Questions & Answers · 2020. 9. 28. · 1 Questions & Answers for JEE (Advanced)-2020 Time : 3 hrs....

1

Questions & Answersforforforforfor

JEE (Advanced)-2020

Time : 3 hrs. Max. Marks: 198

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

Ph.: 011-47623456 Fax : 011-47623472

PAPER - 2

DATE : 27/09/2020

PART - I : PHYSICS

[k.M[k.M[k.M[k.M[k.M - 1 (vf/dre vadvf/dre vadvf/dre vadvf/dre vadvf/dre vad : 18)

bl [k.M esa N% (06) ç'u gSaA

izR;sd iz'u dk mÙkj 0 ls 9 (nksuksa lfEfyr) rd dh ijkl esa ,dy vadh; iw.kk±d,dy vadh; iw.kk±d,dy vadh; iw.kk±d,dy vadh; iw.kk±d,dy vadh; iw.kk±d gS

çR;sd ç'u ds fy,] mÙkj ds lgh vkafdd eku dks ekÅl rFkk vkWu&LØhu opqZvy U;weSfjd dh–iSM ds mi;ksx ls mÙkj çfo"V

djus ds LFkku ij gh çfo"V dhft;sA

çR;sd ç'u ds mÙkj dk ewY;kadu uhps nh xbZ vadu ;kstuk ds vuqlkj fd;k tk;sxk:

iw.kZ vad : +3 ;fn dsoy lgh iw.kk±d eku dks çfo"V fd;k x;k gS_

'kwU; vad : 0 ;fn iz'u dk mÙkj ugha fn;k x;k gS

½.kkRed vad : –1 vU; lHkh fLFkfr;ksa esaA

1. iryh ikjn'khZ ÅèokZ/j nhokjksa okys rFkk ty

4

3vioruZ kad ls Hkjs gq, ,d cM+s oxkZdkj ik=k dks {kSfrt est ij j[kk tkrk

gSA ,d Nk=k blds fdlh ,d dksus ls 12 cm ij ,d irys lh/s rkj dks ty ds vUnj ÅèokZ/j :i ls j[krk gS] tSlk fdfp=k esa ;kstukc¼ :i ls fn[kk;k x;k gSA bl dksus ls rkj dks ns[krs gq,] vU; Nk=k rkj ds nks izfrfcEc ns[krk gS] tks n'kkZ,vuqlkj nf"V js[kk ds izR;sd Nksj ij lefer :i ls fLFkr gSA bu izfrfcEcksa ds eè; nwjh (cm esa) ____________ gSA

mÙkj (4)

JEE (ADVANCED)-2020 (Paper-2)

2

gygygygygy

12 ( , 0)0

S i

(x, y)

i

y

x

45°

izFke izfrfcEc S dk y funsZ'kkad 'y' gS

4 1 3sini sin45 sini

3 42

23

4 23

i i 32

= 45 – i

blfy, (–12, 0) ls xqtjus okyh js[kk dh <ky m = tan(45 – i) =

1 tani

1 tani gS

23 3

m23 3

lehdj.k

23 3 x 12 23 3y

23 3 23 3

vU; lehdj.k y = –x

vc

23 3 y 12 23 3y

23 3 23 3

12 23 32 23y

23 3 23 3

2y =

12 23 3

4.5 423

blfy,] nks izfrfcEcksa ds eè; nwjh 4 cm gSA

2. vuqizLFk&dkV {ks=kiQy St okyh ,d jsyxkM+h vuqizLFk&dkV {ks=kiQy S

0(S

0 = 4S

t) dh ,d yEch lqjax ds vUnj v

t pky ls

xfr'khy gSA ekuk dh jsyxkM+h ds lkeus yxHkx lHkh ok;q (?kuRo ) blds fljksa (fdukjksa) rFkk lqjax dh nhokjksa ds eè; ihNs

dh vksj izokfgr gksrh gSA rFkk] jsyxkM+h ds lkis{k ok;q izokg fLFkj rFkk iVyh; gSA ifjos'kh nkc ysrs gq, rFkk jsyxkM+h ds vUnj

ifjos'kh nkc p0 gSA ;fn jsyxkM+h ds fljksa rFkk lqjax dh nhokjksa ds eè; {ks=k esa nkc p gS] rc

2

0 t

7p p v

2N gSA N dk eku

________ gSA

mÙkj (9)


3

gygygygygy jsyxkM+h ds lkis{k osx ysus ij

v

vt

4Stv

t = 3S

tv

t

4v v

3

blh izdkj ifjos'kh ok;q jsyxkM+h ds lkis{k vt pky ls xfr'khy gksxh

2 2

0 t t

1 1 16p v P v

2 2 9

2

0 t

1 7p p v

2 9

N = 9

3. 0.01 m dh nwjh }kjk iFkDÑr nks cM+h oÙkh; pdfr;ksa dks fp=k esa n'kkZ, vuqlkj ,d fLop }kjk ,d cSVjh ls la;ksftr fd;k

tkrk gSA ?kuRo 900 kg m–3 dh vkosf'kr rsy dh cw¡nksa dks 'kh"kZ pdrh ds dsUnz ij ,d efgu fNnz ls NksM+k tkrk gSA tc

dqN rsy dh cw¡nsa VfeZuy osx izkIr djrh gaS] rc pdfr;ksa ds fljksa ij 200 V dh oksYVrk vkjksfir djds fLop dks cUn fd;k

tkrk gSA ftlds ifj.kkeLo:i] f=kT;k 8 × 10–7 m dh rsy dh ,d cw¡n ÅèokZ/j :i ls xfr djuk cUn dj nsrh gS rFkk

pdfr;ksa ds eè; rSjrh gSA bl rsy dh cw¡n esa mifLFkr bysDVªkWuksa dh la[;k ________ gSA (mRIykou cy dks ux.; ekus]

xq:Roh; Roj.k = 10 ms–2 rFkk bysDVªkWu (e) ij vkos'k = 1.6 × 10–19 C ysa)

mÙkj (6)

gygygygygy

E 0.01 m

v 200E 100

l 1


4

4E 2 10 v/m

mg

Eq

blh izdkj q = Ne (N, bysDVªkWuksa dh la[;k gS)

NeE = mg

NeE = 34r g

3

37

4 19

8 10 900 104N

3 2 10 1.6 10

N = 6

4. ,d gkWV ,;j cSywu dqN ;kf=k;ksa rFkk dqN jsr ds cLrs] ftuesa ls izR;sd dk nzO;eku 1 kg gS] bl izdkj ogu fd, gq, gSa

rkfd bldk dqy nzO;eku 480 kg gSA xqCckjs dks bldh mRIykodr iznku djus okyk bldk izHkkoh vk;ru V gSA xqCckjk

100 m dh lkE; Å¡pkbZ ij pyk;eku gSA tc N jsr ds cLrksa dks ckgj isaQdk tkrk gS] rc xqCckjk yxHkx 150 m dh uohu

lkE; Å¡pkbZ rd mB tkrk gS rFkk bldk vk;ru V vifjofrZr jgrk gSA ;fn /jkry ls Å¡pkbZ h ds lkFk ok;q ds ?kuRo dk

ifjorZu 0

h

h

0(h) eh

gS] tgk¡

0 = 1.25 kg m–3 rFkk h

0 = 6000 m, N dk eku _________ gSA

mÙkj (4)

gygygygygy ?kuRo dk ifjorZu (h) = 0h/h

0e

ekuk xqCckjs ds vUnj ok;q dk ?kuRo] ckg~; ok;q ds ?kuRo dh rqyuk esa ux.; gS] rc

480 g = Vg(100)

(480 – N)g = Vg(150)

480 (100)

480 N (150)

0

0

100

h

0

150

h

0

e480

480 N

e

0

50

h x480 50e e x

480 N 6000

480 501

480 N 6000

N 4


5

5. m nzO;eku ds ,d fcUnq vkos'k q dks l yEckbZ dh ,d Mksjh }kjk ÅèokZ/j :i ls yVdk;k tkrk gSA vc f}/zqo vk?kw.kZ �

p ds

,d fcUnq f}/zqo dks vuUr ls q dh vksj bl izdkj yk;k tkrk gS fd vkos'k nwj xfr djrk gSA f}/zqo dh fn'kk] dks.kksa rFkk

nwfj;ksa lfgr fudk; dh vfUre lkE; fLFkfr uhps fp=k esa n'kkZ;h xbZ gSA ;fn f}/zqo dks bl fLFkfr rd ykus esa fd;k x;k dk;Z

N × (mgh) gS] tgk¡ g xq:Roh; Roj.k gS] rc N dk eku _________ gSA (è;ku nhft, dh fcUnq nzO;eku dks lkE;koLFkk esa

cuk, j[krs gq, rhu leryh; cyksa ds fy,, F

sin lHkh cyksa ds fy, leku gS] tgk¡ F dksbZ ,d cy gS rFkk vU; nks cyksa

ds eè; dks.k gS)

l

l

�

p

q h

2lsin

2

mÙkj (2)

gygygygygy

P mg90 –

2

Fd

90 – 2

T

Li"V :i ls d

Fmg T

sin 180sin 90 sin 90

2 2

blh izdkj d 3

0

q2pF Eq

4 2lsin2

izkjfEHkd fLFkfrt ÅtkZ = (0)

vfUre fLFkfrt ÅtkZ =

2

0

pqmgh W

4 2lsin2

blh izdkj 2

2lsin sin 2lsin h2 2 2

d

3

0

F mg 2pq mgsin

sincos cos4 2lsin2 22


6

3

0

mg 2lsin sinpq 2

42 cos

2

3

2

mg 2lsin sin2

W mgh

2cos 2lsin2 2

mgsin 2lsin2W mgh

2cos2

mg 2sin cos lsin2 2 2W mgh

cos2

2W mg 2lsin mgh 2mgh2

N 2

6. 8 m Å¡pkbZz ds rkih; :i ls foyfxr csyukdkj cUn ik=k dks ÅèokZ/j :i ls j[kk x;k gSA bls 8.3 kg nzO;eku ds ,d

Å"ek&ik;Z (vkn'kZ rkih; pkyd) ?k"kZ.kghu foHkktd }kjk nks cjkcj Hkkxksa esa foHkkftr fd;k tkrk gSA vr% foHkktd izkjEHk esa

'kh"kZ ls 4 m dh nwjh ij fLFkr gS] tSlk fd uhps fp=k esa ;kstukc¼ :i ls fn[kk;k x;k gSA ik=k ds nksuksa Hkkxksa esa ls izR;sd

Hkkx esa 300 K rki ij ,d vkn'kZ xSl ds 0.1 eksy gSA vc foHkktd dks NksM+k tkrk gS rFkk ;g xSl dk fjlko gq, fcuk

ik=k ds ,d Hkkx ls nwljs Hkkx rd xfr djrk gSA tc lkE;koLFkk izkIr gksrh gS] rc 'kh"kZ ls foHkktd dh nwjh (m esa) _______

gksxh (xq:Roh; Roj.k = 10 ms–2 rFkk lkoZf=kd xSl fu;rkad = 8.3 J mol–1K–1 ysa).

