys dks dkyk djsaA

• izR;sd iz'u ds fy, vad fuEufyf[kr ifjfLFkfRk;ksa esa ls fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%

iw.kZ vad % +3 ;fn fliQZ lgh mÙkj ds vuq:i cqycqys dks dkyk fd;k gSA

'kwU; vad % 0 vU; lHkh ifjfLFkfr;ksa esaA

26. fuEufyf[kr oxZ (species) esa çR;sd dsUnzh; ijek.kq ij ,dkdh bysDVªku ;qXeksa dh la[;k dk ;ksx gS

[TeBr6]2–, [BrF

2]+, SnF

3 rFkk [XeF

3]–

(ijek.kq la[;k % N = 7, F = 9, S = 16, Br = 35, Te = 52, Xe = 54)

mÙkjmÙkjmÙkjmÙkjmÙkj (6)

gygygygygy

Te

Br

Br

Br

F

Br

Br

Br

Br

F

esa ,d ,dkdh ;qXe gS

esa nks ,dkdh ;qXe gS

19


Xe

S N≡

F

F

F

F

F

esa rhu ,dkdh ;qXe gS

esa ,dkdh ;qXe ugha gSF

27. fuEufyf[kr esa ls ,jksesfVd ;ksfxd (;ksfxdksa) dh la[;k gS

++

+


gygygygygy +

+

, ,, rFkk ,jksesfVd gS

rFkk vu,jksesfVd gSa tcfd rFkk + çfr,jksesfVd ;kSfxd gaS

28. H2, He

2, Li

2, Be

2, B

2, C

2, N

2, –

2O vkSj F

2 esa çfrpqEcdh; Lih'kht (diamagnetic species) dh la[;k gS

(ijek.kq la[;k % H = 1, He = 2, Li = 3, Be = 4, B = 5, C = 6, N = 7, O = 8, F = 9)


gygygygygy H2 2

1σ

sçfrpqEcdh;

2

2 * 1

1 1He

s s

+ σ σ vuqpqEcdh;

2 * 2 2

2 1 1 2Li

s s sσ σ σ çfrpqEcdh;

2 * 2 2 * 2

2 1 1 2 2Be

s s s sσ σ σ σ çfrpqEcdh;

2 * 2 2 * 2 1 1

2 1 1 2 2 2 2B

x ys s s s p p

σ σ σ σ π = π vuqpqEcdh;

2 * 2 2 * 2 2 2

2 1 1 2 2 2 2C

x ys s s s p p

σ σ σ σ π = π çfrpqEcdh;

2 * 2 2 * 2 2 2 2

2 1 1 2 2 2 2 2N

x y zs s s s p p p

σ σ σ σ π = π σ çfrpqEcdh;

2 * 2 2 * 2 2 2 2 * 1 * 1

2 1 1 2 2 2 2 2 2 2O

z x y x ys s s s p p p p p

σ σ σ σ σ π = π π = π vuqpqEcdh;

2 * 2 2 * 2 2 2 2 * 2 * 2

2 1 2 2 2 21 2 2 2F ,

z x y x ys s p p ps s p p

σ σ σ σ σ π = π π = π çfrpqEcdh;

20


29. ,d 'kq¼ inkFkZ ds ,d fØLVyh; Bksl dh iQyd&dsfUnzr ?ku (face-centred cubic) lajpuk ds lkFk dksfLBdk dksj (cell

edge) dh yEckbZ 400 pm gSA ;fn fØLVy ds inkFkZ dk ?kuRo 8 g cm–3 gS] rks fØLVy ds 256 g esa mifLFkr ijek.kqvksa dhdqy la[;k N × 1024 gSA N dk eku gS


gygygygygy a = 4 × 10–8 cm (a = dksj yEckbZ)

d = 8 g cm–3 (?kuRo)

3

A

ZMd

N a= M = vk.kfod nzO;eku (g/mol)

Z → 1 ,dd dksf"Bdk esa ijek.kqvksa dh la[;k3 23 –24

AdN a 8 6 10 64 10

M 76.8 g/molZ 4

× × × ×= = =

256 g esa Bksl ds eksy = 3.33 eksyijek.kqvksa dh la[;k = 3.33 × N

A = 20 × 1023 = 2 × 1024

30. ,d nqcZy ,d{kkjdh; vEy ds 0.0015 M tyh; foy;u dh pkydRo (conductance) ,d IykfVfuÑr Pt (platinized Pt)

bysDVªksM okys pkydrk lSy dk mi;ksx dj ds fu/kZfjr dh x;hA 1 cm2 vuqçLFk dkV ds {ks=kiQy okys bysDVªksM+ks ds chp dh

