answers & solutions · izr;sd iz'u ds fy, vad fueufyf[kr ifjflfkfr;ksa esa ls fdlh ,d ds...

1

CODE

3

Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005

Ph.: 011-47623456 Fax : 011-47623472

DATE : 21/05/2017

PAPER - 1 (Code - 3)

Answers & Solutions

forforforforfor

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;efgUnh ekè;efgUnh ekè;efgUnh ekè;efgUnh ekè;e

2

JEE (ADVANCED)-2017 (PAPER-1) CODE-3

PHYSICS

[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 fodYi,d ;k ,d ls vf/d fodYi,d ;k ,d ls vf/d fodYi,d ;k ,d ls vf/d fodYi,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;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 : +4 ;fn fliQZ lkjs 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 dkyk

ugha 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 dkys 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 cqycqyksa

dks dkyk djus ij –2 vad feysaxs D;ksafd ,d xyr fodYi ds vuq:i cqycqys dks Hkh dkyk fd;k x;k gSA

1. ,d likV IysV (flat plate) vYi ncko ds xSl (gas at low pressure) esa] vius ry dh vfHkyac fn'kk esa] ckã cy F ds

izHkko esa vxzlfjr gSA IysV dh xfr ,xSl v.kqvksa ds vkSlr xfr u ls cgqr de gSA fuEu esa ls dkSulk (ls) dFku lgh gS@gSa?

(A) IysV kjk vuqHko gqvk izfrjksèkd cy ds lekuqikrh gS

(B) dqN le; ds ckn ckã cy F vkSj izfrjksèkd cy larqfyr gks tk,axs

(C) IysV loZnk 'kqU;srj fLFkj Roj.k (constant non-zero acceleration) ls pyrh jgsxh

(D) izfrxkeh ,oa vuqxkeh i"B ds ncko dk varj uvds lekuqikrh gS

mÙkjmÙkjmÙkjmÙkjmÙkj (A, B, D)

gygygygygy n = izfr bdkbZ vk;ru v.kqvksa dh la[;k

u = xSl v.kqvksa dh vkSlr pky

tc IysV v pky ls xfr'khy gS] rc eq[; cy ds lkisk v.kqvksa dh lkisfkd pky = v + u

lEeq[k VDdj esa izfr VDdj IysV dks LFkkukUrfjr laosx = 2m(u + v)

le; t esa VDdj dh la[;k 1( )

2u v n tA

tgk¡ A = i"Bh; ks=kiQy

blfy, vkxs dh lrg ls le; t esa LFkkukUrfjr laosx = m(u + v)2nAt

ihNs dh lrg ls le; t esa LFkkukUrfjr laosx = m(u – v)2nAt

usV cy = mnA[(v + u)2 – (u – v)2]

= mnA[4vu]

F v

Pvxz

– Pi'p

= mn[u + v]2 – mn[u – v]2

= mn[4uv] = 4mnuv

P uv

3


2. ekuoh; i"Bh; ks=kiQy yxHkx 1 m2 gksrk gSA ekuo 'kjhj dk rkieku ifjos'k ds rkieku ls 10 K vfèkd gksrk gSA ifjos'k

rkieku T0 = 300 K gS] bl ifjos'k rkieku ds fy,

4 2

0460WmT

gSA tgk¡ LVhiQku&cksYV~teku fu;rkad (Stefan-

Boltzmann constant) gSA fuEu esa ls dkSu lk (ls) dFku lgh gS@gSa\

(A) ifjos'k rkieku vxj T0 ls ?kVrk gS (T

0 << T

0) rc ekuo ds 'kjhj dks rkieku dk vuqjk.k djus ds fy,

3

0 04W T T vfèkd ÅtkZ fofdfjr djuh iM+rh gS

(B) ekuoh; 'kjhj ds rkieku esa vxj lkFkZd of¼ gks rc izdk'k pqEcdh; fodj.k LiSDVªe dh f'k[kj rjax&nSè;Z (peak in the

electromagnetic spectrum) nh?kZ rjax&nSè;Z dh vksj foLFkkfir gksrh gS

(C) i"Bh; ks=kiQy ?kVkus (tSls% fldqM+us ls) ls ekuo vius 'kjhj ls fofdfjr ÅtkZ ?kVkrs gSa ,oa vius 'kjhj dk rkieku vuqjfkr