8 m

mÙkj (6)

gygygygygy ekuk xSl vkn'kZ ,dy&ijek.kqd xSl gS

V

3C R

2

rFkk vuqizLFk dkV {ks=kiQy A gSA

izkjEHk esa


7

0

P 4A 0.1R 300 (nksuksa ds fy,)

ekuk fiLVu x ehVj rd foLFkkfir gks tkrk gS rFkk vfUre rki T gSA

rc

2

P A(4 x) 0.1RT

2

0.1RT RTP A

4 x 10(4 x)P

1

P2

1 x

rFkk 1

RTP A

10(4 x)

ysfdu vUr esa 2 1

RT 1 1P A P A 83

10 4 x 4 x

2

RT 2x83

10 16 x

V V

0.2 C 300 mg x 0.2C T

3 2 2 3R 300 83x RT

2 10 10 2

900R RT83x 3

10 10

300R 83 RTx

10 3 10

2

83x 2x30R 83

3 16 x

2

x 2x83 3 83

3 16 x

2

(9 x) (2x)1

3 16 x

18x + 2x2 = 48 – 3x2

5x2 + 18x – 48 = 0

x 1.78 2

blfy,] 'kh"kZ ls nwjh 6 m gSA


8



çR;sd ç'u ds pkjpkjpkjpkjpkj fodYi gSaA bu pkj fodYiksa eas ls ,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 lgh mÙkj gS@gSaA

çR;sd ç'u ds fy, (lHkh) lgh mÙkjksa ls lacaf/r fodYi@fodYiksa dk p;u dhft;sA

çR;sd ç'u ds mÙkj dk ewY;kadu uhps nh xbZ vadu ;kstuk ds vuqlkj fd;k tk,xk:

iw.kZ vad : +4 ;fn dsoy lHkh lgh fodYi@fodYiksa dk p;u fd;k x;k gS_

vakf'kd vad : +3 ;fn lHkh pkj fodYi lgh gSa ijUrq dsoy rhu fodYiksa dk p;u fd;k x;k gS_

vakf'kd vad : +2 ;fn rhu ;k vf/d fodYi lgh gSa ijUrq dsoy nks fodYiksa dk p;u fd;k x;k gS rFkk nksuksa lgh

gks_

vkaf'kd vad : +1 ;fn nks ;k vf/d fodYi lgh gSa ijUrq dsoy ,d fodYi dk p;u fd;k x;k gS rFkk og lgh

fodYi gks_

'kwU; vad : 0 ;fn fdlh Hkh fodYi dk p;u ugha fd;k x;k gSA (vFkkZr~ ç'u dk mÙkj ugha fn;k x;k gks);

½.kkRed vad : –2 lHkh vU; fLFkfr;ksa esaA

7. r f=kT;k ds ,d chdj dks ty viorZukda 4

3

ls H Å¡pkbZ rd Hkjk tkrk gS] tSlk fd ck;ha vksj fp=k esa fn[kk;k x;k gSA

chdj dks dks.kh; pky ls ?kwf.kZr ,d {kSfrt est ij j[kk tkrk gSA ;g ty dh lrg dks bl izdkj oØkdkj cukrk gS fd

chdj ds dsUnz ij ty dh Lrj dh Å¡pkbZz rFkk bldh ifjf/ ij ty ds Lrj dh Å¡pkbZ esa vUrj h (h << H,

h << r) gS] tSlk fd nk;ha vksj fp=k esa fn[kk;k x;k gSA bl lrg dks yxHkx xksyh; ys] ftldh oØrk f=kT;k R gSA fuEu esa

ls dkSulk@dkSuls lgh gS@gSa\ (g, xq:Roh; Roj.k gS)

H

h

(A)

2 2h r

R2h

(B) 2

3rR

2h

(C) chdj ds iSans dh vkHkklh xgjkbZ yxHkx

123H H

12 2g

gS

(D) chdj ds iSans dh vkHkklh xgjkbZ yxHkx

123H H

14 4g

gS

mÙkj (A, D)


9

gygygygygy2

2h r2g

R2 = r2 + (R – h)2

2 2

2

r h g hR

2h 2

r

R (R – h)

H

oØkdkj lrg ds viorZu ds fy, lw=k iz;qDr djus ij

4 41

1 3 3

v H R

1 4 1

v 3H 3R

3HRv

4R H

vkHkklh xgjkbZ =

13H H

| v | 14 4R

=

123H H

14 4g

8. ,d Nk=k ,d jsEi ds Åij dh vksj LdsfVax djrk gS] tks {kSfrt ds lkFk 30° dks.k cukrk gSA og v0 pky ls jsEi ds vk/kj

ls izkjEHk (tSlk dh fp=k esa fn[kk;k x;k gS) gksrk@gksrh gS rFkk R f=kT;k ds ,d v¼ZoÙkh; iFk xyz ds Åij ?kweuk pkgrk@pkgrh

gS ftlds nkSjku og /jkry ls vf/dre Å¡pkbZ h (fcUnq y ij) igq¡prk@igq¡prh gS tSlk fd fp=k esa fn[kk;k x;k gSA ekuk fd

ÅtkZ gkfu ux.; gS rFkk mPpre fcUnq ij bl ?kqeko ds fy, vko';d cy dsoy mlds Hkkj }kjk iznku fd;k tkrk gSA rc

(g xq:Roh; Roj.k gS)

Rz

y

x

30°

h

(A) 2

0

1v 2gh gR

2

(B) 2

0

3v 2gh gR

2

(C) fcUnq x rFkk z ij vko';d vfHkdsUnzh; cy 'kwU; gS

(D) vko';d vfHkdsUnzh; cy fcUnq x rFkk z ij vf/dre gS

mÙkj (A, D)


10

gygygygygy fcUnq y ij

mg sing30° =

2mv

R

2 gRv

2

y

mg sin 30°xz

;kaf=kd ÅtkZ laj{k.k iz;qDr djus ij

2 2

0

1 1mv mgh mv

2 2

2

0

gRv 2gh

2

iFk xyz ds vuqfn'k] pky fcUnq x RkFkk z ij vf/dre gS] blfy, vf/dre vfHkdsUnzh; cy dh vko';drk gksxhA

9. nzO;eku m rFkk yEckbZ L dh ,d NM+] tks vius fdlh ,d fljs ij dhyfdr gS] ÅèokZ/j :i ls yVdk;h xbZ gSA v pky ls

xfr'khy leku nzO;eku dh ,d xksyh NM+ ls blds dhyfdr fljs ls x nwjh ij Vdjkrh gS rFkk blesa /¡l tkrh gSA vc la;qDr

fudk; dhyd ds lkis{k dks.kh; pky ls ?kw.kZu djrk gSA vf/dre dks.kh; pky M

, x = xM ds fy, izkIr gksrh gSA rc

L

x

v

(A) 2 2

3vx

L 3x(B)

2 2

12vx

L 12x

(C) M

Lx

3(D)

M

v3

2L

mÙkj (A, C, D)

gygygygygy dhyd ds lkis{k dks.kh; laosx laj{k.k iz;qDr djus ij

L, m

x

m

v

2

2 mLmvx mx

3

2 2

3vx

L 3x

M

d L0 x

dx 3

rFkk M

3v

2L


11

10. ,d X-fdj.k uyh esa] /kjk I xzg.k fd, gq, ,d rUrq (dSFkksM+) ls mRlftZr bysDVªkWu dSFkksM+ ls d nwjh ij ,d y{; (,uksM) lsVdjkrs gSaA y{; dks dSFkksM+ ls mPp foHko V ij j[kk tkrk gS ftlds ifj.kkeLo:i lrr~ rFkk vfHkyk{kf.kd X-fdj.kksa dk

mRltZu gksrk gSA ;fn rUrq dh /kjk I dks I

2 rd ?kVk;k tkrk gS] rc foHkokUrj V, 2V rd c<+ tkrk gS rFkk iFkDdj.k nwjh

d, d

2 rd ?kV tkrh gS] rc

(A) vard rjaxnSè;Z vk/s rd ?kV tk,xh rFkk vfHkyk{kf.kd X-fdj.kksa dh rjaxnSè;Z leku jgsxh

