Regd. Office : Aakash Tower, Plot No.-4, Sec-11, MLU ... · Regd. Office : Aakash Tower, Plot No.-4, Sec-11, MLU, Dwarka, New Delhi-110075 Ph.: 011-47623456 Fax : 011-47623472 funsZ'k@INSTRUCTIONS

(1)

Answers & Solutionsforforforforfor

JEE (Advanced)-2013

Time : 3 hrs. Max. Marks: 180

DATE : 02/06/2013

PAPER - 1 (Code - 0)

CODE

0

Regd. Office : Aakash Tower, Plot No.-4, Sec-11, MLU, Dwarka, New Delhi-110075Ph.: 011-47623456 Fax : 011-47623472

funs Z'k@funs Z'k@funs Z'k@funs Z'k@funs Z'k@INSTRUCTIONSiz'u&i=k dk izk:iiz'u&i=k dk izk:iiz'u&i=k dk izk:iiz'u&i=k dk izk:iiz'u&i=k dk izk:i

bl iz'u&i=k ds rhu Hkkx (HkkSfrd foKku] jlk;u foKku vkSj xf.kr) gSaA gj Hkkx ds rhu [k.M gSaA

[kaM[kaM[kaM[kaM[kaM 1 esa 10 cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u gSaA gj iz'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls dsoy ,ddsoy ,ddsoy ,ddsoy ,ddsoy ,d ghghghghgh

lghlghlghlghlgh gSA

[kaM[kaM[kaM[kaM[kaM 2 esa 5 cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u gSaA gj iz'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls ,d ,d ,d ,d ,d ;k vf/d;k vf/d;k vf/d;k vf/d;k vf/d

lghlghlghlghlgh gSaA

[kaM[kaM[kaM[kaM[kaM 3 esa 5 iz'u iz'u iz'u iz'u iz'u gSaA izR;sd iz'u dk mÙkj 0 ls 9 rd (nksuksa 'kkfey) ds chp dk ,dy vadh; iw.kk±d gSA

vadu ;kstukvadu ;kstukvadu ;kstukvadu ;kstukvadu ;kstuk

[kaM[kaM[kaM[kaM[kaM 1 esa gj iz'u esa dsoy lgh mÙkj okys cqycqys (BUBBLE) dks dkyk djus ij 2 vadvadvadvadvad vkSj dksbZ Hkh cqycqyk

dkyk ugha djus ij 'kwU;'kwU;'kwU;'kwU;'kwU; (0) vad vad vad vad vad iznku fd, tk;saxsA bl [kaM ds iz'uksa esa xyr mÙkj nsus ij dksbZ ½.kkRed½.kkRed½.kkRed½.kkRed½.kkRed

vad ugha vad ugha vad ugha vad ugha vad ugha fn;s tk;saxsA

[kaM [kaM [kaM [kaM [kaM 2 ds gj iz'u esa dsoy lgh mÙkjksa (mÙkj) okys lHkh lHkh lHkh lHkh lHkh cqycqyksa (cqycqys) dks dkyk djus ij 4 vadvadvadvadvad iznku

fd, tk;saxs vkSj dksbZ Hkh cqycqyk dkyk ugha djus ij 'kwU; 'kwU; 'kwU; 'kwU; 'kwU; (0) vad iznku fd, tk;saxsA vU; lHkh fLFkfr;ksa esa

½.kkRed ,d½.kkRed ,d½.kkRed ,d½.kkRed ,d½.kkRed ,d (–1) vad vad vad vad vad iznku fd;k tk;sxkA

[kaM [kaM [kaM [kaM [kaM 3 ds gj iz'u esa dsoy lgh mÙkj okys cqycqys dks dkyk djus ij 4 vad iznku fd;s tk;saxs vkSj dksbZ Hkh

cqycqyk dkyk ugha djus ij 'kwU; 'kwU; 'kwU; 'kwU; 'kwU; (0) vad vad vad vad vad iznku fd, tk;saxsA vU; lHkh fLFkfr;ksa esa ½.kkRed ,d ½.kkRed ,d ½.kkRed ,d ½.kkRed ,d ½.kkRed ,d (–1) vadvadvadvadvad

iznku fd;k tk;sxkA

(2)