djrs gSa

(D) ekuoh; 'kjhj ls 1 lsadM esa fudVre fofdfjr ÅtkZ 60 twy (60 Joules) gS

mÙkjmÙkjmÙkjmÙkjmÙkj (A, C, D)

gygygygygy 'kjhj dk mikip; ra=k rki dks cuk;s j[kus ds fy, vfrfjDr ÅtkZ mRiUu djsxk

ekuk vkUrfjd mikip; ds dkj.k] mRiUu 'kfDr I gS]

tc 'kjhj dk rki fu;r gS] Q usV

= 0

I = A [T4 – 4

0T ]

dI = 4A3

0T (dT

0) = 4A

3

0T .T

0(fodYi A)

= 4× 460 × 10

300 60 J/s (fodYi D)

3. ,d leku jSf[kd ?kurkokys (uniform mass per unit length) mèokZèkj Mksj ds fupys fljs ij ,d xqVdk M yVdk gqvk gSA

Mksj dk nwljk fljk n<+ vkèkkj (fcanq O) ls layXu gSA rajx&nSè;Z 0 dh vuqizLFk rjax Lian (Lian 1, Pulse 1) fcanq O ij

mRiUu dh xbZ gSA ;s rjax LiUn fcanq O ls fcanq A rd TOA

le; esa igq¡prh gSA xqVds M dks fcuk fokksfHkr fd;s gq, fcanq A

ij fuekZ.k dh xbZ rjax&nSè;Z 0 dh vuqizLFk rjax Lian (Lian 2, pulse 2), fcanq A ls fcanq O rd T

AO le; esa igq¡prh gSA

fuEu esa ls dkSulk (ls) dFku lgh gS@gSa\

MA

O Pulse 1

Pulse 2

(A) le; TAO

= TOA

(B) Mksj ds eè; fcanq ij Lian 1(pulse 1) ,oa Lian 2(pulse 2) dk osx leku gS

(C) Lian 1(pulse 1) dh rjax&nSè;Z fcanq A rd igqapus esa yEch gks tk,xh

(D) Mksj ds vuqfn'k izsf"kr fdlh Hkh Lian dk osx mldh vkofÙk ,oa rjax&nSè;Z ij fuHkZj ugha gS


4


gygygygygy fdlh fcUnq ij Tv ,

pw¡fd osx ,d fcUnq ij ruko rFkk izfr bdkbZ yEckbZ nzO;eku ij fuHkZj djrk gS] blfy, O ls A rd

rFkk A ls O rd le; leku gksxkA

TOA

= TAO

pw¡fd vkofÙk lHkh fcUnq ij fu;r jgrh gSA bls leku cuk;s j[kus ds fy, v

leku gksxk rFkk v, O(T mPpre) ij mPpre

gS blfy, , O ij mPpre gksxkA

4. ,d lefckgq fizTe dk fizTe dks.k A gS (isosceles prism of angle A)A bl fizTe dk viorZukad gSA bl fizTe dk

U;wure fopyu dks.k (angle of minimum deviation) m

= A gSA fuEu esa ls dkSu lk (ls) dFku lgh gS@gSa\

(A) U;wure fopyu esa vkifrr dks.k i1 ,oa izFke viorZd ry ds viorZd dks.k r

1 = (i

1/2) kjk lacafèkr gS

(B) fizTe dk viorZukad ,oa fizTe dks.k (A), 11

cos2 2

A ⎛ ⎞ ⎜ ⎟⎝ ⎠

kjk lacafèkr gS

(C) tc igys ry ij vkiru dks.k 2

1sin 1 sin 4cos 1 cos

2

Ai A A

⎡ ⎤ ⎢ ⎥

⎢ ⎥⎣ ⎦ gS] rc bl fizTe ds fy, frh; ry ls fuxZr

fdj.k fizTe ds i"B ls Li'khZ; gksxh (tangential to the emergent surface)