nwjh 120 cm gSA bl foy;u dh pkydRo dk eku 5 × 10–7 S ik;k x;kA foy;u dk pH 4 gSA bl nqcZy ,d{kkjdh; vEy

dh tyh; foy;u esa lhekUr eksyj pkydrk (limiting molar conductivity) ( )o

mΛ dk eku Z × 102 S cm–1 mol–1 gSA Z

dk eku gS


gygygygygyl

CA

⎛ ⎞κ = ⎜ ⎟⎝ ⎠

7 1205 10

1

−= × ×

= 6 × 10–5 S cm–1

M

1000

M

κ ×λ =

5

4

6 10 1000

15 10

−

−× ×=

×

⇒ λM

= 40 S cm2 mol–1.

[H+] = Cα = 10–4

4

4

10 1

1515 10

−

−α = =×

⇒ M

º

M

λα =

λ

⇒M

°λ = 40 × 15 = 600 = 6 × 102 S cm–1 mol–1

21



• bl [kaM esa lqesy izdkj ds NgNgNgNgNg iz'u gSaA

• bl [kaM esa nks Vscy gSa (izR;sd Vscy esa 3 dkye vkSj 4 iafDr;k¡ gSa)A

• izR;sd Vscy ij vkèkkfjr rhurhurhurhurhu iz'u gSaA

• çR;sd iz'u ds pkjpkjpkjpkjpkj mÙkj fodYi (A), (B), (C) vkSj (D) gSa ftuesa fliZQ ,d ,d ,d ,d ,d fodYi lgh gSA

• izR;sd iz'u ds fy, vks-vkj-,l- ij lgh mÙkj ds vuq:i cqycqys dks dkyk djsaA

• izR;sd iz'u ds fy, vad fuEufyf[kr ifjfLFkfr;ksa esa ls fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %

iw.kZ vad : +3 ;fn lgh fodYi ds vuq:i cqycqys dks dkyk fd;k gSA

'kwU; vad : 0 ;fn fdlh cqycqys dks dkyk ugha fd;k gSA


uhps nh x;h Vscy ds rhu dk¡yeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa vkSj ds mÙkj nhft;sA

rjax iQyu] ,d xf.krh; iQyu gS ftldk eku bysDVªkWu ds xksyh; /zqoh; funsZ'kkad ij fuHkZj djrk gS vkSj DokaVe la[;k

vkSj ls vfHkyf{kr gksrk gSA ;gk¡ uwfDyvl ls nwjh gS] dksfV'kj gS] vkSj fnUx'k gSA Vscy esa fn, x;s xf.krh; iQyuksa esa ijek.kq Øekad gS vkSj cksj f=kT;k gSA

vkfcZVy

Q.31, Q32

ψl m r (colatitude) (azimuth)

z a (Bohr radius)

(I) 1s (orbital) (i) (P)

(II) 2s (orbital) (ii) (radial) (Q)

(Probability density)

(III) 2p (orbital) (iii) (R)

(Probability density)

(IV) 3d (orbital) (iv) xy- (S) n = 2 n = 4

n = 2

n = 6

1

0

z

z

θ φ

dkye dkye dkye 3I 2

vkfcZVy ,d f=kT;kRed uksM uwfDyvl ij izkf;drk ?kuRo

vkfcZVy uwfDyvl ij izkf;drk ?kuRo

vf/dre gS

vkfcZVy lery ,d uksMh; ry gS bysDVªksu dks voLFkk ls voLFkk rd

mÙksftr djus dh mQtkZ] bysDVªksu dks

voLFkk ls voLFkk rd mÙksftr djus ds

fy, vko';d mQtkZ ls xquk gS

2

Q.33

(r, , ) n, n,j,m θ φ

0

3

2

, ,0

zr

an j m

Ze

a

⎛ ⎞− ⎜ ⎟⎝ ⎠⎛ ⎞ψ ∝ ⎜ ⎟

⎝ ⎠

ψn,l,m

1(r)

a r/a0

0

30

1

a∝

0

5 zr

2 an,j,m

0

Ze cos

a

⎛ ⎞− ⎜ ⎟⎝ ⎠⎛ ⎞ψ ∝ θ⎜ ⎟⎝ ⎠

27

32

31. He+ vk;u ds fy, fuEufyf[kr fodYiksa esa ls dsoy xyrxyrxyrxyrxyr (INCORRECT) la;kstu gS

(A) (I) (i) (S) (B) (II) (ii) (Q)