PART–I : PHYSICS

[k.M[k.M[k.M[k.M[k.M - 1 : (dsoy (dsoy (dsoy (dsoy (dsoy ,dy lgh fodYi çdkj,dy lgh fodYi çdkj,dy lgh fodYi çdkj,dy lgh fodYi çdkj,dy lgh fodYi çdkj)))))

bl [k.M esa 10 cgqfodYi ç'u cgqfodYi ç'u cgqfodYi ç'u cgqfodYi ç'u cgqfodYi ç'u gSaA çR;sd ç'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls dsoy ,d dsoy ,d dsoy ,d dsoy ,d dsoy ,d lgh gSA

1. ,d cy] ⎡ ⎤

+⎢ ⎥+ +⎣ ⎦

2 2 3/2 2 2 3/2ˆ ˆ

( ) ( )x yK i j

x y x y (K ,d lfpr foek dk fLFkjkad gS)] ,d m nzO;eku ds d.k dks (a, 0) fcUnq

ls (0, a) fcUnq rd ,d a f=kT;k ds o`Ùkh; iFk ij ys tkrk gS] ftldk dsUnz x – y ry dk ewy fcUnq gSA bl cy }kjk fd;kx;k dk;Z fuEu gS%

(A)π2K

a (B)πK

a

(C)π

2K

a (D) 0

mÙkjmÙkjmÙkjmÙkjmÙkj (D)

ladsrladsrladsrladsrladsr :⎡ ⎤

= +⎢ ⎥+ +⎢ ⎥⎣ ⎦

2 2 3/2 2 2 3/2

ˆ ˆ( ) ( )

xi yjF kx y x y

= .dW F dx

⎡ ⎤+= ⎢ ⎥

+⎣ ⎦2 2 3/2( )

xdx ydydW kx y

ekuk x2 + y2 = r2

xdx + ydy = r2

⇒ = =3 2krdr kdW drr r

−⎡ ⎤= ⎢ ⎥⎣ ⎦

2

1

r

r

kWr

vc r1 = a, r2 = a ⇒ W = 0

2. nks le:ih vk;rkdkj xqVdksa dks n'kkZ;s fp=kkuqlkj nks foU;klksa-I vkSj II esa O;ofLFkr fd;k x;k gSA xqVdksa dh Å"ek pkydrk κo 2κ gSA nksuksa foU;klksa esa x-v{k ds nksuksa Nksjksa ij rkieku dk vUrj leku gSA foU;kl-I esa Å"ek dh ,d fuf'pr ek=kk xjeNksj ls BaMs Nksj rd vfHkxeu esa 9 s ysrh gSA foU;kl-II esa leku ek=kk dh Å"ek ds vfHkxeu ds fy, le; gS%

foU;kl-I foU;kl-II

κ 2κx

2κκ

(A) 2.0 s (B) 3.0 s(C) 4.5 s (D) 6.0 s

(3)

mÙkjmÙkjmÙkjmÙkjmÙkj (A)

ladsrladsrladsrladsrladsr : −=

+κ κ

1 2

2

T TQl ltA A

[fLFkfr (i) esa]

κ κ⎡ ⎤= − +⎢ ⎥⎣ ⎦1 2

2( )'

Q A AT Tt l l

⇒ =' 2

9tt

⇒ t' = 2 s3. nks vufHkfØ;k'khy ,d&ijek.kqd vkn'kZ xSlksa dk ijek.kq nzO;eku 2 : 3 ds vuqikr esa gSA tc budks ,d fLFkjrkih; crZu esa

ifjc¼ fd;k tkrk gS] rc buds vakf'kd nkcksa dk vuqikr 4 : 3 gSA buds ?kuRo dk vuqikr gS%