(D) Tkc fizTe dk vkiru dks.k i1 = A gS rc fizTe ds Hkhrj izdk'k fdj.k fizTe ds vkèkkj ds lekukUrj gksxhA


gygygygygy 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

i1

c

A

r A1 = /2

A A

r A2 = /2

2sin cos2 2

sin2

A A

A

1sinA = sinr

2cos2

A sinA = 2cos .sin2

Ar

1sini1 = × sin(A –

C)

2sin cos2 2

sin

2cos2

A A

rA

= sin

2

A

= 2cos sin cos – cos sin2

C C

AA A

2

Ar

= 2 1

2cos sin 1– sin – cos2

C

AA A

⎡ ⎤⎢ ⎥⎣ ⎦

= 2

1 cos2cos sin 1– –

22cos

2

A AA

A

⎡ ⎤⎢ ⎥⎢ ⎥

⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

5


= 2

1 cos2cos sin 1– –

24cos 2cos

2 2

A AA

A A

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

i1 =

–1 2

sin sin 4cos –1– cos2

AA A

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥

⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

U;wure fopyu ds fy,] 12

Ar

–1

cos2 2

A ⎡ ⎤ ⎢ ⎥⎣ ⎦

–1

2cos2

A⎡ ⎤ ⎢ ⎥⎣ ⎦

5. fp=k esa fn[kk;s x, ifjiFk esa L = 1H, C = 1F, R = 1kgSA ,d ifjorhZ oksYVrk (V = V0sint) lzksr ls Jzs.kh lacaèk gSA

fuEu esa ls dkSu lk (ls) dFku lgh gS@gSa\

V t0 sin

L = 1 Hμ C = 1 Fμ R = 1 Ωk

(A) tc >>106 rad. s–1, ifjiFk laèkkfj=k (capacitor) dh rjg O;ogkj djrk gS

(B) tc ~ 0 gksxh rc ifjiFk esa cgrh èkkjk 'kwU; ds fudV gksxh

(C) tc = 104 rad.s–1 gksxh rc fo|qr èkkjk (electric current) oksYVrk dh ledyk esa gksxh

(D) tc fo|qr èkkjk oksYVrk dh ledyk esa gksxh rks og vkofÙkZ R ij fuHkZj ugha djsxh

mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)

gygygygygy >> 106 ij

XL

= L

= = 106 ds fy, 106 × 10–6

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

XC = 6 –6

1 11

10 10C

= 106 ij >> 106 ds fy, X

c << 1

R = 1 k

RLC –6 –6

1 1

10 10

= 106 rad.s–1

0 Xc ij /kjk 0

6. oÙkkdkj pki okys ,d xqVds dk nzO;eku M gSA ;s xqVdk ,d ?k"kZ.k jfgr est ij fLFkr gSA est ds lkis; (in a coordinate

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

mass) dks oÙkkdkj pki ds mPpre fcanq ls fojkekoLFkk ls NksM+k tkrk (released from rest) gSA ;s fcanq d.k oÙkkdkj iFk ij

uhps dh vkSj ljdrk gSA tc fcanq d.k xqVds ls laidZ foghu gks tkrk gS] rc mldh rkRkf.kd fLFkfr x vkSj xfr gSA fuEu

esa ls dkSulk (ls) dFku lgh gS@gSa\

6


R

Rm

M

y

x

x = 0

(A) xqVds (M) dk osx 2m

V gRM

gS

(B) xqVds (M) ds lagfr dsUnz ds foLFkkiu dk X ?kVd (X co-ordinate) mR

M m

gS

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

xM m

gS

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

1

gR

m

M

gS


gygygygygy v = M dk osx

u = m dk osx

mu = –MV ...(i)

2 21 1

2 2mgR mu MV ...(ii)

2 – 2

1 1

gR m gRu v

m mM

M M

rFkk Mx = m(R – x) tgk¡ x = CykWd M dk foLFkkiu

x(M + m) = mR mR

xm M

ck;ha vksj

;k –

mRx

m M

7. ,d xksykdkj fo|qr&jks/h rkez rkj (insulated copper wire) dks A ,oa 2A okys nks ks=kiQyksa ds oy;ksa esa O;kofrZr fd;k x;k

gSA rkjksa ds vfrØe.k fcanq fo|qrjks/h jgrs gS (tSlk fp=k esa n'kkZ;k x;k gS)A laiw.kZ oy; dkxt+ ds ry esa fLFkr gSA dkxt ds

ry ds vfHkyEcor fLFkj rFkk ,dleku pqEcdh; ks=k B

loZ=k mifLFkr gSA oy; vius lkeqnkf;d O;klksa ls cus vk ds ifjr%

le; t = 0 ls dks.kh; osx (angular velocity) ls ?kweuk 'kq: djrk gSA fuEu esa ls dkSulk(ls) dFku lgh gS@gSa\