(C) (I) (iii) (R) (D) (I) (i) (R)

mÙkjmÙkjmÙkjmÙkjmÙkj (C)

gygygygygy 1s d{kd vfn'kkRed gS vr ψ cosθ ij fuHkZj ugha djsxh

vr% C xyr gS

22


32. dkye&1 esa fn, x;s vkfcZVy (Orbital) ds fy, fuEufyf[kr fodYiksa esa ls fdlh Hkh gkbMªkstu&leku Lih'kht (species) ds

fy, dsoy lghlghlghlghlgh la;kstu gS

(A) (II) (ii) (P) (B) (I) (ii) (S)

(C) (IV) (iv) (R) (D) (III) (iii) (P)

mÙkjmÙkjmÙkjmÙkjmÙkj (A)

gygygygygy H- ds leku Lih'kht ds fy, dsoy A lgh gS

D;kasfd

(B) esa] 1s d{kd esa f=kT;h; uksM ugha gS

(C) esa] 2

z3d , ds fy, xy ry uksMh; ry ugha gS

(D) esa] 2pz d{kd esa f=kT;h; uksM ugha gS

33. gkbMªkstu ijek.kq ds fy, fuEufyf[kr fodYiksa esa ls dsoy lghlghlghlghlgh la;kstu gS

(A) (I) (i) (P) (B) (I) (iv) (R)

(C) (II) (i) (Q) (D) (I) (i) (S)

mÙkjmÙkjmÙkjmÙkjmÙkj (D)

gygygygygy H-ijek.kq ds fy,

1s-d{kd 0

3 Zr

2 a

0

Ze

a

⎛ ⎞−⎜ ⎟⎝ ⎠

⎛ ⎞ψ ∝⎜ ⎟

⎝ ⎠

vkSj, 4 2

3E E

16− =

6 2

2E E

9− =

vr% 6 2 4 2

27(E E ) E E

32− × = −

uhps nh x;h Vscy ds rhu dk¡yeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dk¡yeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dk¡yeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dk¡yeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dk¡yeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa 34, 35 ,oa ,oa ,oa ,oa ,oa 36 ds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sA

dkye 1, 2 3 vkSj esa Øe'k% vkjfEHkd inkFkZ] vfHkfØ;k voLFkk,a] vkSj vfHkfØ;kvksa ds izdkj gSaA

(I) (Toluene)VkyqbZu

(II) (Acetophenone)vflVksiQsuksau

(III) (Benzaldehyde)csfUtYMgkbM

(IV) (Phenol)isQuksy

(i) NaOH/Br2

(ii) Br /hv2

(iii) (CH CO) O/CH COOK3 2 3

(iv) NaOH/CO2

(P) (Condensation)la?kuu

(Q) (Carboxylation)dkcksZfDLydj.k

(R) (Substitution)izfrLFkkiu

(S) (Haloform)gkyksiQeZ

23


34. fuEufyf[kr fodYiksa esa ls dsoy lghlghlghlghlgh la;kstu ftlesa vfHkfØ;k ewyd (Radical) izfØ;k }kjk c<+rh gS] gS

(A) (IV) (i) (Q) (B) (III) (ii) (P)

(C) (II) (iii) (R) (D) (I) (ii) (R)


gygygygygy

CH3

CH2

CH2

CH – Br2

+ Br

+ Br2

+ HBr

+ Br

Br2

Br + Brhν

35. csUtksbZd vEy ds la'ys"k.k (synthesis) ds fy, fuEufyf[kr fodYiksa esa ls dsoy lghlghlghlghlgh la;kstu gS

(A) (II) (i) (S) (B) (I) (iv) (Q)

(C) (IV) (ii) (P) (D) (III) (iv) (R)


gygygygygy C CH3

O

NaOH

Br2

COO Na– +

+ CHBr3

H O/H2

+

COOH

;g gSyksiQkWeZ vfHkfØ;k gS

36. fuEufyf[kr fodYiksa esa ls dsoy lghlghlghlghlgh la;kstu tks fd nks fHkUu dkcksZfDlfyd vEy nsrk gS] gS

(A) (IV) (iii) (Q) (B) (II) (iv) (R)