(A) 1 : 4 (B) 1 : 2(C) 6 : 9 (D) 8 : 9


ladsrladsrladsrladsrladsr :ρ

=RTPM

ρ= ×

ρ1 1 2

2 1 2

P MP M

⇒ρ

=ρ

1

2

4 33 2

⇒ρ

=ρ

1

2

89

4. ,d m nzO;eku ds d.k dks çkjfEHkd xfr u0 ls {kSfrt ls α dks.k ij ç{ksfir fd;k tkrk gSA ;g d.k ç{ksI; iFk ds mPprefcUnq ij ,d leku nzO;eku ds d.k ds lkFk iw.kZr% vçR;kLFk la?kêð djrk gS] tks fd Hkwry ls ÅèokZ/j fn'kk esa leku çkjfEHkdxfr u0 ls isQadk x;k FkkA la;qDr fudk; la?kêð ds rRdky ckn {kSfrt ls fuEu dks.k cuk,xk%

(A) π4 (B) π

+ α4

(C) πα–

2 (D) π2


ladsrladsrladsrladsrladsr : mPpre fcUnq ij igys d.k dh pky = u0cosα

α

u0cosαu0

Hu0

mPpre fcUnq ij nwljs d.k dh pky = −20 2u gH

vc, α

=2 20 sin

2uH

g⇒ nwljs d.k dh pky = u0cosα

∴ vfUre laosx = α + α0 0ˆ ˆcos cosmu i mu j

∴ dks.k = π4

(4)

5. ,d NksVh oLrq tks çkjEHk esa fojke voLFkk esa gS] çdk'k dh 100 ns dh ,d Lian dks iw.kZr;k vo'kksf"kr djrh gSA Lian dh'kfDr 30 mW gS o çdk'k dh xfr 3 × 108 ms–1 gSA oLrq dk vfUre laosx gS(A) 0.3 × 10–17 kg ms–1 (B) 1.0 × 10–17 kg ms–1

(C) 3.0 × 10–17 kg ms–1 (D) 9.0 × 10–17 kg ms–1

mÙkjmÙkjmÙkjmÙkjmÙkj (B)

ladsrladsrladsrladsrladsr :× × × ×

= = = =×

–3 –9–17 –1

830 10 100 10 10 kg ms

3 10E P tpC C

6. ,d ;ax f}&fLyV ç;ksx esa λ rjax&nSè;Z ds ,do.khZ çdk'k dk ç;ksx fd;k tkrk gSA ,sls fcUnq dk ftl ij çdk'k dh rhozrkf'k[kj rhozrk dh vk/h gS] iFkkUrj gS (iw.kk±d n ds inksa esa)

(A) λ+(2 1)

2n (B) λ

+(2 1)4

n

(C)λ

+(2 1)8

n (D)λ

+(2 1)16

n


ladsrladsrladsrladsrladsr : φ⎛ ⎞= ⎜ ⎟⎝ ⎠2cos

2I Ivf/dre

φ⎛ ⎞= ⎜ ⎟⎝ ⎠

21 cos2 2

⇒ cos φ = 0

⇒π π π π

φ =3 5 7, , ,

2 2 2 2

⇒λ λ λ

Δ =3 5, ,

4 4 4x

⇒λ

Δ = +(2 1)4

x n

7. ,d lery mÙky ysal ,d okLrfod çfrfcEc ysal ds 8 m ihNs cukrk gS tks fd oLrq ds vkdkj dk ,d&frgkbZ gSA ySal ds

vUnj çdk'k dh rjaxnSè;Z fuokZr dh rjaxnSè;Z ls 23 xquk gSA ysal ds xksyh; ofØr i"B dh oØrk f=kT;k gS%