7


× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

× × × × × × × ×

area 2A

area A

B

× × × × × × × ×

(A) nksuks a oy;ks a ls mRiUu vf/dre dqy çsfjr fo|qr okgd cy (net emf) dk vk;ke] NksVs oy; esa mRiUu

vfèkdre çsf"kr fo|qr okgd cy ds vk;ke ds cjkcj gksxk

(B) tc oy;ks a dk ry dkxt+ ds ry ls vfHkyac fn'kk es a gksrk gS rc vfHkokg ds ifjorZu ds nj vf/dre

gksrh gS

(C) çsfjr fo|qr okgd cy (emf induced) oy;ksa ds ks=kiQyksa ds ;ksx ds lekuqikfrd gS

(D) nksuksa oy;ksa ls mRiUu dqy çsfjr fo|qr okgd cy (emf induced) cos t ds lekuqikrh gS

mÙkjmÙkjmÙkjmÙkjmÙkj (A, B)

gygygygygy = BA cost

e = B sint

ywi 1 ds fy,

1 = BA cost : |e

1| = AB sint

2 = 2BA sint |e

2| = 2BA sint

e1 o e

2 ,d nwljs ds foijhr gSa] blfy, usV çsfjr fo-ok-c- dk vk;ke = 2BA – BA = BA

e1 rFkk e

2, t = /2 ;k = 90° ij f'k[kj gksaxsA

[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 ikap ç'uikap ç'uikap ç'uikap ç'uikap ç'u gSaA

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

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

çR;sd ç'u ds fy, vad fuEufyf[kr ifjfLFkfr;ksa esa ls fdl ,d ds vuqlkj fn;s tk;saxs%

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

'kwU; vad : 0 vU; lHkh ifjfLFkfr;ksa esa

8


8. ,d fLFkj L=kksr vko`fÙk f0 = 492 Hz dh èofu mRlftZr djrk gSA 2 ms–1 ds xfr ds vixeuh dkj ls ;g èofu ijkofrZr

gksrh gSA èofu L=kksr ijkofrZr ladsr dks çkIr dj ds ewy ladsr ij vè;kjksfir (superpose) djrk gSA rc ifj.kkeh flXuy

dh foLian&vkofÙk (beat frequency) gS

(èofu dh xfr 330 ms–1 gSA dkj èofu dks mldh çkIr gqbZ vkofÙk ij ijkofrZr djrh gSA)

mÙkjmÙkjmÙkjmÙkjmÙkj (6)

gygygygygy fLFkj L=kksr kjk mRlftZr èofu dh vko`fÙk = f0 = 492 Hz

xfr'khy dkj kjk ijkofrZr èofu dh vkofÙk

0'

⎡ ⎤ ⎢ ⎥⎣ ⎦

c

c

c vf f

c v

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

vc = dkj dh pky = 2 m/s

gy djus ij

332' 492 498 Hz

328 f

ifj.kkeh flXuy dh foLiUn vkofÙk = f ' – f0

= (498 – 492) Hz

= 6 Hz

9. ,d gkbMªkstu ijek.kq dk ,d bysDVªkWu ni DokaVe la[;k (quantum number) okys dk ls n

f DokaVe la[;k (quantum number)