(C) (I) (i) (S) (D) (III) (iii) (P)


gygygygygy

CH = O

CH – C3

CH – C3

O

O

O

CH COOK3

CH = CH – COOH

flusfed vEy

;g nks T;kferh; :iksa esa ik;k tkrk gS

C –H

H – C – COOH

C –H

HOC – C – H

O

lei{k leko;ofoi{k leko;o

;g i£du vfHkfØ;k dk ewy mnkgj.k gS

24


MATHEMATICS


bl [kaM esa lkrlkrlkrlkrlkr iz'u gSaA

çR;sd iz'u ds pkjpkjpkjpkjpkj mÙkj fodYi (A), (B), (C) vkSj (D) gSa ftuesa ,d ;k ,d ls vf/d,d ;k ,d ls vf/d,d ;k ,d ls vf/d,d ;k ,d ls vf/d,d ;k ,d ls vf/d fodYi lgh gSaA

izR;sd iz'u ds fy, vks-vkj-,l- ij lkjs lgh mÙkj (mÙkjksa) ds vuq:i cqycqys (cqycqyksa) dks dkyk djsaA

izR;sd iz'u ds fy, vad fuEufyf[kr ifjfLFkfr;ksa esa ls fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %fdlh ,d ds vuqlkj fn;s tk,axs %

iw.kZ vad : +4 ;fn fliQZ lkjs lgh fodYi (fodYiksa) ds vuq:i cqycqys (cqycqyksa) dks dkyk fd;k gSA

vkaf'kd vad : +1 izR;sd lgh fodYilgh fodYilgh fodYilgh fodYilgh fodYi ds vuq:i cqycqys dks dkyk djus ij] ;fn dksbZ xyr fodYi dkykugha fd;k gSA

'kwU; vad : 0 ;fn fdlh cqycqys dks dkyk ughaughaughaughaugha fd;k gSA


mnkgj.k % ;fn ,d iz'u ds lkjs lgh mÙkj fodYi (A), (C) vkSj (D) gSa] rc bu rhuksa ds vuq:i cqycqyksa dks dkyk djus ij+4 vad feysaxs_ fliQZ (A), (D) ds vuq:i cqycqyksa dks dkyk djus ij +2 vad feysaxsa_ rFkk (A) vkSj (B) ds vuq:i cqycqyksadks dkyk djus ij –2 vad feysaxs D;ksafd ,d xyr fodYi ds vuq:i cqycqys dks Hkh dkyk fd;k x;k gSA

37. ekuk fd X vkSj Y bl izdkj dh nks ?kVuk;sa (events) gSa fd P(X) = 1 1, |

3 2P X Y vkSj 2

|5

P Y X gSA rc

(A)4

( )15

P Y (B) 1|

2P X Y

(C)2

( )5

P X Y ∪ (D)1

( )5

P X Y ∩

mÙkjmÙkjmÙkjmÙkjmÙkj (A, B)

gygygygygy1 1 2

( ) , ,3 2 5

X YP X P P

Y X

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( ) 1

( ) 2

P X YXP

P YY

⎛ ⎞ ⎜ ⎟⎝ ⎠

( ) 2

( ) 5

P X YYP

P XX

⎛ ⎞ ⎜ ⎟⎝ ⎠

4 1 2( ) , ( ) , ( )

15 3 15P Y P X P X Y⇒

( ) ( ) ( )

( ) ( )

P X Y P Y P X YXP

P Y P YY

⎛ ⎞ ⎜ ⎟⎝ ⎠

4 2

115 15

4 2

15

P(X Y) = 1 4 2

3 15 15

9 2 7

15 15

25


38. ekuk fd : (0,1)f � ,d lrr iQyu (continuous function) gSA rc fuEu iQyuksa esa ls dkSuls iQyu(uksa) dk(ds) eku

vUrjky (interval) (0, 1) ds fdlh fcUnq ij 'kwU; gksxk\

(A)0

( )sinx

x

e f t t dt ∫ (B) 2

0

( ) ( )sinf x f t t dt

∫

(C) 2

0

( )cosx

x f t t dt

∫ (D) x9 – f(x)

mÙkjmÙkjmÙkjmÙkjmÙkj (C, D)

gygygygygy 2

0

( ) ( )cosx

g x x f t t dt

∫

2

0

(0) 0 ( )cos 0g f t t dt

∫ pw¡fd 0 ( ) cos < 1f t t

12

0

(1) 1 ( )cos 0g f t t dt

∫

9( ) ( ) (0) 0 (0) 0g x x f x g f

(1) 1 (1) 0g f

rFkk (0, 1) ds fy,

0( )sin 0

xx

e f t t dt ∫

rFkk

/2

0( ) ( )sin 0f x f t t dt

∫ (0,1)x

39. ekuk fd a, b, x vkSj y bl izdkj dh okLrfod la[;k;sa (real numbers) gSa fd a – b = 1 vkSj y 0 gSaA ;fn lfEeJ la[;k

(complex number) z = x + iy, Im1

az by

z

⎛ ⎞ ⎜ ⎟⎝ ⎠ dks larq"V djrh gS] rc fuEu esa ls dkSulk(ls) x dk(ds) lEHkkfor eku

gS(gSa)?