(A) 1 m (B) 2 m(C) 3 m (D) 6 m

mÙkjmÙkjmÙkjmÙkjmÙkj (C)

ladsrladsrladsrladsrladsr :λ

μ = = =ν λ

32

fcf

ok;q

ekè;e

vc, ν = +8 m, − ν= = ⇒ = −

1 24m3

m uu

= −ν

1 1 1f u

= + =1 1 1 4

8 24 24ff = 6 m

=μ −1

Rf

= ⇒ =6 3 m0.5R R

(5)

8. ,d 2L yEckbZ o 2R f=kT;k ds eksVs {kSfrt rkj ds ,d fljs dks L yEckbZ o R f=kT;k okys ,d irys {kSfrt rkj ls osfYMaxds }kjk tksM+k x;k gSA bl O;oLFkk ds nksuksa fljksa ij cy yxkdj rkuk tkrk gSA irys o eksVs rkjksa esa nSè;Zo`f¼ dk vuqikrfuEu gS%(A) 0.25 (B) 0.50(C) 2.00 (D) 4.00


ladsrladsrladsrladsrladsr : cy leku gksxk

vc, =FLAY

= × =2

12

2

(2 ) 22

L RLR

9. ,d lery niZ.k ij vkifrr çdk'k fdj.k dh çxkeh fn'kk +1 ˆ ˆ( 3 )2

i j gSA ijkorZu ds ckn çxkeh fn'kk 1 ˆ ˆ( – 3 )2

i j gks

tkrh gSA fdj.k dk vkiru dks.k gS(A) 30° (B) 45°(C) 60° (D) 75°


ladsrladsrladsrladsrladsr :

⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠° − θ =

+

ˆ ˆ3 – 3.2 2

cos(180 2 ) ˆ ˆ3 – 32 2

i j i j

i j i j

θ θ

+i 32

j –i 32

180° – 2θ

^ j

−

− θ =

(1 3)4cos21

−− θ =

1cos22

θ =1cos22

θ = °2 60θ = 30°

10. ,d csyu dk O;kl ekius ds fy, 'kwU; =kqfV jfgr ,d o£u;j dSfyilZ dk mi;ksx gksrk gSA ekius ds nkSjku o£u;j iSekus dk'kwU;] eq[; iSekus ds 5.10 cm vkSj 5.15 cm ds chp esa ik;k tkrk gSA o£u;j iSekus ds 50 Hkkx 2.45 cm ds rqY; gSaA blo£u;j iSekus dk pkSchlok¡ (24th) Hkkx eq[; iSekus ds ,d Hkkx ls lVhd laikrh gksrk gSA csyu dk O;kl gS(A) 5.112 cm (B) 5.124 cm(C) 5.136 cm (D) 5.148 cm

mÙkjmÙkjmÙkjmÙkjmÙkj (B)ladsrladsrladsrladsrladsr : 1 MSD = 5.15 – 5.10 = 0.05 cm

1 VSD = 2.4550 = 0.049 cm

LC = 1 MSD – 1 VSD = 0.001 cm

iBu = 5.10 + L.C. × 24 = 5.10 + 0.024 = 5.124 cm

(6)

[k.M[k.M[k.M[k.M[k.M - 2 : (((((,d ;k vf/d lgh fodYi çdkj,d ;k vf/d lgh fodYi çdkj,d ;k vf/d lgh fodYi çdkj,d ;k vf/d lgh fodYi çdkj,d ;k vf/d lgh fodYi çdkj)))))

bl [k.M esa 5 cgqfodYi ç'u cgqfodYi ç'u cgqfodYi ç'u cgqfodYi ç'u cgqfodYi ç'u gSaA çR;sd ç'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls dsoy ,d dsoy ,d dsoy ,d dsoy ,d dsoy ,d ;k vf/dvf/dvf/dvf/dvf/d lgh gaSA