ds dk esa ços'k djrk gSA Vi rFkk V

f çkFkfed ,oa vafre fLFkfrt ÅtkZ,a gSaA ;fn 6.25,

i

f

v

v

rc nf dh U;wure lEHkkoh

la[;k (smallest possible nf) gS


gygygygygy2

2PE 27.2

Z

n H ds fy, Z = 1

2

27.2PE

n

1 2

27.2PE

i

i

V

n

2 2

27.2PE

f

f

v

n

22

26.25

i f f

f ii

V n n

V nn

⎛ ⎞ ⎜ ⎟⎝ ⎠

nf = 2.5 n

i

5

2

f

i

n

n

ni feuV = 2

nf feuV = 5

nf = 5

9


10. i"B&ruko (surface tension) –1

0.1Nm

4S

ds æo ds ,d cwan dh f=kT;k R = 10–2 m gS] ftls K le:i cwanksa esa foHkkftr

fd;k x;k gSA i"B&ÅtkZ dk cnyko U = 10–3 Joules gSA ;fn K = 10 gS rc dk eku gksxk


gygygygygy ekuk NksVh cw¡n dh f=kT;k = r

3 34 4

3 3R K r

1

3R K r ...(i)

S(K4r2 – 4R2) = 10–3

22 3

2

3

0.14 10

4

Rk R

K

⎛ ⎞ ⎜ ⎟

⎜ ⎟⎝ ⎠

1

2 23 1 10R K

1

3 1 100K

1

3 101K

310 101

a 6

11. vk;ksMhu dk leLFkkfud (isotope)1311, ftldh v/Z&vk;q 8 fnu gS] -k; ds dkj.k tsuksau (xenon) ds leLFkkfud esa kf;r

gksrk gSA vYi ek=k dk 1311 fpfëòr (labelled) lhje (serum) ekuo 'kjhj esa vUr%fkIr (inject) fd;k x;k] ftl ek=kk dh

vWfDVork (activity) 2.4 × 105 csdsjsy (Becquerel) gSA ;g lhje :f/j /kjk esa vk/s ?kaVs esa ,dleku forfjr gksrk gSA vxj

11.5 ?kaVs ckn 2.5 ml jDr 115 csdsjsy dh vWfDVork n'kkZrk gS] rc ekuo 'kjhj esa jDr vk;ru (yhVj esa) gS (vki ex 1

+ x for |x| << 1 ,oa ln 2 0.7 dk mi;ksx dj ldrs gSaA)


gygygygygy ekuk jDr dk dqy vk;ru V ml gS

∵ 2.5 ml jDr lfØ;rk = 115 Bq

1 ml jDr lfØ;rk = 115

2.5

lEiw.kZ jDr dh lfØ;rk = 115

V Bq2.5

⎛ ⎞⎜ ⎟⎝ ⎠

51152.4 10

2.5

tVe⎛ ⎞ ⎜ ⎟

⎝ ⎠

ln211.5

5 8 242.4 10 e

0.7 11.5

5 8 242.4 10 e

10


1

5 242.4 10 e

5 12.4 10 1

24

⎛ ⎞ ⎜ ⎟⎝ ⎠

5 23 2.5V 2.4 10

24 115 5000 ml

jDr dk dqy vk;ru = 5 L

12. ,do.khZ çdk'k (monochromatic light) viorZukad n = 1.6 okys ekè;e esa çxkeh gSA ;g çdk'k dk¡p dh phrh (stack of

glass layers) ij fupys lrg ls = 30° dks.k ij vkifrr gksrk gS (tSlk fd fp=k esa n'kkZ;k x;k gS)A dk¡pksa ds Lrj ijLij

lekarj gSA dk¡p ds phrh ds viorZukad ,dfn"V nm

= n – mn, Øe ls ?kV jgs gSaA ;gk¡ m Lrj dk viorZukad n

m gS

vkSj n = 0.1 gSA çdk'k fdj.k (m – 1) ,oa m Lrj ds i`"Bry ls lekarj fn'kk esa nkbZa vksj ls ckgj fudyrk gSA rc m dk

eku gksxk

m n m n – n m n – ( – 1)m – 1

3

2

1

n n– 3n n– 2n n – n


gygygygygy izFke ijr rFkk m oha ijr ds eè; Lusy fu;e yxkus ij

sin sin90º n n m n

11.6 1.6 0.1 1

2 m

0.88

0.1 m


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

bl [k.M esa nks Vscynks Vscynks Vscynks Vscynks Vscy gSa (izR;sd Vscy esa 3 dkWye vkSj 4 iafDr;ka gSa)

izR;sd Vscy ij vk/kfjr rhurhurhurhurhu iz'u gSaA

çR;sd iz'u ds pkjpkjpkjpkjpkj mÙkj fodYi (A), (B), (C) vkSj (D) gSa ftuesa fliQZ ,d fodYifliQZ ,d fodYifliQZ ,d fodYifliQZ ,d fodYifliQZ ,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 fliQZ lgh fodYi ds vuq:i cqycqys dks dkyk fd;k gSA