(A) 21 1 y (B) 2

1 1 y

(C) 21 1 y (D) 2

1 1 y


gygygygygy

a – b 1, y 0, z = x + iy

Im1

az by

z

⎛ ⎞ ⎜ ⎟⎝ ⎠

( ) 1lm

( 1) 1

ax b ayi x iyy

x iy x iy

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠

2

2 2

( )( 1) ( 1) ( )lm

( 1)

ax b x ay ay x i iy ax by

x y

⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠

26


ay(x + 1) – y(ax + b) = y(x + 1)2 + y3

ax + a – ax – b = x2 + 2x + 1 + y2

a – b = x2 + y2 + 2x + 1

1 = x2 + y2 + 2x + 1

(x + 1)2 = 1 – y2

21 1x y

21 1x y

40. ;fn 2x – y + 1 = 0 vfrijyo; (hyperbola) 2 2

21

16

x y

a

dh Li'kZjs[kk (tangent) gS rks fuEu esa ls dkSu lh ledks.kh; f=kHkqt

(right angled triangle) dh Hkqtk;sa ugha gks ldrh gS(gSa)\

(A) a, 4, 1 (B) 2a, 4, 1

(C) a, 4, 2 (D) 2a, 8, 1


gygygygygy y = 2x + 1, 2 2

21

16

x y

a

dh Li'kZ js[kk gS

y = 2x + 1 dh 2 216y mx a m ls rqyuk djus ij

m = 2 rFkk a2m2 – 16 = 1 4a2 = 17

2a, 4, 1 ledks.kh; f=kHkqt dh Hkqtk,¡ gSa

41. ekuk fd x ls NksVk ;k x ds leku lcls cM+k iw.kk±d (integer) [x] gSA rc f(x) = xcos((x + [x])) fuEu esa ls fdu fcUnq(vksa)ij vlrr (discontinuous) gS\

(A) x = –1 (B) x = 1

(C) x = 0 (D) x = 2

mÙkjmÙkjmÙkjmÙkjmÙkj (A, B, D)

gygygygygy f(x) = xcos((x + [x]))

cos ,[ ]

cos , [ ]

x x x

x x x

⎧ ⎪ ⎨ ⎪⎩

le gS

fo"ke gS

Li"Vr% f(1+) f(1), f(2+) f(2), f(–1+) f(–1–)

ijUrq f(0) = f(0+) = f(0) = 0 vr% f, x = 1, –1, 2 ij vlrr~ gS ijUrq x = 0 ij lrr~ gS

42. fuEu esa ls dkSu lk(ls) okLrfod la[;kvksa ds 3 × 3 vkO;wg (matrix) dk oxZ (square) ugha gS(gSa)?

(A)

1 0 0

0 1 0

0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(B)

1 0 0

0 1 0

0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(C)

1 0 0

0 1 0

0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(D)

1 0 0

0 1 0

0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

mÙkjmÙkjmÙkjmÙkjmÙkj (A, C)

27


gygygygygy pw¡fd fodYi (B) o (D) esa vkO;wg ds lkjf.kd /ukRed gSa

vr% ;g vkO;wg ds oxZ ds :i esa fu:fir fd, tk ldrs gSa ijUrq fodYi (A) o (C) esa vkO;wg ds lkjf.kd eku ½.kkRed gSa]vr% bls vkO;wg ds oxZ ds :i esa fu:fir ugha fd;k tk ldrkA

43. ;fn ijoy; (parabola) y2 = 16x dh ,d thok (chord), tks Li'kZjs[kk (tangent) ugha gS] dk lehdj.k 2x + y = p rFkk eè;fcUnq(midpoint) (h, k) gS] rks fuEu esa ls p, h ,oe~ k ds lEHkkfor eku gS(gSa)\

(A) p = –1, h = 1, k = –3

(B) p = 2, h = 3, k = –4

(C) p = –2, h = 2, k = –4

(D) p = 5, h = 4, k = –3

mÙkjmÙkjmÙkjmÙkjmÙkj (B)

gygygygygy thok dh lehdj.k 2x + y = p gS

ijoy; ds lkFk izfrPNsnh fcanq ds fy,

2( 2 ) 16p x x

2 24 (4 16) 0x p x p⇒

p = 128p + 256

vr% p = –2

eè; fcanq (h, k) ds fy, thok dk lehdj.k

28( ) 16yk x h k h

28 8yk x k h⇒

–8x + ky = k2 – 8h

rqyuk djus ij]