(B) vard rjaxnSè;Z rFkk vfHkyk{kf.kd X-fdj.kksa dh rjaxnSè;Z leku jgsxh

(C) vard rjaxnSè;Z vk/s rd ?kV tk,xh rFkk lHkh X-fdj.kksa dh rhozrk,sa ?kVsxh

(D) vard rjaxnSè;Z nks xquk vf/d gks tk,xh rFkk lHkh X-fdj.kksa dh rhozrk,sa ?kVsxh

mÙkj (A, C)

gygygygygy tc vkifrr bysDVªkWuksa dh ÅtkZ dks nksxquk fd;k tkrk gS] rc vard rjaxnSè;Z vk/s rd ?kV tk,xhA vfHkyk{kf.kd X-fdj.kdh rjaxnSè;Z izHkkfor ugha gksxhA rUrq dh /kjk esa deh ds dkj.k rhozrk ?kVsxhA

11. leku nzO;eku rFkk vkos'k ds nks ,dleku vpkyd Bksl xksyksa dks leku yEckbZ dh nks vpkyd] nzO;ekughu Mksfj;ksa }kjk ,d

mHk;fu"B fcUnq ls ok;q esa yVdk;k tkrk gSA lkE;koLFkk ij] Mksfj;ksa ds eè; dks.k gSA vc xksyksa dks 800 kg m–3 ?kuRo rFkk

ijkoS|qrkad 21 ds ,d ijkoS|qr nzo esa Mqck;k tkrk gSA ;fn Mqckus ds ckn Mksfj;ksa ds eè; dks.k leku jgrk gS] rc

(A) xksyksa ds eè; fo|qr cy vifjofrZr jgrk gS

(B) xksyksa ds eè; fo|qr cy ?kVrk gS

(C) xksyksa dk nzO;eku ?kuRo 840 kg m–3 gS

(D) xksyksa dks lEHkkyus okyh Mksfj;ksa esa ruko vifjofrZr jgrk gS

mÙkj (A, C)

gygygygygy xksys ij usV cy ?kVsxk] ysfdu xksyksa ds eè; cy leku jgsxkA

tc ok;q esa

mg

TFe

eF

tan2 mg

rFkk 2 2

eT F (mg)

tc nzo esa Mqck;k tkrk gS

rFkke

2

2e

FFktan T (mg B)

2 mg B k

b

mg kmg 1

�

b

k 21 800

k 1 21 1

�

= 840 kg/m3


12

12. 1 ms–1 dh pky ls ewy fcUnq ls le; t = 0 ij izkjEHk gksus okyk] ,d d.k x-y ry esa nks&foeh; iz{ksi&iFk dk bl izdkj

vuqlj.k djrk gS fd blds funsZ'kkad lehdj.k 2x

y2

}kjk lEcfU/r gksrs gSaA blds Roj.k ds x rFkk y ?kVdksa dks Øe'k%

ax rFkk a

y }kjk iznf'kZr fd;k tkrk gSA rc

(A) ax = 1 ms–2 n'kkZrk gS fd tc d.k ewy fcUnq ij gS] rc a

y = 1 ms–2

(B) ax = 0 n'kkZrk gS fd lHkh le; ij a

y = 1 ms–2

(C) t = 0 ij] d.k dk osx x-fn'kk esa bafxr djrk gS

(D) ax = 0 n'kkZrk gS fd t = 1 s ij] d.k ds osx rFkk x v{k ds eè; dks.k 45° gS

mÙkj (A, B, C, D)

gygygygygy2x

y2

dy dxx

dt dt

2xy

2

22 2

2 2

d y d x dxx

dtdt dt

t = 0 ij, x = 0 vx = 1 m/s rFkk v

y = 0 m/s

x = 0 ij ax = 1 ds fy,

ay = 1 m/s2

ax = 0 ds fy,

x ds lHkh ekuksa ds fy, ay = 1 m/s2

t = 1 s ij ax = 0 ds fy,

x yˆ ˆv v i v j

�

ˆ ˆ1i 1j


bl [k.M esa N% (06) ç'u gSaA çR;sd ç'u dk mÙkj ,d vkafdd eku gSA


djus ds LFkku ij gh çfo"V dhft;sA ;fn vakfdd eku esa nks ls vf/d n'keyo LFkku gks] rks çkIr eku dk nks n'keyo LFkku rd

lfUudVu djsaA


iw.kZ vad : +4 ;fn dsoy lgh vakfdd eku dks çfo"V fd;k x;k gS_

'kwU; vad : 0 vU; lHkh fLFkfr;ksa esaA


13

13. ty ds vUnj ,d xksyh; cqycqys dh f=kT;k R gSA cqycqys ds vUnj nkc rFkk ty dk nkc p0 ds :i esa ysaA vc cqycqyk

:¼ks"e izdkj ls vjh; :i ls bl izdkj lEihfM+r gks tkrk gS fd bldh f=kT;k (R – a) gks tk,A a << R ds fy,] izfØ;k esa

fd, x, dk;Z ds ifjek.k dks (4p0Ra2)X }kjk O;Dr fd;k tkrk gS] tgk¡ X ,d fu;rkad gS rFkk = C

p/C

v = 41/30 gSA

X dk eku ________ gSA

mÙkj (1.00)

gygygygygy fd;k x;k dk;Z = IE esa of¼ = Uf – U

i =

0 0PV P V

1...(i)

blh izdkj 0 0

P V PV

0

0

VP P

V...(ii)

(i) rFkk (ii) ls

010 0 0

0

0 0

VP V P V

V30W P V 1

1 11

3( 1)

0 0 0 0 3( 1)

30 R 30 1W P V 1 P V 1

11 R a 11 a1

R

3(1 )

0 0 0 0

30 a 30 aP V 1 1 P V 1 3(1 ) 1

11 R 11 R

3

0

30 4 11 aW P R 3

11 3 30 R

2 2

0 0W (4 R P a)x (4 R P )a x 01

14. lUrqyu dh fLFkfr esa] ,d OghVLVksu lsrq dh pkj Hkqtkvksa ds izfrjks/ksa ds eku uhps fp=k esa n'kkZ,s x, gSA izfrjks/ R3 dk rki

xq.kkad 0.0004 ºC–1 gSA ;fn R3 ds rki dks 100 ºC rd c<+k;k tkrk gS] rc S rFkk T ds eè; mRiUu oksYVrk __________

oksYV gksxhA

V

P

Q

TS

R = 60 1

R = 100 2

R = 500 4

R = 300 3

50

V

mÙkj (0.27)


14

gygygygygy 4 2 2

3 3 3 3R R [1 E) R [1 4 10 10 ) R [1 4 10 ]

3

R 300[1 0.04] 300 1.04 3 104 312

1

50I A

372

P S 1

50V V I 60 60

372

500

60 100

312 = R3

I1 I

2

TS

P

50 V

okYs VP S

500V V

62

2

50 50 5I A

100 500 600 60

oksYVP T

5 50V V 100

60 6

P T P S S T

50 500 500 500(V V ) (V V ) V V

6 62 60 62

S T

1 1 2 1000 100V V 500 500

60 62 60 62 60 62 6 62 okYs V100

0.2688372

okYs VS T

V V 0.27

15. /kfjrk eku C1 = 2000 10 pF rFkk C

2 = 3000 15 pF okys nks la/kfj=kksa dks Js.khØe esa la;ksftr fd;k tkrk gSA bl

la;kstu ds fljksa ij vkjksfir oksYVrk V = 5.00 0.02 V gSA la/kfj=kksa ds bl la;kstu esa laxzfgr ÅtkZ dh x.kuk esa izfr'kr

=kqfV _______ gSA

mÙkj (1.30)

gygygygygy

1 2

1 2

C C 2000 3000 6000C pf

C C 5000 5

1 2

2 2 21 2 1 2

C C1 1 1 C

C C C C C C

2 1 2

2 2

1 2

C C 9 4C C 10 15 6nF

25 25C C

22

2 2

1

C 6000 1 9

5 25C (2000)


15

2 22

2

2

C 6000 1 4

5 3000 25C

21 U C 2 VU CV 100 100 100

2 U C V

U 6 5 0.02

100 100 2 100U 6000 5

= 0.5 + 0.8 = 1.30

16. iFoh dh lrg ij ,d ?kuh; Bksl ,yqfefu;e (vk;ru izR;kLFkrk xq.kkad = dP

VdV

= 70 GPa) ds ,d xqVds dh Hkqtk dh

yEckbZ 1 m gSA bls 5 km xgjs egklkxj dh ryh ij j[kk tkrk gSA ty dk vkSlr ?kuRo rFkk xq:Roh; Roj.k Øe'k%

103 kg m–3 rFkk 10 ms–2 ysus ij] xqVds dh Hkqtk dh yEckbZ esa ifjorZu (mm esa) _____ gSA

mÙkj (0.24)

gygygygygy dPV BdV

dV dP

V B

3( dl) dP

l B

3 34

9

l dP l gh 1 10 10 5 10l 2.4 10 0.24 mm

3 B 3 B 3 70 10

17. nks LR ifjiFkksa ds izsjdksa dks ,d nwljs ds lehi j[kk tkrk gS] tSlk fd fp=k esa fn[kk;k x;k gSA izsjdksa ds Lo&izsjdRo] izfrjks/]

vU;ksU;&izsjdRo rFkk vkjksfir oksYVrkvksa ds eku fn, x, ifjiFk esa iznf'kZr gSaA nksuksa fLopksa dks ,d lkFk cUn djus ds ckn]

/kjkvksa ds vius LFkk;h voLFkk eku izkIr djus ds le; rd izsjdksa esa izsfjr fo-ok-c- ds fo:¼ cSVfj;ksa }kjk fd;k x;k dqy dk;Z