11


uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa Q.13, Q.14 vkSj vkSj vkSj vkSj vkSj Q.15 dsdsdsdsds

mÙkj nhft;sAmÙkj nhft;sAmÙkj nhft;sAmÙkj nhft;sAmÙkj nhft;sA

,d pktZ;qDr d.k (bysDVªkWu ;k izksVksu) vkjafHkd xfr

ls ewy fcUnq (x = 0, y = 0, z = 0) ij izLrqr (introduced) gksrk

gSA fLFkj rFkk ,dleku fo|qr~ ks=k

E ,oa pqEcdh; ks=k

B loZ=k mifLFkr 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, B

0 ds eku /ukRed gSaA

(I) bysDVªkWu (i) (P)

(II) bysDVªkWu (ii) (Q)

(IV) izksVksu (iv) (S)

(III) izksVksu (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

ls

ls

ls

ls

13. fdl fLFkfr esa d.k lh/h js[kk esa ½.kkRed y-vk (negative y-axis) dh fn'kk esa pysxk\

(A) (III) (ii) (P) (B) (II) (iii) (Q)

(C) (IV) (ii) (S) (D) (III) (ii) (R)

mÙkjmÙkjmÙkjmÙkjmÙkj (D)

gygygygygy izksVkWu ½.kkRed y-fn'kk ds vuqfn'k ljy js[kk esa xfr djsxk tc

0

v ,

0

E E y rFkk

0B B y

14. fdl fLFkfr esa d.k +z–vk vuqfn'k dqaMfyuh iFk (helical path along positive z-axis) dk vuqlj.k djsxk\

(A) (II) (ii) (R) (B) (III) (iii) (P)

(C) (IV) (ii) (R) (D) (IV) (i) (S)


gygygygygy izksVkWu dq.Myhnkj iFk esa xfr djsxk rFkk vk /ukRed z-fn'kk ds vuqfn'k gSA

;fn 0

0

2 Ev x

B,

0

E E z rFkk

0

B B z

15. fdl fLFkfr esa d.k vpy vpy vpy vpy vpy xfr ls lh/h js[kk esa pyu djrk gS\

(A) (II) (iii) (S) (B) (III) (ii) (R)

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

mÙkjmÙkjmÙkjmÙkjmÙkj (A)

gygygygygy bysDVªkWu fu;r osx ls ljy js[kk esa xfr djsxk ;fn

0

0

Ev y

B,

0

E E x ,

0

B B z

12


uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa uhps nh x;h Vscy ds rhu dkyeksa esa miyC/ lwpuk dk mi;qDr <ax ls lqesy dj iz'uksa Q.16, Q.17 vkSj vkSj vkSj vkSj vkSj Q.18

ds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sAds mÙkj nhft;sA

P

V

12

P

V

1

2

P

V

1 2

P

V

1

2

(i) lerkih;

levk;rfud (Isochoric)

lenkch; (Isobaric)

:¼ks"e (adiabatic)

1 2 2 2 1 1

1–

–1W P V PV

1 2 2 1–W PV PV

1 20W

2

1 2

1

–V

W nRT lnV

⎛ ⎞

⎜ ⎟⎝ ⎠

(I)

,d vkn'kZ xSl fofHkUu pØh; Å"ekikfrd izØeksa ls xqtjrk gSA ;g fuEu dkWye esa vkjs[k kjk n'kkZ;k x;k gSA dsoy fLFkfr ls fLFkfr tkusokys iFk dh vksj è;ku nsaA bl iFk ij fudk; ij gqvk dk;Z gS A ;gk¡ fu;r nkc ,oa fu;r vk;ru Å"ek&/kfjrkvksa dk vuqikr gS A xSl ds eksyksa dh la[;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)

16. fuEu fodYiksa esa ls dkSu lk la;kstu vkn'kZ xSl esa èofu dh xfr dh eki ds la'kks/u esa iz;qDr Å"ekxfrd izfØ;k dks lghn'kkZrk gS\