11. fp=k esa n'kkZ;s ifjiFk esa nks lekukUrj IysVksa okys la/kfj=kksa esa çR;sd dh /kfjrk C gSA çkjEHk esa fLop S1 dks nck;k tkrk gS rkfdla/kfj=k C1 iw.kZ :i ls vkosf'kr gks tk,A blds ckn S1 dks NksM+ fn;k tkrk gSA blds i'pkr~ la/kfj=k C2 dks vkosf'kr djus dsfy, fLop S2 dks nck;k tkrk gSA dqN le; ds ckn S2 dks NksM+ fn;k tkrk gS rFkk S3 dks nck;k tkrk gSA dqN le; ckn

S1

2V0

C1V0

S2 S3

C2

(A) C1 dh Åijh IysV ij 2CV0 vkos'k gS (B) C1 dh Åijh IysV ij CV0 vkos'k gS

(C) C2 dh Åijh IysV ij 'kwU; vkos'k gS (D) C2 dh Åijh IysV ij –CV0 vkos'k gS

mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)ladsrladsrladsrladsrladsr : tc S1 dks nck;k tkrk gS rFkk NksM+k tkrk gS

2V0 C1 V0

S2 S3

C2+–

+2CV0

tc S2 dks nck;k tkrk gS rFkk NksM+k tkrk gS

2V0 V0

S2 S3

C2+–

CV0

S1

+CV0

tc S3 dks nck;k tkrk gS

2V0 C1 V0C2+–

CV0 CV0

CV0

–+

12. ,d M nzO;eku rFkk Q /u vkos'k dk d.k] tks −= 11

ˆ4 msu i ds ,dleku osx ls xfr'khy gS] ,dleku fLFkj pqEcdh; {ks=k

esa ços'k djrk gSA ;g pqEcdh; {ks=k x-y ry ds vfHkyEcor~ gS rFkk bldk foLrkj {ks=k x = 0 ls x = L rd çR;sd y ds eku

ds fy, gSA bl pqEcdh; {ks=k dks ;g d.k 10 feyh lsd.M esa ikj dj nwljh vksj −= + 12

ˆ ˆ2( 3 )msu i j osx ls çdV gksrk

gSA lgh çdFku gS@gSa%

(A) pqEcdh; {ks=k –z fn'kk esa gSA(B) pqEcdh; {ks=k +z fn'kk esa gSA

(C) pqEcdh; {ks=k dk ifjek.k π50

3M

Q bdkbZ gS

(D) pqEcdh; {ks=k dk ifjek.k π100

3M

Q bdkbZ gS

(7)

mÙkjmÙkjmÙkjmÙkjmÙkj (A, C)

ladsrladsrladsrladsrladsr :θ θ

= =ω

MtqB

Li"Vr% θ = 30° = π6

C

F θ

x L=

⊕

ˆ2 j v

ˆ2 3i

ˆ4ix = 0−

π π π= = =

× × 3100 50

6 36 10 10M M MB

q Qq

B –z fn'kk esa gksxkA

13. nksuksa fljksa ij ifjc¼ {kSfrt rfur Mksjh ik¡poh xq.kko`fÙk lehdj.k y(x, t) = (0.01 m) sin [(62.8 m–1)x] cos[(628 s–1)t] }kjkdfEir gks jgh gSA ;fn π = 3.14 ekuk tk; rc fuEu çdFku lgh gS@gSa

(A) fuLianksa dh la[;k 5 gS

(B) Mksjh dh yEckbZ 0.25 m gS

(C) lkE;koLFkk ls Mksjh ds eè; fcUnq dk vf/dre foLFkkiu 0.01 m gS

(D) ewy vkofÙk 100 Hz gS

mÙkjmÙkjmÙkjmÙkjmÙkj (B, C)