2

2

8 8

2 1

x y p

k k h

p

28 4

16 8 4

2 4

k h p

h p

p h

⇒

k = –4, p = 2, h = 3

[kaM[kaM[kaM[kaM[kaM 2 (vf/dre vad(vf/dre vad(vf/dre vad(vf/dre vad(vf/dre vad % % % % % 15)))))

• bl [kaM esa ik¡p iz'u gSaA

• izR;sd iz'u dk mÙkj 0 ls 9 rd (nksuksa 'kkfey) ds chp dk ,d ,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad,dy vadh; iw.kkZ ad gSA

• izR;sd iz'u ds fy, vks-vkj-,l- ij lgh iw.kkZad ds vuq:i cqycqys dks dkyk djsaA

• izR;sd iz'u ds fy, vad fuEufyf[kr ifjfLFkfRk;ksa esa ls fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%fdlh ,d ds vuqlkj fn;s tk,axs%

iw.kZ vad % +3 ;fn fliQZ lgh mÙkj ds vuq:i cqycqys dks dkyk fd;k gSA

'kwU; vad % 0 vU; lHkh ifjfLFkfr;ksa esaA

28


44. okLrfod la[;k (real number) α ds fy;s] ;fn jSf[kd lehdj.k fudk; (system of linear equations)

x

y

z

2

2

1 1

1 –1

11

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦

ds vuUr gy (infinitely many solutions) gSa] rc 1 + α + α2 =


lehdj.k dks AX = B ds :i esa iqu% fy[kk tkrk gS

tgk¡

2

2

1 1

, ,1 1

11

x

A X By

z

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦

adj( ).

det( )

A BX

A

vifjfer gy ds fy,] det(A) = 0 rFkk adj(A).B = 0

= 1, –1

ijUrq , 1 ds cjkcj ugha gS D;ksafd bl fLFkfr esa lehdj.k vlaxr gSA

blfy,] = –1 i.e., 21 1

45. ,d ledks.kh; f=kHkqt (right angled triangle) dh Hkqtk;sa lekUrj Js<h (arithmetic progression) esa gSaA ;fn bldk {ks=kiQy24 gS rc bldh lcls NksVh Hkqtk dh yEckbZ D;k gS\


gygygygygy ()2 = 2 + (–)2 dk iz;ksx djus ij

–

+

4

rFkk] 1

( ) 242

8

2

⎫⎬ ⎭

lcls NksVh Hkqtk = 6

46. ekuk fd f : � � bl izdkj dk vodyuh; iQyu (differentiable function) gS fd f(0) = 0, f 32

⎛ ⎞ ⎜ ⎟⎝ ⎠

,oe~ f ' (0) = 1

gSaA ;fn x 0,2

⎛ ⎤ ⎜ ⎥⎝ ⎦

ds fy;s x

g x f t t t t f t dt

2

( ) [ '( )cosec – cot cosec ( )]

∫ gS] rc x

g x0

lim ( )


gygygygygy/2

( ) ((cosec )( ( )))

x

g x d t f t

∫ /2{(cosec ). ( )}x

t f t

blfy,] 0 0

lim ( ) lim ( ).cosec2x x

g x f f x x

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ 0

( )3 lim

sinx

f x

x

0

( )3 lim 2

cosx

f x

x

29


47. p ds fdrus ekuksa ds fy;s o`Ùk (circle) x2 + y2 + 2x + 4y – p = 0 ,oe~ funsZ'kkad v{kksa (coordinate axes) esa dsoy rhu

fcUnq mHk;fu"B (common) gSa\


gygygygygy laHko fLFkfr;k¡

p = –1 p = 0

48. v{kjksa A, B, C, D, E, F, G, H, I, J ls 10 yEckbZ ds 'kCn cuk;s tkrs gSaA ekuk fd x bl rjg ds mu 'kCnksa dh la[;k gS

ftuesa fdlh Hkh v{kj dh iqujko`fr ugha gksrh gS] rFkk y bl rjg ds mu 'kCnksa dh la[;k gS ftu esa dsoy ,d v{kj dh

iqujkofr nks ckj gksrh gS o fdlh vU; v{kj dh iqujkofr ugha gksrh gSA rc y

x9


gygygygygy Li"Vr% x = 10!

y dh x.kuk djus ds fy, ,d v{kj lfEefyr ugha djuk gS] ,slk 10 rjhdksa ls gks ldrk gSA

'ks"k 9 v{kjksa esa ls ,d dh iqujkofÙk gksxh

10 9

1 1

10! 5 9 10!