________ mJ gSA

V =

5 V

1 L =

10

mH

1

R = 5 1

I1

M =

5 m

H

R = 10 2

I2

V =

20

V2L

= 2

0 m

H2

mÙkj (55)


16

gygygygygy 1

5I 1 A

5

2

20I 2 A

10

izsfjr fo-ok-c- ds fo:¼ cSVjh }kjk fd;k x;k dqy dk;Z = pqEcdh; ÅtkZ esa of¼

= U1 + U

2 + U

12

2 2

1 1 2 2 1 2

1 1W L I L I MI I

2 2

3 2 2 3 31 110 10 1 20 (2) 10 5 10 1 2

2 2

3(5 40 10) 10

= 55 mJ

18. ,d ik=k esa 1 kg ty Hkjk gqvk gS] bls lw;Z ds izdk'k esa j[kk tkrk gS] ftlds dkj.k ty ifjos'k dh rqyuk esa xeZ gks tkrk

gSA lw;Z ds izdk'k ds dkj.k izkIr dh xbZ izfr bdkbZ {ks=kiQy izfr bdkbZ le; vkSlr ÅtkZ 700 Wm–2 gS rFkk bls

0.05 m2 ds izHkkoh {ks=kiQy ij ty }kjk vo'kksf"kr fd;k tkrk gSA ekuk fd ty ls ifjos'k dh vksj Å"ek gkfu U;wVu ds

'khryu ds fu;e }kjk lapkfyr gksrh gS] yEcs le; ds ckn ty rFkk ifjos'k ds rki esa vUrj (ºC esa) _____________ gksxkA

(ik=k ds izHkko dks ux.; ekusa rFkk U;wVu ds 'khryu ds fu;e ds fy, fu;rkad = 0.001 s–1 ysa] ty dh Å"ek /kfjrk

= 4200 J kg–1 K–1)

mÙkj (8.33)

gygygygygy lkE;koLFkk esa

izfr lsd.M izkIr dh xbZ Å"ek = izfr lsd.M Å"ek gkfu

E × A = S

0K ms[ ] M

( – 0) =

3

E A 700 0.05 ºC

K S M 10 4200 1

=

2

3 2

700 5 10 35 350 508.33ºC

4.2 42 610 42 10


17

PART-II : CHEMISTRY



izR;sd iz'u dk mÙkj 0 ls 9 (nksuksa lfEfyr) rd dh ijkl esa ,dy vadh; iw.kk±d,dy vadh; iw.kk±d,dy vadh; iw.kk±d,dy vadh; iw.kk±d,dy vadh; iw.kk±d gS


djus ds LFkku ij gh çfo"V dhft;sA


iw.kZ vad : +3 ;fn dsoy lgh iw.kk±d eku dks çfo"V fd;k x;k gS_

'kwU; vad : 0 ;fn iz'u dk mÙkj ugha fn;k x;k gS

½.kkRed vad : –1 vU; lHkh fLFkfr;ksa esaA

1. ijek.kq Øekad n, n + 1, n + 2 rFkk n + 3 tgk¡ n < 10, okys pkj ijek.kqvksa dh 1st, 2nd, rFkk 3rd vk;uu ,UFkSYih I1, I

2, rFkk

I3] uhps lkj.kh esa nh x;h gSA n dk eku D;k gS\

ijek.kqØekad

vk;uu ,UFkSYih (kJ/mol)

I1

I2 I

3

n

n + 1

n + 2

n + 3

1681

2081

738

496

3374

1451

3952

4562

6050

6910

6122

7733

mÙkj (9)

gy %gy %gy %gy %gy % n = 9 (Ýyqvksjhu)

n + 1 = 10 (Ne)

n + 2 = 11 (Na)

n + 3 = 12 (Mg)

2. nzzo :i esa fuEufyf[kr ;kSfxdksa ij fopkj dhft,: O2, HF, H

2O, NH

3, H

2O

2, CCl

4, CHCl

3, C

6H

6, C

6H

5Cl.

tc ,d vkosf'kr daèks dks budh fuEufyf[kr èkkjk ds lehi yk;k tkrk gS] rks buesa ls fdrus fuEufyf[kr fp=k ds vuqlkjfopyu n'kkZrs gSa\

mÙkj (7)

gy %gy %gy %gy %gy % vuqpqEcdh; nzo rFkk èkzqoh; nzo vkosf'kr da?ks dh vksj vkdf"kZr gksaxsA vr% fopyu n'kkZus okys nzo O2, HF, H

2O, NH

3,

H2O

2, CHCl

3, C

6H

5Cl gSa


18

3. nqcZy {kkjh; foy;u esa KMnO4 rFkk KI dh jllehdj.kferh; fHkrh; ek=kkvksa ds eè; jklk;fud vfHkfØ;k esa 4 eksy KMnO

4

iz;qDr gksus ds fy, eqDr I2 ds eksyksa dh la[;k D;k gS\

mÙkj (0)

gy %gy %gy %gy %gy % 4 2 2 3

2MnO H O I 2MnO 2OH IO

4. iksVSf'k;e ØksesV ds vEyhdr foy;u dks ,fey ,YdksgkWy ds leku vk;ru ds lkFk feyk;k tkrk gSA tc bls 1 mL, 3%

H2O

2 foy;u feykus ds ckn foyksfMr fd;k tkrk gS rks uhys ,YdksgkWy dh ijr izkIr gksrh gSA uhyk jax Øksfe;e (VI) ;kSfxd

‘X’ ds cuus ds dkj.k gksrk gSA X ds ,d v.kq eas Øksfe;e ds lkFk dsoy ,dy cUèk }kjk cfUèkr vkWDlhtu ijek.kqvksa dhla[;k D;k gS\

mÙkj (4)

gy %gy %gy %gy %gy % 2

4 2 2 5 2CrO 2H 2H O CrO 3H O

X, CrO5 gSA

Cr

O

O

O

O

O

dsoy ,dy ca/ }kjk Øksfe;e ls caf/r vkWDlhtu ijek.kqvksa dh la[;k 4 gSA

uhyk jax Øksfe;e isUVkvkWDlkbM dh mifLFkfr ds dkj.k gksrk gSA

5. uhps ,d isIVkbM dh lajpuk nh x;h gS

HO

H N2

O

H

NN

H

O

OH

O

CO H2

NH2

;fn pH = 2, pH = 6, rFkk pH = 11 ij isIVkbM ij usV vkos'k ds fujis'k eku Øe'k% |z1|, |z

2|, rFkk |z

3| gSa] rks |z

1| + |z

2|

+ |z3| dk eku D;k gS\

mÙkj (5)

gy %gy %gy %gy %gy % pH = 2 ij

|Z1| = 2 (lHkh ,ehuks vERy

3NH ds :i esa ik, tkrs gSa)

pH = 6 ij

|Z2| = 0 (TohVj vk;u :i)

pH = 11 ij

|Z3| = |–3| = 3 (lHkh COOH o OH, –COO o –O ds :i esa ik, tkrs gSa)

|Z1| + |Z

2| + |Z

3| = 2 + 0 + 3 = 5

6. ,d dkcZfud ;kSfxd (C8H

10O

2) lery èkzqfor izdk'k dks ?kwf.kZr dj nsrk gSA ;g mnklhu FeCl

3 foy;u ds lkFk xqykch jax

nsrk gSA bl ;kSfxd ds fy, lHkh laHko leko;fo;ksa dh dqy la[;k D;k gS\

mÙkj (6)


19

gy %gy %gy %gy %gy %

OH

C* — CH3

OH

H

OH

C* — CH3

OH

HOH

HC* — CH3

OH

iQhuksfyd lewg mnklhu FeCl3 ds lkFk /ukRed ijh{k.k nsrk gSA



çR;sd ç'u ds pkjpkjpkjpkjpkj fodYi gSaA bu pkj fodYiksa eas ls ,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 lgh mÙkj gS@gSaA

çR;sd ç'u ds fy, (lHkh) lgh mÙkjksa ls lacaf/r fodYi@fodYiksa dk p;u dhft;sA

çR;sd ç'u ds mÙkj dk ewY;kadu uhps nh xbZ vadu ;kstuk ds vuqlkj fd;k tk,xk:

iw.kZ vad : +4 ;fn dsoy lHkh lgh fodYi@fodYiksa dk p;u fd;k x;k gS_

vakf'kd vad : +3 ;fn lHkh pkj fodYi lgh gSa ijUrq dsoy rhu fodYiksa dk p;u fd;k x;k gS_

vakf'kd vad : +2 ;fn rhu ;k vf/d fodYi lgh gSa ijUrq dsoy nks fodYiksa dk p;u fd;k x;k gS rFkk nksuksa lghgks_

vkaf'kd vad : +1 ;fn nks ;k vf/d fodYi lgh gSa ijUrq dsoy ,d fodYi dk p;u fd;k x;k gS rFkk og lghfodYi gks_

'kwU; vad : 0 ;fn fdlh Hkh fodYi dk p;u ugha fd;k x;k gSA (vFkkZr~ ç'u dk mÙkj ugha fn;k x;k gks);

½.kkRed vad : –2 lHkh vU; fLFkfr;ksa esaA

7. ,d iz;ksx eas ,d ik=k esa fy, x, m xzke ;kSfxd X (xSl/nzo/Bksl) dks uhps fp=k I esa n'kkZ, vuqlkj ,d rqyk ij j[kk tkrk gSApqEcdh; {ks=k dh mifLFkfr esa X okyk iyM+k ;kSfxd X ij fuHkZjrk ds vkèkkj ij ;k rks Åij dh (fp=k II) ;k uhps dh vksj(fp=k III) fopfyr gksrk gSA lgh dFku@dFkuksa dks igpkfu;sA

pqEcd

Nm x

S SN

(I)

larqfyrpqEcdh;