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

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

mÙkjmÙkjmÙkjmÙkjmÙkj (B)

gygygygygy :¼ks"e izØe dks ,d vkn'kZ xSl esa èofu dh pky ds fu/kZj.k esa la'kks/u ds :i esa iz;qDr fd;k tkrk gSA

lgh fodYi (B) gS

:¼ks"e izØe esa fudk; ij fd;k x;k dk;Z = 2 2 1 1

– –

–1

P V PV

13


17. fuEu fodYiksa esa dkSu lk la;kstu lgh gS\

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

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

mÙkjmÙkjmÙkjmÙkjmÙkj (C)

gygygygygy fodYi (C) lgh la;kstu gSA

W1 2

= 0 levk;rfud izØe

18. fuEu fn, fodYiksa esa dkSulk la;kstu U = Q – P V izfØ;k dk vdsys lgh izfrfuf/Ro djrk gS\

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

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


gygygygygy fodYi (B) lgh izn'kZu gSA

W1 2

= –PV2 + PV

1 lenkch;

END OF PHYSICS

14


CHEMISTRY


• 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



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

ClH C3

(A) 1-Dyksjks-4-eSfFky csathu (B) 1-eSfFky-4-Dyksjkscsathu

(C) 4-Dyksjks Vksyqbu (D) 4-eSfFkyDyksjks csathu

mÙkjmÙkjmÙkjmÙkjmÙkj (A, C)

gygygygygy IUPAC uke

(A)

Cl

CH3

1- -4-Dyksjks esfFkycsathu

1

2

3

4

(C)

CH3

Cl

4-DyksjksVksyqbu

1

2

3

4

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

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

(B) HClO4 dk la;qXeh kkj (conjugate base) H

2O ls nqcZy kkj gS

(C) Cl2 dh H

2O ds lkFk vfHkfØ;k gksus ij HClO

4 curk 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

(B) HClO4 > H

2O vEyh; yk.k

ClO4

– < OH– la;qXeh kkj

çcy vEyksa ds la;qXeh kkj nqcZy gksrs gSaA

15


(D) HClO4 > HClO vEyh; lkeF;Z

+

4 4HClO H + ClO

−⎯⎯→+

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

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 tc xL → 0 rks jkmYV dk fu;e (Raoult's law) dk ikyu

gksrk gS

(B) 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

(C) '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

(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 (C, D)

gygygygygy Z

x = 0M

x = 1L

'kq¼ L

xM

→x = 1M

M'kq¼

pL

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, visfkr) 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

16


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

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

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

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

mÙkjmÙkjmÙkjmÙkjmÙkj (A, B, C)

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.

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

3 6 2[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

23. 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

17


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

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

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

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

mÙkjmÙkjmÙkjmÙkjmÙkj (A, C)

gygygygygy C = CH

CH3

H C3

H

foik

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)

leik izfr jslhfedleik izfr jslhfedleik izfr jslhfedleik izfr jslhfedleik izfr jslhfed

M rFkk N eslks gS (le:i)

O rFkk P çfrfcEc leko;oh ;qXe gS

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

(C) (M rFkk O) rFkk (N rFkk P) vçfrfcEc leko;fo;ksa ds nks ;qXe gSA

24. ,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 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

(B) 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

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

(D) 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;

mÙkjmÙkjmÙkjmÙkjmÙkj (A, B, C)

gygygygygy (A) 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

P1

P2

P

(P , V )1 2

(P , V )2 2

V1

V2

V

(P , V )1 1

vuqRØe.kh;mRØe.kh;

18


(B)

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

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

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

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

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

mÙkjmÙkjmÙkjmÙkjmÙkj (B, C)

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. ,d nqcZy ,dkkjdh; 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 dhnwjh 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 ,dkkjdh; 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

−= × ×

19


= 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

27. 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)


gygygygygy

Te

Br

Br

Br

F

Br

Br

Br

Br

F

esa ,d ,dkdh ;qXe gS

esa nks ,dkdh ;qXe gS

Xe

S N≡

F

F

F

F

F

esa rhu ,dkdh ;qXe gS

esa ,dkdh ;qXe ugha gSF

28. ,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


20


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

29. 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

30. 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;