2!y C C

5 9 10!

59 9 10!

y

x


bl [kaM esa lqesy izdkj ds NgNgNgNgNg iz'u gSaA

bl [kaM esa nks Vscy gSa (izR;sd Vscy esa 3 dkye vkSj 4 iafDr;k¡ gSa)A

izR;sd Vscy ij vkèkkfjr rhurhurhurhurhu iz'u gSaA

çR;sd iz'u ds pkjpkjpkjpkjpkj mÙkj fodYi (A), (B), (C) vkSj (D) gSa ftuesa fliZQ ,d ,d ,d ,d ,d fodYi lgh gSA

izR;sd iz'u ds fy, vks-vkj-,l- ij lgh mÙkj ds vuq:i cqycqys dks dkyk djsaA

izR;sd iz'u ds fy, vad fuEufyf[kr ifjfLFkfr;ksa esa ls fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %fdlh ,d ds vuqlkj fn;s tk,saxs %

iw.kZ vad : +3 ;fn lgh fodYi ds vuq:i cqycqys dks dkyk fd;k gSA

'kwU; vad : 0 ;fn fdlh cqycqys dks dkyk ugha fd;k gSA


30


uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa 49, 50 ,oa ,oa ,oa ,oa ,oa 51 ds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sA

dkWye 1, 2 3 (conic), (tangent)

(point of contact)

rFkk esa Øe'k% dkWfud dkWfud ij Li'kZjs[kk dk lehdj.k rFkk Li'kZfcUnq fn;s x;s gSa

(I) + = x y a2 2 2

(II) + = x y a2 2 2

a2

(III) = 4y ax2

(IV) x a y a2 2 2 2

– =

(i)

(ii)

(iii)

(iv)

2my m x a

2 1y mx a m

2 2 – 1y mx a m

2 2 1y mx a m

(P)

(Q)

(R)

(S)

2

2,

a a

mm

⎛ ⎞⎜ ⎟⎝ ⎠

2 2,

1 1

ma a

m m

⎛ ⎞⎜ ⎟

⎝ ⎠

2

2 2 2 2

1,

1 1

a m

a m a m

⎛ ⎞⎜ ⎟⎜ ⎟ ⎝ ⎠

2

2 2 2 2

–1,

– 1 – 1

a m

a m a m

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

49. 2a ds fy, mi;qDr dkWfud (dkWye 1) ij ,d Li'kZjs[kk [khph tkrh gS ftldk Li'kZfcUnq (–1, 1), rc fuEu esa ls dkSulk

fodYi bl Li'kZ js[kk dk lehdj.k izkIr djus dk dsoy lgh la;kstu gS\

(A) (I) (ii) (Q) (B) (I) (i) (P)

(D) (II) (ii) (Q) (C) (III) (i) (P)


50. ;fn mi;qDr dkWfud (dkWye 1) ds fcUnq1

3,2

⎛ ⎞⎜ ⎟⎝ ⎠

ij Li'kZjs[kk 3 2 4,x y gS] rc fuEu esa ls dkSulk fodYi dsoy lgh

la;kstu gS\

(A) (IV) (iv) (S) (B) (II) (iv) (R)

(C) (IV) (iii) (S) (D) (II) (iii) (R)

mÙkjmÙkjmÙkjmÙkjmÙkj (B)

51. ;fn mi;qDr dkWfud (dkWye 1) ds Li'kZfcUnq (8, 16) ij Li'kZjs[kk 8y x gS] rc fuEu esa ls dkSulk fodYi dsoy lgh

la;kstu gS\

(B) (I) (ii) (Q) (A) (III) (i) (P)

(C) (II) (iv) (R) (D) (III) (ii) (Q)


iz- la-iz- la-iz- la-iz- la-iz- la- 49 lslslslsls 51 ds fy, gyds fy, gyds fy, gyds fy, gyds fy, gy

lgh la;kstu fuEu gS

I, ii, Q

II, iv, R

III, i, P

IV, iii, S

49. (–1, 1), 2a ds fy, x2 + y2 = a2 ij fLFkr gS] blfy, lgh mÙkj dsoy (A) gSA

31


50. fn, x, fodYiksa esa ls A, B lgh la;kstu gSa ftudh tk¡p vko';d gSA blesa ls (ftudh tk¡p de djuh gS)

x2 + a2y2 = a2,

2

2 2 2 2

31 14, 12 , 21 12 2

a m

a m a m

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

tks fd fn;k x;k gS] blfy, (B) lgh gSA

51. pw¡fd y = x + 8 rFkk bldk Li'kZ fcanq (8, 16) dsoy y2 = 4ax dks larq"V djrk gS

uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkWyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa 52, 53 ,oa ,oa ,oa ,oa ,oa 54 ds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sA