{ks=k vuqifLFkr

(II)

Åij dh vksjfopyu pqEcdh;{ks=k mifLFkr

(III)

uhps dh vksjfopyu pqEcdh;{ks=k mifLFkr

(A) ;fn X, H2O(l) gks rks iyMs+ dk fopyu Åij dh vksj gksxk

(B) ;fn X, K4[Fe(CN)

6](s), gks rks iyMs+ dk fopyu Åij dh vksj gksxk

(C) ;fn X, O2 (g) gks rks iyMs+ dk fopyu uhps dh vksj gksxk

(D) ;fn X, C6H

6(l) gks rks iyMs+ dk fopyu uhps dh vksj gksxk

mÙkj (A,B,C)


20

gy gy gy gy gy : H2O(l), K

4[Fe(CN)

6](s) ,oa C

6H

6(l) çfrpqEcdh; ;kSfxd gSA ;s ;kSfxd pqEcdh; {ks=k ls Åij dh vksj fopfyr gksxs

O2(g) vuqpqEcdh; gS ;g pqEcdh; {ks=k dh vksj (uhps dh vksj) vkd£"kr gksxhA

8. nh x;h vfHkfØ;k ds fy, fuEu esa ls dkSuls vkjs[k lgh gS@gSa\

(P dh izkjfEHkd lkanzrk [P]0 gS ) H C

3Br

CH3

CH3

+ NaOH H C3

OH

CH3

CH3

+ NaBr

P Q

(A)

[P]0

t1/2

(B)

[P]0

izkjfE

Hkd o

sx

(C)

le;

0

[Q]

[P]

(D)

le;

0

[P]ln

[P]

mÙkj (A)

gy gy gy gy gy : osx [fØ;k/kj]1 (SN

1 vfHkfØ;k ds fy,)

osx = k[[P]0 – X]

vfHkfØ;k dh dksfV = 1

izFke dksfV vfHkfØ;k ds fy, Q = [P]0(1 – e–kt)

t1/2

izkjfEHkd lkanzrk ij fuHkZj ugha djrkA


21

9. ckWDlkbV ls ,yqfefu;e ds fu"d"kZ.k ds fy, fuEu esa ls dkSulk@dkSuls dFku gS@gSa\

(A) tc lksfM;e ,yqfeusV ds foy;u esa CO2 izokfgr dh tkrh gS] rks ty;ksftr Al

2O

3 vo{ksfir gksrk gS

(B) Na3AlF

6 feykus ij ,yqfeuk dk xyukad de gks tkrk gS

(C) fo|qr vi?kVu ds nkSjku ,uksM ij CO2 eqDr gksrh gS

(D) dSFkksM ,d LVhy dk ik=k gksrk gS ftlesa dkcZu dk vLrj yxk gksrk gS

mÙkj (A,B,C,D)

gy gy gy gy gy : Al2O

3 + 2NaOH + 3H

2O 2Na[Al(OH)

4]

2Na[Al(OH)4] + 2CO

2 Al

2O

3xH

2O + 2NaHCO

3

Al2O

3xH

2O 1473K

Al2O

3(s) + xH

2O

10. fuEufyf[kr esa ls lgh dFku@dFkuksa dk p;u dhft,A

(A) SnCl2.2H

2O ,d vipk;d gS

(B) SnO2 rFkk KOH fØ;k djds K

2[Sn(OH)

6] cukrs gS

(C) HCl esa PbCl2 ds foy;u esa Pb2+ rFkk Cl– vk;u gksrs gS

(D) Pb3O

4 dh xeZ o ruq ukbfVªd vEYk ds lkFk vfHkfØ;k ij PbO

2 izkIr gksrk gS tks ,d fjMkWDl vfHkfØ;k gS

mÙkj (A,B)

gy gy gy gy gy : LFkkf;Ro Øe Sn4+ > Sn2+ ...

Sn2+ vipk;d ds :i esa dk;Z djsxk

SnO2 + 2KOH + 2H

2O K

2[Sn(OH)

6]

lkanz HCl esa PbCl2 DyksjksIyEcl vEy ds :i esa ik;k tkrk gS] H

2[PbCl

4] PbCl

2 + 2HCl H

2[PbCl

4]

Pb3O

4 + 4HNO

3 2Pb(NO

3)2 + PbO

2 + 2H

2O

(2PbO + PbO2)

11. uhps fn, x, pkj ;kSfxdksa I, II, III rFkk IV ij fopkj dhft,A

N

H C3

CH3

NH2

NH2

NO2

O N2

NO2

N

NO2

O N2

NO2

H C3

CH3

I II

III IV


22

lgh dFku@dFkuksa dk p;u dhft,A

(A) {kkjdrk dk Øe II > I > III > IV gS

(B) I rFkk II ds eè; pKb ds ifjek.k dk vUrj III rFkk IV ds eè; pK

b ds ifjek.k ds vUrj ls vfèkd gSS

(C) III esa vuqukn izHkko IV eas vuqukn izHkko ls vfèkd gS

(D) f=kfoe izHkko ;kSfxd IV dks III dh vis{kk vfèkd {kkjh; cuk nsrk gS

mÙkj (C,D)

gy gy gy gy gy : {kkjdrk dk Øe: II > I > IV > III

IV esa f=kfoe izHkko ds dkj.k–N(Me)2 ry dss ckgj dh vksj gksxk

blfy, oy; ds lkFk N dk vuqukn ?kVsxk rFkk {kkjh; izdfr c<+sxhA

12. ;kSfxd P ds fuEufyf[kr :ikUrj.kksa ij fopkj dhft,

R( )izdkf'kd lfØ;

(i) NaNH2

(ii) C H COCH36 5

(iii) H O /3

+

P(C H )

9 12

(i) X ( )vfHkdeZd

(ii) KMnO /H SO /4 2 4

Q(C H O )

8 12 6

( )izdkf'kd lfØ; vEy

CH3

Pt/H2

lgh fodYi dk p;u dhft,A

(A) P; gS

(B) X, Pd-C/fDouksyhu/H2 gS

(C) P; gS

(D) R; gS

mÙkj (B,C)

Sol. P

X Pd-C/fDoukWy/H2

R

*

O


23

(P)

X

(ii) KMnO /H SO /4 2 4

COOHCOOH

* COOH

(Q)


bl [k.M esa N% (06) ç'u gSaA çR;sd ç'u dk mÙkj ,d vkafdd eku gSA


djus ds LFkku ij gh çfo"V dhft;sA ;fn vakfdd eku esa nks ls vf/d n'keyo LFkku gks] rks çkIr eku dk nks n'keyo LFkku rd

lfUudVu djsaA


iw.kZ vad : +4 ;fn dsoy lgh vakfdd eku dks çfo"V fd;k x;k gS_

'kwU; vad : 0 vU; lHkh fLFkfr;ksa esaA

13. 0.1 M nqcZYk {kkj (B) ds foy;u dks izcy vEy (HA) ds 0.1 M foy;u ds lkFk vuqekfir fd;k tkrk gSA feyk, x, vEy

HA ds vk;ru ds lkFk foy;u dh pH esa ifjorZu dks uhps fp=k n'kkZ;k x;k gSA {kkj dk pKb D;k gSaA mnklhuhdj.k vfHkfØ;k

B + HA BH+ + A– gSA HA

12

10

8

6

4

2

0 2 4 6 8 10 12

pH

HA (mL)dk vk;ru

mÙkj (3)

gy gy gy gy gy : B + HA BH+ + A–

rqY;rk fcUnq rd iz;qDr HA dk vk;ru = 6 mL (fn;s x, vkjs[k ls)

vèkZ rqY;rk fcUnq ij foy;u B rFkk BH+ okyk {kkjh; ciQj gSA

b

[BH ]pOH pK log

[B]

∵


24

vèkZ rqY;rk fcUnw ij : [BH+] = [B]

pOH = pKb = 14 – 11 = 3

bpK 3

14. nzo A rFkk B, 25°C ij A rFkk B ds lHkh la?kVu ij vkn'kZ foy;u cukrs gSaA A ds 0.25 rFkk 0.50 eksy izHkkt okys ,sls nksfoy;uksa dk dqy ok"i nkc Øe'k% 0.3 rFkk 0.4 ckj gSa] 'kq¼ nzo B dk ckj esa ck"i nkc D;k gS\

mÙkj (0.2)

gy gy gy gy gy : jkÅYV fu;e ds mi;ksx ls

o o

T A A B BP x P x P

o o

A B0.3 0.25 P 0.75 P ... (i)

o o

A B0.4 0.5 P 0.5 P ... (ii)

(i) o (ii) ls

o

AP 0.6 bar

o

BP 0.2 bar

Ans 0.20

15. uhps fn;k x;k fp=k fLFkfrt ÅtkZ rFkk bysDVªkWuh; vkè;koLFkk esa H2 v.kq dh vUrj ukfHkdh; nwjh ds eè; vkjs[k dks n'kkZrk

gSAd = d0 ds fy, usV fLFkfrt ÅtkZ E

0 (fp=k esa fufn"V) dk eku kJ mol–1 esa D;k gS tgk¡ bysDVªkWu&bysDVªkWu izfrd"kZ.k rFkk

ukfHkd&ukfHkd izfrd"kZ.k ÅtkZ vuqifLFkr gSA lanHkZ ds vuqlkj tc bysDVªkWu rFkk ukfHkd vuUr :i ls cgqr nwj gks] rks Hijek.kq dh fLFkfrt ÅtkZ 'kwU; yh tkrh gS

vkoksxknzks la[;k 6.023 × 1023 mol–1 gS

H Hd

d0

vUrjukfHkdh; nwjh (d)

fLFkfr

t] ÅtkZ

(kJ

mo

l)