ladsrladsrladsrladsrladsr : fuLiUnksa dh la[;k = 6, π

λ =2k =

×=

2 3.14 0.1 m62.8

yEckbZ = λ=

5 0.25m2

eè; fcUnq çLiUn gSA bldk vf/dre foLFkkiu = 0.01 m

f = ω

= =×

20 Hz2 2vl k l

14. ,d f=kT;k R o ?kuRo ρ okys Bksl xksyd dks ,d nzO;eku jfgr fLçax ds ,d fljs ls tksM+k x;k gSA bl fLçax dk cyfu;rkad k gSA fLçax ds nwljs fljs dks nwljs Bksl xksyd ls tksM+k x;k gS ftldh f=kT;k R o ?kuRo 3ρ gSA iw.kZ foU;kl dks 2ρ?kuRo ds nzo esa j[kk tkrk gS vkSj bldks lkE;koLFkk esa igq¡pus fn;k tkrk gSA lgh çdFku gS@gSa

(A) fLçax dh usV nSè;Zo`f¼ π ρ34

3R g

k gS

(B) fLçax dh usV nSè;Zo`f¼ π ρ383R g

k gS

(C) gYdk xksyd vakf'kd :i ls Mwck gqvk gS

(D) gYdk xksyd iw.kZ :i ls Mwck gqvk gS

mÙkjmÙkjmÙkjmÙkjmÙkj (A, D)

ladsrladsrladsrladsrladsr : lkE;koLFkk ij] Åijh xksys ds fy,

R ρ

R 3ρ

2ρ+ π ρ = π ρ3 34 4 (2 )

3 3kx R g R g

⇒π ρ

=34

3R gx

k

fudk; dks iw.kZr% Mqck;k tkrk gS pw¡fd dqy Hkkj = dqy mRIykou cy

(8)

15. nks R o 2R f=kT;k okys vpkyd Bksl xksydksa dks ftu ij Øe'k% ρ1 rFkk ρ2 ,dleku vk;ru vkos'k ?kuRo gS] ,d nwljs lsLi'kZ djrs gq, j[kk x;k gSA nksuksa xksydksa ds dsUnzksa ls xqtjrh gqbZ js[kk [khaph tkrh gSA bl js[kk ij NksVs xksyd ds dsUnz ls

2R nwjh ij usV fo|qr {ks=k 'kwU; gSA rc vuqikr ρρ

1

2 dk eku gks ldrk gS%

(A) –4 (B) −3225

(C)3225 (D) 4

mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)

ladsrladsrladsrladsrladsr : EP = 0

⇒ρ π ρ

=επε

31 2

200

4( )3

34 (2 )

R RR P' P

R 2R

2R⇒

ρ=

ρ1

24

=' 0PE

⇒ρ π −ρ π

=πε πε

3 31 2

2 20 0

4 4 (2 )3 3

4 (2 ) 4 (5 )

R R

R R

⇒ρ

= −ρ

1

2

3225

[k.M[k.M[k.M[k.M[k.M - 3 : (iw.kk±d eku lgh çdkj)(iw.kk±d eku lgh çdkj)(iw.kk±d eku lgh çdkj)(iw.kk±d eku lgh çdkj)(iw.kk±d eku lgh çdkj)

bl [k.M esa 5 iz'uiz'uiz'uiz'uiz'u gSaA izR;sd iz'u dk mÙkj 0 ls 9 rd (nksuksa 'kkfey) ds chp dk ,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 gSA

16. ,d m nzO;eku dk xksyd l1 yEckbZ dh Mksjh ls yVdk gqvk gSA bls ,d osx fn;k tkrk gS tks fd ÅèokZ/j ry esa ,d o`Ùkiwjk djkus ds fy, U;wure gSA vius mPpre fcUnq ij ;g xksyd nwljs m nzO;eku ds xksyd ls izR;kLFk la?kV~V djrk gSAnwljk xksyd l2 yEckbZ dh Mksjh ls yVdk gqvk gS rFkk izkjaHk esa fojkekoLFkk ij gSA nksuksa Mksfj;k¡ nzO;eku jfgr o vforkU; gSaA;fn la?kV~V ds ckn nwljs xksyd dks ,slh xfr izkIr gksrh gS tks fd ÅèokZ/j ry esa iw.kZ oÙk iwjk djus ds fy, U;wure gS] rc