(P) (0, 1) f o/Zeku gS(i) lim ( ) 0x

f x

(ii) lim ( )x

f x

(iii) lim ( )x

f x

(iv) lim ( ) 0x

f x

(Q) ( , ) f e e2 esa ßkleku gS

(R) (0, 1) f esa o/Zeku gS

(S) f esa ßkleku gS( , ) e e2

(I) ( ) = 0 (1, ) f x x e 2 ds fy, fdlh

(II) ( ) = 0 (1, ) f x x e ds fy, fdlh

(III) = 0 (0, 1) x ds fy, f x( ) fdlh

(IV) = 0 (1, e) x ds fy, f x( ) fdlh

1

2 (limiting behaviourat infinity)

3 (increasing/decreasing) (nature)

ekuk fd gSaA

dkWye esa ,oe~ ds 'kwU;ksa dh lwpuk nh xbZ gSaA

dkWye esa ,oe~ ds vuUr dh rjiQ lhek ij O;ogkj dh lwpuk nh xbZ gSA

dkWye esa ,oe~ ds o/Zeku@ßkleku gksus dh izÑfr dh lwpuk nh xbZ gSA

f x x x x( ) = + log , (0, ) e

f x f x f x

f x f x f x

f x f x

( ), ( ) ( )

( ), ( ) ( )

( ) ( )

52. fuEu esa ls dkSulk fodYi dsoy xyr xyr xyr xyr xyr la;kstu (only INCORRECT combination) gS\

(A) (I) (iii) (P) (B) (II) (iv) (Q)

(C) (II) (iii) (P) (D) (III) (i) (R)


53. fuEu esa ls dkSulk fodYi dsoy lgh la;kstu gS\

(A) (I) (ii) (R) (B) (III) (iv) (P)

(C) (II) (iii) (S) (D) (IV) (i) (S)


54. fuEu esa ls dkSulk fodYi dsoy lgh la;kstu gS\

(A) (III) (iii) (R) (B) (IV) (iv) (S)

(C) (II) (ii) (Q) (D) (I) (i) (P)


iz- la-iz- la-iz- la-iz- la-iz- la- 52 lslslslsls 54 ds fy, gyds fy, gyds fy, gyds fy, gyds fy, gy

( ) log loge e

f x x x x x

(I) (1) 1 log1 1log1 1 0f

2 2 2 2 2 2 2 2( ) log log 2 2 2 0e e

f e e e e e e e e

(I) lR; gSA

32


(II)1

( ) 1 log 1e

f x xx

(1) 1f

1( ) 1 0f e

e

(II) lR; gSA

(III)2

1 1( ) 0f x

xx

lHkh (0, )x ds fy,

blfy,] ( )f x ßkleku gS

blfy, (0, 1) esa ( )f x dk U;wure eku (1) 1f gS

blfy, (III) vlR; gSA

(IV) (1) 2 0f

2

1 1( ) 0f e

ee

blfy, (IV) vlR; gSA

dkWyedkWyedkWyedkWyedkWye 2

0

log1lim ( ) lim log log lim log e

e e ex x t

t

f x x x x x tt t

0

1 log log

lim0

e e

t

t t t

t

(i) vlR; gS

(ii) lR; gS

(iii) 0

1lim ( ) lim log lim log

e ex x t

f x x t tx

(lR;)

(iv) 2 2

1 1 1lim ( ) lim 0x x

xf x

xx x

(lR;)

dkWyedkWyedkWyedkWyedkWye 3

(P)1

( ) log 0e

f x xx

lR; gS

(0, 1)x ds fy,] 1

(1, )x

log ( , 0)e

x

(Q)1

( ) loge

f x xx

ßkleku iQyu gS

1( ) 1 0f e

e

2

2

1( ) 2 0f e

e

lR; gSA

(R)2

1 1( ) 0f x

x x

blfy, ( )f x ßkleku gS

blfy,] (R) vlR; gSA

(S) lR; gSA

iz'u i=k lekIriz'u i=k lekIriz'u i=k lekIriz'u i=k lekIriz'u i=k lekIr

Documents

Time : 3 hrs. Max. Marks: 183 Answers & Solutions...(B) dNq l e; d sckn ckã cy F vkjS ifzrjksèkd cy l rafqyr gk t k,xas (C) IysV }kjk vuHqko gvk ifzrjkèkd cy v d sl ekuikrh gS (D)