–1

E0

mÙkj (–5242.42)


25

gy gy gy gy gy : tc bysDVªkWu rFkk ukfHkd vuUr nwjh ij gks rks H-ijek.kq dh fLFkfrt ÅtkZ 'kwU; ekuh tkrh gS (nh x;h gSZ).

vkè;koLFkk esa bysDVªkWu okys H-ijek.kq dh fLFkfrt ÅtkZ

= – 27.2 eV (cksgj izfr:i ls)

vUrj ukfHkdh; nwjh 'd0 ij bysDVªkWu&bysDVªkWu izfrd"kZ.k rFkk ukfHkd&ukfHkd izfrd"kZ.k vuqifLFkr gksrs gSA

nks H-ijek.kq dh fLFkfrt ÅtkZ = – 2 × 27.2 eV = – 54.4 eV

=

19 2354.4 1.6 10 6.023 10

kJ/mol1000

= – 5242.42 kJ/mol

16. uhps n'kkZ;h x;h P ls Q rd vfHkfØ;k vuqØe ij fopkj dhft,A P ls izkIr eq[; mRikn Q dh lEiw.kZ yfCèk 75% gSA

9.3 mL P ls izkIr Q dh xzke esa ek=kk D;k gS\ (P dk ?kuRo = 1.00 g mL–1; eksyj nzO;eku : C = 12.0, H = 1.0,

O = 16.0 rFkk N = 14.0 g mol–1)

NH2

P

(i) NaNO + HCl/0-5°C2

OHQ

+ NaOH(ii)

(iii) CH CO H/H O3 2 2

mÙkj (18.6)

gy gy gy gy gy :

NH2

N

N2Cl

(P)

(Q)

(i) NaNO + HCl/0-5°C 2

+ –

(ii) OH

,

N = N

O

CH CO H/H O3 2 2

OH

NNaOH

M.M. = 248

9.3 mL P 9.3 g P

P ds eksy = 9.30.1

93

izkIr Q ds eksy = 0.1 × 0.75 = 0.075

izkIr Q dk nzO;eku = 248 × 0.1 × 3

4 = 18.6 g


26

17. dSflVsjkbV ls fVu dks dksd ds lkFk vip;u }kjk izkIr fd;k tkrk gSA uhps fn, x, vkadM+ks ls ml U;wure rki (K esa) dk

fuèkkZj.k dhft, ftl ij dksd }kjk dSflVsjkbM dk vip;u gksxk

298 K ij: fH°(SnO

2(s)) = –581.0 kJ mol–1,

fH°(CO

2(g)) = –394.0 kJ mol–1,

S°(SnO2(s)) = 56.0 J K–1 mol–1, S°(Sn(s)) = 52.0 J K–1 mol–1,

S°(C(s)) = 6.0 J K–1 mol–1, S°(CO2(g)) = 210.0 J K–1 mol–1.

ekuk ,UFkSYih rFkk ,UVªkWih rki ij fuHkZj ugha djrh

mÙkj (935)

gy gy gy gy gy : SnO2(s) + C(s) Sn(s) + CO

2(g)

o

rH 394 581 187 kJ/mol

o

rS 52 210 56 6 200 J/K mol

Tmin

(ftl ij vip;u gksxk) =

o

r

o

r

H935K

S

18. 0.05 M Zn2+ dk vEyhdr foy;u 0.1 M H2S }kjk larIr gSA ZnS ds vo{ksi.k dks jksdus ds fy, vko';d H+ dh U;wure

eksyj lkanzrk (M) D;k gS\

Ksp

(ZnS) = 1.25 × 10–22, H2S dk lEiw.kZ fo;kstu fu;rkad K

usV = K

1K

2= 1 × 10–21.

mÙkj (0.2)

gy gy gy gy gy : ZnS(s) 2 2Zn (aq) S (aq)

[Zn2+] = 0.05 M

[S2–]max

(vo{ksi.k jksdus ds fy,) = sp

2

K (ZnS)

[Zn ]

=

22211.25 10

2.5 10 M0.05

H2S 2H+ + S2–

2 2

1 2

2

[H ] [S ]K K

[H S]

2 2121 [H ] 2.5 10

1 100.1

2 1[H ]

25

1[H ] 0.2 M

5

Ans 0.20


27

PART-III : MATHEMATICS

[ k.M - 1 (vf/ dre vad : 18)

bl [ k.M esa N% (06) ç' u gSaA

i zR; sd i z' u dk mÙkj 0 l s 9 (nksuksa l fEefyr ) rd dh i jkl esa , dy vadh; i w.kk±d gS

çR; sd ç' u ds fy, ] mÙkj ds l gh vkafdd eku dks ekÅl rFkk vkWu&LØhu opqZvy U; weSfjd dh–i SM ds mi ; ksx l s mÙkj çfo"Vdjus ds LFkku i j gh çfo"V dhft ; sA

çR; sd ç' u ds mÙkj dk ewY; kadu uhps nh xbZ vadu ; kst uk ds vuql kj fd; k t k; sxk:

i w.kZ vad : +3 ; fn dsoy l gh i w.kk±d eku dks çfo"V fd; k x; k gS_

' kwU; vad : 0 ; fn i z' u dk mÙkj ugha fn; k x; k gS

½.kkRed vad : –1 vU; l Hkh fLFkfr ; ksa esaA

1. , d l fEeJ l a[ ; k z ds fy, ] ekuk Re(z), z ds okLr fod Hkkx dks n'kkZrk gSA ekuk S l Hkh l fEeJ l a[ ; kvksa z dk l eqPp;

gS t ks z4 – |z|4 = 4i z2 dks l arq"V djrh gS] t gk¡ i 1 , r c |z1 – z2|2 dk U; wure l aHkkfor eku] t gk¡ z1, z2 S

rFkk Re(z1) > 0 ,oa Re(z2) < 0] gS_____

mÙkj (8)

gy% ekuk z2 = a + ib

z4 – |z|4 = 4iz2

(a2 – b2) + 2abi – (a2 + b2) = 4i(a + ib)

b2 = 2b rFkk ab = 2a

vr% z2 = 0 ;k z2 = + 2i, R

ekuk z = x + iy

x2 – y2 + 2xy = + 2i

xy = 1

z dk fcanqi Fk ,d l edks.kh; vfr i joy; xy = 1 gS

; fn Re(z1) > 0 rFkk Re(z2) < 0, rc

z1 o z2 ds eè; U; wure nwjh] bl vfr i joy; ds nks ' kh"kksZa dks feykus okyk js[ kk[ k.M gSA

2 21 2 min

z z (1 1) (1 1) 8

21 2 minz z 8

2. , d fel kbZy }kjk ,d y{; i j l i Qyrki woZd Vdjkus dh i zkf; drk 0.75 gSA y{; dks i w.kZr ; k u"V djus ds fy, fel kbZy ds

de l s de rhu ckj l i Qyrki woZd Vdjkus dh vko' ; drk gSA rc U;wure fdruh fel kbZy nkxh t k,a r kfd y{; dks i w.kZr ; k

u"V djus dh i zkf; drk 0.95 l s de u gks _____

mÙkj (6)


28

gy% y{; i j Vdjkus dh i zkf; drk 3H

4

y{; l s pwd t kus dh i zkf; drk 1M

4

vHkh"V i zkf;drk de l s de 3 fel kbysa Vdjkrh gSa

; gk¡ 1 – (P(x = 0) + P(x = 1) + P(x = 2)) 0.9

; gk¡ P(x = r) n'kkZrk gS fd r fel kbysa Vdjkrh gSaA

; gk¡ n n 1 2 n 2

n n n0 1 2

1 3 1 3 11 C C C 0.95

4 4 4 4 4

n

2 2n 3 9n(n 1) 1202 4

...(1)

n ds fofHkUu eku j[ kus i j]

n dk y?kqÙke eku 6 i zkIr gksrk gSA

n = 5 ds fy, , l ehdj.k (1) dk L.H.S. gS 2120.10703125

2 1024

n = 6 ds fy, , l ehdj.k (1) dk L.H.S. gS 308 0.0750.037

2 4096 2

; gk¡ n = 6 fel kbysa nkxh t krh gSaA

3. ekuk O oÙk x2 + y2 = r2 dk dsUnz gS, t gk¡ 5

r2

, ekuk PQ bl oÙk dh t hok gS rFkk P o Q l s xqt jus okyh js[ kk dk

l ehdj.k 2x + 4y = 5 gSA ; fn f=kHkqt OPQ ds i fjoÙk dk dsUnz] js[ kk x + 2y = 4 i j fLFkr gS, rc r dk eku gS _____

mÙkj (2)

gy% P

O

Q

R(x , y )1 1M

2x + 4y = 5

OPQ dk i fjoÙk] OR dk eè; fcanq]

O ds funsZ' kkad = (0, 0)

ekuk R (x1, y1) gS] rc PQ dk l ehdj.k gS

xx1 + yy1 = r2

2x + 4y = 5 ds l kFk rqyuk djus i j , ; gk¡ :