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

(A) (II) (i) (Q)

(B) (I) (iv) (R)

(C) (I) (i) (S)

(D) (I) (i) (P)

22



gygygygygy H-ijek.kq ds fy,

1s-dkd 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− × = −

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

(A) (I) (iii) (R) (B) (II) (ii) (Q)

(C) (I) (i) (S) (D) (I) (i) (R)


gygygygygy 1s dkd vfn'kkRed gS vr ψ cosθ ij fuHkZj ugha djsxh

vr% A xyr gS

33. 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) (IV) (iv) (R) (B) (III) (iii) (P)

(C) (II) (ii) (P) (D) (I) (ii) (S)


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

D;kasfd

(A) esa] 2

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

(B) esa] 2pz dkd esa f=kT;h; uksM ugha gS

(D) esa] 1s dkd esa f=kT;h; uksM ugha gS

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 tks fd nks fHkUu dkcksZfDlfyd vEy nsrk gS] gS

(A) (II) (iv) (R) (B) (III) (iii) (P)

(C) (I) (i) (S) (D) (IV) (iii) (Q)


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

leik leko;ofoik leko;o

;g i£du vfHkfØ;k dk ewy mnkgj.k gS

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

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

(C) (IV) (i) (Q) (D) (II) (iii) (R)


gygygygygy

CH3

CH2

CH2

CH – Br2

+ Br

+ Br2

+ HBr

+ Br

Br2

Br + Brhν

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

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

(C) (IV) (ii) (P) (D) (II) (i) (S)


gygygygygy C CH3

O

NaOH

Br2

COO Na– +

+ CHBr3

H O/H2

+

COOH

;g gSyksiQkWeZ vfHkfØ;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



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. ;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 = 5, h = 4, k = –3

(C) p = –2, h = 2, k = –4 (D) p = 2, h = 3, k = –4


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

25


38. 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) 21 1 y

(C) 21 1 y

(D) 21 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

⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠

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

39. 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)2

( )5

P X Y ∪

(B)4

( )15

P Y

(C) 1|

2P X Y

(D)1

( )5

P X Y ∩

mÙkjmÙkjmÙkjmÙkjmÙkj (B, C)

26


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

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) 2a, 4, 1

(B) a, 4, 1

(C) a, 4, 2

(D) 2a, 8, 1

mÙkjmÙkjmÙkjmÙkjmÙkj (B, C, D)

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 = 2

(B) x = 0

(C) x = 1

(D) x = –1


27


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, D)

gygygygygy pw¡fd fodYi (B) o (C) 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 (D) 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. 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) 2

0

( ) ( )sinf x f t t dt

∫ (B)0

( )sinx

x

e 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

28


[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

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. p ds fdrus ekuksa ds fy;s o`Ùk (circle) x2 + y2 + 2x + 4y – p = 0 ,oe~ funsZ'kkad vkksa (coordinate axes) esa dsoy rhufcUnq mHk;fu"B (common) gSa\


gygygygygy laHko fLFkfr;k¡

p = –1 p = 0

29


46. ,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

47. vkjksa 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 gSftuesa fdlh Hkh vkj dh iqujko`fr ugha gksrh gS] rFkk y bl rjg ds mu 'kCnksa dh la[;k gS ftu esa dsoy ,d vkj dh

iqujkofr nks ckj gksrh gS o fdlh vU; vkj dh iqujkofr ugha gksrh gSA rc y

x9


gygygygygy Li"Vr% x = 10!

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

'ks"k 9 vkjksa 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

48. 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


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. ;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\

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

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


50. 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)

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


51. ;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) (iii) (S) (B) (II) (iv) (R)

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


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

50. (–1, 1), 2a ds fy, x2 + y2 = a2 ij fLFkr gS] blfy, lgh mÙkj dsoy (A) gSA

31


51. 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

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 lgh la;kstu gS\

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

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


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

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

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


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

(A) (II) (iii) (S) (B) (I) (ii) (R)

(C) (III) (iv) (P) (D) (IV) (i) (S)


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

answers & solutions · izr;sd iz'u ds fy, vad fueufyf[kr ifjflfkfr;ksa esa ls fdlh ,d ds...

Documents