21 1x y r

2 4 5


29

2 2

1 12r 4r

R (x ,y ) ,5 5

i fjdsUnz 2 2r 2r

M ,5 5

M, x + 2y = 4 i j fLFkr gS

2 2r 4r

45 5

r2 = 4

r = 2

4. , d oxZ vkO; wg dk vuqjs[ k bl ds fod.kZ vo; oksa ds ; ksxi Qy ds : i esa i fjHkkf"kr gSA ; fn A , d 2 × 2 vkO; wg gS t cfd A

dk vuqjs[ k 3 gS rFkk A3 dk vuqjs[ k –18 gS, r c A ds l kjf.kd dk eku gS

mÙkj (5)

gy% ekuk 2x yA A 3x x yz

z 3 x

vc

2 x y x y

Az 3 x z 3 x

2 2

2 2

xy 3y xy 3yx yz x yz

xz 3z xz 3zyz 9 x 6x yz 9 x 6x

2

32

3y x yx yzA

3z z 3 xyz 9 x 6x

3 2 2

2 2 2 3 2

x xyz 3yz x y y z 9y 3xy

3xz yz 9z x z 6xz 6yz 27 3x 18x xyz 9x x 6x

vuqjs[ k 3 2 3 2x xyz 3yz 3yz 27 3x 18x xyz 9x x 6x 18

9yz + 9x2 – 27x + 27 = –18

yz + x2 – 3x + 3 = –2

x2 – 3x + yz = –5

3x – x2 – yz = 5

|A| = 5

5. ekuk i Qyu f : (–1, 1) rFkk g : (–1, 1) (– 1, 1) ,

f(x) = |2x – 1| + |2x + 1| rFkk g(x) = x - [x] }kjk i fjHkkf"kr gSA

t gk¡ [x], x l s NksVs ; k cjkcj egÙke i w.kkZad dks n' kkZrk gSA ekuk f o g: (–1, 1) , (f o g) (x) = f(g(x)) }kjk i fjHkkf"kr

l a; qDr i Qyu gSA ekuk c vUr jky (–1, 1) esa fcanqvksa dh l a[ ; k gS t gk¡ f o g l r r ugha gS rFkk ekuk d vUr jky (–1, 1) esa

fcanqvksa dh l a[ ; k gS t gk¡ f o g vodyuh; ugha gSA rc c + d dk eku gS _____

mÙkj (4)


30

gy% ; gk¡ f(x) = |2x – 1| + |2x + 1|

(fog)(x) = |2{x} – 1| + |2{x} + 1|

12x 1 2x 3 , x 1,

2

12x 1 2x 3 , x , 0

2fog x

12x 1 2x 1, x 0,

2

12x 1 2x 1, x , 1

2

12, x 1,

2

14x 4, x , 0

2fog (x)

12, x 0,

2

14x, x , 1

2

f(g(x)), x = 0 i j vl r r gS

rFkk

10, x 1,

2

14, x ,0

2fog x

10, x 0,

2

14, x ,1

2

f(g(x)), 1 1

x , 0,2 2

i j vuvodyuh; gS

c = 1 rFkk d = 3.

c + d = 4

6. l hek

x2

4 2 sin3x sinxlim

3x 5x 3x2sin2xsin cos 2 2 cos2x cos

2 2 2

dk eku gS_____

mÙkj (8)

gy%x

2

4 2(2sin2x cos x)lim

x 7x 5x 3xcox – cos cos – cos – 2(1 cos2x)

2 2 2 2


31

2

2x2

16 2 cos x sinxlim

3x 5x2cos .cos x – 2cox .cosx – 2 2 cos x

2 2

2

2x2

16 2 cos x sinxlim

x2cosx 2sin2x sin – 2 2 cos x

2

x

2

16 2 sinx 16 2lim

x 4 2 – 2 28sinx sin – 2 22

= 8

[ k.M - 2 (vf/ dre vad : 24)

bl [ k.M esa N% (06) ç' u gSaA

çR; sd ç' u ds pkj fodYi gSaA bu pkj fodYi ksa eas l s ,d ;k ,d l s vf/ d l gh mÙkj gS@gSaA

çR;sd ç'u ds fy, (l Hkh) l gh mÙkjksa l s l acaf/ r fodYi @fodYi ksa dk p; u dhft ; sA

çR; sd ç' u ds mÙkj dk ewY; kadu uhps nh xbZ vadu ; kst uk ds vuql kj fd; k t k, xk:

i w.kZ vad : +4 ; fn dsoy l Hkh l gh fodYi @fodYi ksa dk p; u fd; k x; k gS_

vakf' kd vad : +3 ; fn l Hkh pkj fodYi l gh gSa i jUrq dsoy rhu fodYi ksa dk p; u fd; k x; k gS_

vakf' kd vad : +2 ; fn rhu ; k vf/ d fodYi l gh gSa i jUrq dsoy nks fodYi ksa dk p; u fd; k x; k gS rFkk nksuksa l ghgks_

vkaf' kd vad : +1 ; fn nks ; k vf/ d fodYi l gh gSa i jUrq dsoy ,d fodYi dk p; u fd; k x; k gS rFkk og l ghfodYi gks_

' kwU; vad : 0 ; fn fdl h Hkh fodYi dk p; u ugha fd; k x; k gSA (vFkkZr~ ç' u dk mÙkj ugha fn; k x; k gks);

½.kkRed vad : –2 vU; l Hkh fLFkfr ; ksa esaA

7. ekuk b , d v&' kwU; okLr fod l a[ ; k gSA ekuk f : , d vodyuh; i Qyu gS t cfd f(0) = 1, ; fn f dk vodyt f

l ehdj.k

2 2

f(x)f (x)

b x

dks l Hkh x ds fy, l arq"V djrk gS, r c fuEu esa l s dkSul k dFku l R; gS@gSa\

(A) ; fn b > 0, rc f ,d o/ Zeku i Qyu gS

(B) ; fn b < 0, rc f ,d gzkl eku i Qyu gS

(C) l Hkh x ds fy, f(x)f(–x) = 1

(D) l Hkh x ds fy, f(x) – f(–x) = 0

mÙkj (A, C)

gy% 2 2

f x 1f x b x

2 2

f (x) dxdx c

f(x) b x


32

ln 11 xf(x) tan c

b b

( f(0) = 1 c = 0)

11 x

tanb bf(x) e

vr% 11 x

tanb bf(x) e

; k

11 xtan

b bf(x) e

ysfdu f(0) = 1, bl fy, 11 x

tanb bf(x) e

f(x) > 0, b ds l Hkh ekuksa ds fy,

8. ekuk a o b / ukRed okLr fod l a[ ; k, a gSa t cfd a > 1 r Fkk b < a, ekuk P i zFke pr qFkkZa' k esa , d fcanq gS t ks

vfr i joy; 2 2

2 2

x y– 1

a b i j fLFkr gSA ekuk P i j vfr i joy; dh Li ' kZ js[ kk fcanq (1, 0) l s xqt jrh gS rFkk ekuk P i j vfr i joy;

dk vfHkyEc funsZ' kkad v{kksa i j l eku vUr%[ k.M dkVrk gSA ekuk P i j Li ' kZ js[ kk] P i j vfHkyEc ,oa x-v{k }kjk fufeZrf=kHkqt dk {ks=ki Qy gSA ; fn e vfr i joy; dh mRdsUnzrk gS] rc fuEu esa l s dkSul k@l s dFku l R; gS@gSa\

(A) 1 e 2 (B) 2 e 2

(C) 4a (D) 4b

mÙkj (A, D)

gy%

P(asec , btan )

(1, 0)A 2 2a b ,0

B

ekuk P (asec, btan)

x yT : sec tan 1

a b

(1, 0), Li ' kZ js[ kk i j fLFkr gS] bl fy, a = sec

2 2ax byN: a b

sec tan

vfHkyEc dh i zo.krk –1 gS, vr% a

sin 1b

b = tan

vr%, a2 – b2 = 1

2 2

2 2 2

b a 1 1e 1 1 2

a a a

D;ksafd a > 1, bl fy, e 1, 2

PAB dk {ks=ki Qy 2 21a b 1 .b tan

2

2 2 4 412tan tan tan b

2


33

9. ekuk i Qyu f : rFkk g:

f(x + y) = f(x) + f(y) + f(x)f(y) rFkk f(x) = xg(x)

dks l Hkh x, y ds fy, l arq"V djr s gSaA ; fn

x 0lim g(x) 1, r c fuEu esa l s dkSul k@l s dFku l R; gS@gSa\

(A) f i zR; sd x i j vodyuh; gS

(B) ; fn g(0) = 1, rc g i zR; sd x i j vodyuh; gS

(C) vodyt f(1), 1 ds cjkcj gS

(D) vodyt f(0), 1 ds cjkcj gS

mÙkj (A, B, D)

gy%

h 0

f(x h) f(x)f (x) lim

h

h 0

f(x) f(h) f(x) f(h) f(x)f (x) lim

h

h 0

f(h)f (x) lim (1 f(x))

h

f (x) 1 f(x)

f (x)

11 f(x)

ln(1 + f(x)) = x + c 1 + f(x) = ex + c ( f(0) = 0 c = 0) f(x) = ex – 1 f(x) = ex

10. ekuk , , , okLr fod l a[ ; k, a gSa t cfd 2 + 2 + 2 0 r Fkk + = 1, ekuk fcanq (3, 2, –1) l er yx + y + z = ds l ki s{k fcanq (1, 0, –1) dk ni Z.k i zfr fcEc gSA rc fuEu esa l s dkSul k@l s dFku l R; gS@gSa\(A) + = 2 (B) – = 3

(C) + = 4 (D) + + =

mÙkj (A, B, C)

gy% P(3, 2, –1) o Q(1, 0, –1), ; fn P dk l ery ds l ki s{k i zfr fcEc Q gSA rc PQ , l ery i j vfHkyEc gS rFkk PQ dk

eè; fcanq l ery i j fLFkr gSA: 1 (x – 2) + 1 (y – 1) + 0 (z + 1) = 0 : x + y = 3

Li "Vr% = 0, vr% = 1 (D;ksafd + = 1)

rc = 1 rFkk = 3

11. ekuk a o b / ukRed okLr fod l a[ ; k,a gSaA ekuk ˆ ˆPQ ai bj

rFkk ˆ ˆPS ai – bj

, d l ekUr j prqHkqZt PQRS dh vkl Uu

Hkqt k, ¡ gSaA ekuk u o v

, Øe'k% P Q P So ds vuqfn'k ˆ ˆw i j

ds i z{ksi l fn' k gSaA ; fn u v w

rFkk ; fn l ekUr j

prqHkqZt PQRS dk {ks=ki Qy 8 gS, r c fuEu esa l s dkSul k@l s dFku l gh gS@gSa\(A) a + b = 4

(B) a – b = 2

(C) l ekUr j prqHkqZt PQRS ds fod.kZ PR dh yEckbZ 4 gS

(D) w

, l fn' kksa PQ PSrFkk dk dks.k v¼Zd gS

mÙkj (A, C)


34

gy%

2 2 2 2

a b a – bv u

a b a b

rFkk

2 2

a b a – bv u w 2

a b

WLOG, a b

bl fy,

2 22a a b

a = b

rFkk PQRS dk {ks=ki Qy PQ PS 2ab 8

vr% a = b = 2, bl fy,

2 2PQ PS a – b 0

vr% PQRS, Hkqt k yackbZ2 2 okyk ,d oxZ gSA

12. v&½.kkRed i w.kkZadksa s o r ds fy, , ekuk

s! r s,

s r!(s –r)!

r0 r s.

; fn

; fn

/ ukRed i w.kkZadksa m o n ds fy, , ekuk

m n

p 0

f(m,n,p)g(m,n)

n p

p

t gk¡ fdl h v&½.kkRed i w.kkZad p ds fy, ,

p

i 0

m n i p nf(m,n,p) ,

i p p – i

r c fuEu esa l s dkSul k@l s dFku l R; gS@gSa\

(A) l Hkh / ukRed i w.kkZadksa m, n ds fy, g(m, n) = g(n, m)

(B) l Hkh / ukRed i w.kkZadksa m, n ds fy, g(m, n + 1) = g(m + 1, n)

(C) l Hkh / ukRed i w.kkZadksa m, n ds fy, g(2m, 2n) = 2g(m, n)

(D) l Hkh / ukRed i w.kkZadksa m, n ds fy, g(2m, 2n) = (g(m, n))2

mÙkj (A, B, D)

gy%

pm

ii 0

p nn if(m,n,p) C

p n i – p n i p – i

p

mi

i 0

n p 1f m,n,p C

p p – i n – (p – i)


35

p

m ni p–i

i 0

n pf m,n,p C C

n p

m n n p m np p p

n pf m,n,p C C C

n p

m nm n m n

pp 0

g(m,n) C 2

[ k.M - 3 (vf/ dre vad : 24)

bl [ k.M esa N% (06) ç' u gSaA çR; sd ç'u dk mÙkj , d vkafdd eku gSA

çR; sd ç' u ds fy, ] mÙkj ds l gh vkafdd eku dks ekÅl rFkk vkWu&LØhu opqZvy U; weSfjd dh–i SM ds mi ; ksx l s mÙkj çfo"Vdjus ds LFkku i j gh çfo"V dhft ; sA ; fn vakfdd eku esa nks l s vf/ d n' keyo LFkku gks] r ks çkIr eku dk nks n' keyo LFkku rdl fUudVu djsaA

çR; sd ç' u ds mÙkj dk ewY; kadu uhps nh xbZ vadu ; kst uk ds vuql kj fd; k t k; sxk:

i w.kZ vad : +4 ; fn dsoy l gh vakfdd eku dks çfo"V fd; k x; k gS_

' kwU; vad : 0 vU; l Hkh fLFkfr ; ksa esaA

13. , d vfHk; ark dks i zR; sd ekg ds i gys 15 fnuksa ds nkSjku Bhd pkj fnu dkj[ kkus dk Hkze.k djuk vko' ; d gS rFkk ; gvfuok; Z gS fd og yxkrkj nks fnu Hkze.k ugha djsxk] r c mu l Hkh l aHkkfor r jhdksa dh l a[ ; k ft uesa vfHk; ark 1-15 t wu2021 ds nkSjku dkj[ kkus esa Hkze.k dj l drk gks] gS _____

mÙkj (495)

gy% ekuk ;g 1st ckj Hkze.k i j x1 fnuksa ds i ' pkr t krk gS

2nd ckj Hkze.k i j x2 fnuksa ds i ' pkr t krk gS

3rd ckj Hkze.k i j x3 fnuksa ds i ' pkr t krk gS

4th ckj Hkze.k i j x4 fnuksa ds i ' pkr t krk gS

rFkk rc ' ks"k x5 fnu gS

x1 + x2 + x3 + x4 + x5 = 11

x1, x5 0 rFkk x2, x3, x4 1

gyksa dh l a[ ; k = 12C4 = 12 11 10 9

55.9 4954 3 2

14. , d gksVy esa] pkj dejs mi yC/ gSaA N% O; fDr bu pkj dejksa esa bl i zdkj j[ ks t krs gSa fd buesa l s i zR; sd dejs esa de l sde ,d o vf/ d l s vf/ d nks O; fDr gksaA rc , sl k dqy fdrus r jhdksa l s fd; k t k l drk gS\

mÙkj (1080)

gy% l ewg gksaxs dsoy 2, 2, 1, 1

O; fDr dk l ewg esa foHkkt u = 22 2

6!

(2!) (1!) 2!

bUgsa O; ofLFkr djus ds r jhdksa dh l a[ ; k = 4

6!4! 1080

(2!)


36

15. nks fu"i {k i kl s ft uesa l s i zR; sd ds i Qydksa i j 1,2,3,4,5 o 6 l a[ ; k, ¡ vafdr gSa, , d l kFk yq<+dk, t krs gSa rFkk i Qydksa i jvk;h gq; h l a[ ; kvksa dk ; ksxi Qy ; kn j[ kk t krk gSA ;g i zfØ; k r c rd nksgjkbZ t krh gS t c rd ; ksxi Qy ; k r ks ,d vHkkT;l a[ ; k ; k ,d i w.kZ oxZ i zkIr u gks t k,A ekuk ,d vHkkT; l a[ ; k i zkIr gksus l s i gys ,d i w.kZ oxZ i zkIr gksrk gSA ; fn bl i w.kZoxZ l a[ ; k ds ,d fo"ke l a[ ; k gksus dh i zkf;drk p gS] rc 14p dk eku gS _____

mÙkj (8)

gy% i fj.kke r jhdksa dh l a[ ; k

i w.kZ oxZ 1, 4, 9 7 (1 dHkh ugha vk,xk)

vHkkT; 2, 3, 5, 7, 11 15

' ks"k 6, 8, 10, 12 14

nh x; h i zfØ; k dk ,d i w.kZ oxZ }kjk l ekIr gksuk = E (ekuk ?kVuk dk uke)

rFkk i zfØ; k dk ,d fo"ke oxZ }kjk l ekIr gksuk = O (ekuk ?kVuk dk uke)

OP

E Kkr djus gsrq

P(O E)

P(E)

P(O E) = 1, 4, 2, 3, 5, 7, 11 l s i gys 9 vkus dh i zkf; drk

24 4 14 14 4......

36 36 36 36 36

P(E) = 2, 3, 5, 7, 11 l s i gys 4 ; k 9 vkus dh i zkf; drk

27 14 7 14 7......

36 36 36 36 36

4O 436P p

7E 736

14p = 8

16. ekuk i Qyu f : [0, 1] fuEu }kjk i fjHkkf"kr gSx

x

4f(x)

4 2

.

rc

1 2 3 39 1f f f ..... f f

40 40 40 40 2

dk eku gS _____

mÙkj (19)

gy% f(x) = x

x

4

4 2

x 1 x

x 1 x

4 4f x f 1 x 1

4 2 4 2

1 2 3 39 1

f f f .... f f40 40 40 40 2


37

= 1 39 2 38 1

f f f f .... f40 40 40 40 2

= 20 11 1 ... 19 f f

40 2

i n

= 19

17. ekuk f : , d vodyuh; i Qyu gS t cfd bl dk vodyt f' l r r gS r Fkk f()=–6, ; fn F : [0, ] ,x

0F(x) f(t)dt }kjk i fjHkkf"kr gS, rFkk ; fn

0

f (x) F(x) cosx dx 2

,

r c f(0) dk eku gS __________

mÙkj (4)

gy% F(x) = f(x) bl fy, F() = –6

0

f (x) F(x) cos x dx

=0 0

F (x)cosx dx F(x)cosx dx

= 0 0 0

F x cos x sinx.F x dx F x cos xdx

= 0 0 0

F F 0 sinx.F x cosx.F x dx F(x)cosx dx

= 6 – F(0)

6 – F(0) = 2

F(0) = 4 = f(0)

18. ekuk i Qyu f : (0, ) , 2 4f( ) (sin cos ) (sin cos ) }kjk i fjHkkf"kr gSA

ekuk i Qyu f dk i j LFkkuh; fufEu"B gS t c 1 r{ ,......, }, t gk¡ 0 < 1 < .... r < 1, rc 1 + .... + r dk eku gS_______

mÙkj (0.50)

gy% f() = (sin + cos)2 + (sin – cos)4

ekuk (sin + cos)2 = x

f() = x + (2 – x)2 = x2 – 3x + 4

23 7

f( ) x –2 4


38

f() y?kqÙke gS t c 3

x2

3

sin cos2

3

sin4 2

2

4 3 3

; k 5

12 12 ; k

vr%, 11

12 rFkk 2

512

1 212

= 0.50

Questions & Answers · 2020. 9. 28. · 1 Questions & Answers for JEE (Advanced)-2020 Time : 3 hrs....

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Transcript of Questions & Answers · 2020. 9. 28. · 1 Questions & Answers for JEE (Advanced)-2020 Time : 3 hrs....