1

2

ll dk vuqikr gS %

mÙkjmÙkjmÙkjmÙkjmÙkj (5)

ladsrladsrladsrladsrladsr : mPpre fcUnq ij igys xksyd dh pky = 1gl

leku nzO;eku dh oLrqvksa ds eè; çR;kLFk VDdj ds fy, osxksa dk fofues; gksrk gS ⇒ nwljs xksyd dh pky 1= gl

ysfdu =1 25gl gl ⇒ =1

2 5l

l

17. ,d 0.2 kg nzO;eku dk d.k ,d cy ds vUrxZr] tks fd ,d fu;r 'kfDr 0.5 W d.k dks nsrk gS] ,d fn'kk esa xfr'khygSA ;fn d.k dh izkjafHkd xfr 'kwU; gS rc 5 s ckn bldh xfr (ms–1 esa) gksxh


ladsrladsrladsrladsrladsr : pw¡fd 'kfDr fu;r gS P × t = ΔkE

⇒ × = × −210.5 5 (0.2) 02

v

⇒ 2.5 = 0.1 v2

⇒ v = 5 m/s

(9)

18. pk¡nh ,oa lksfM;e ds dk;Z iQyu Øe'k% 4.6 rFkk 2.3 eV gSaA pk¡nh o lksfM;e ds fujks/h foHko ,oa vko`fÙk ds chp xzkiQksa ds<+ky dk vuqikr gS


ladsrladsrladsrladsrladsr :φ

= −hfVe e

<ky = .he

;g nksuksa ds fy, leku gSA

19. ,d rqjar rS;kj fd;k gqvk jsfM;ks vkblksVksi izfrn'kZ] ftldh v/Z&vk;q 1386 s gS] dh lfØ;rk 103 fo?kVu izfr lsdaM gSA ;fn ln2 = 0.693 gS] rc izFke 80 s esa fo?kfVr ukfHkdksa o izkjafHkd ukfHkdksa dh la[;kvksa dk vuqikr (izfr'kr fudVre iw.kk±d esa) gS


ladsrladsrladsrladsrladsr : −λ= 0tN N e

⇒ −λ=0

tN eN

⇒ − ×=

ln2 801386

0

N eN

⇒ ×

−=

0.693 801386

0

N eN

⇒ −= 0.04

0

N eN

⇒ ⎛ ⎞= ⎜ ⎟⎝ ⎠

0.04

0

1NN e

{kf;r ukfHkd dk Hkkx = ⎛ ⎞− = − ⎜ ⎟⎝ ⎠

0.04

0

11 1NN e = 0.04 = 4%P

20. ,d 50 kg o 0.4 m f=kT;k dh ,dleku fMLd viuh ÅèokZ/j v{k ds fxnZ 10 rad s–1 ds dks.kh; osx ls ?kwe jgh gSA nks,dleku o`Ùkkdkj NYys /hjs ls fMLd ij lefer rjhds ls ,d nwljs dks Nwrs gq, bl izdkj fMLd ry ij j[ks tkrs gSa fd osfMLd ds v{k dks Hkh Li'kZ djsa A izR;sd NYys dk nzO;eku 6.25 kg o f=kT;k 0.2 m gSA bl fudk; dk u;k dks.kh; osx(rad s–1 esa) fuEu gksxk (eku yhft;s fd fMLd ,oe~ NYys ds chp ?k"kZ.k bruk gS fd fMLd o NYys ds chp lkis{k xfr 'kwU;gS vkSj fudk; ewy v{k ij ?kw.kZu dj jgk gS) %


ladsrladsrladsrladsrladsr : dks.kh; laosx laj{k.k ls

⎡ ⎤× × × = × × + × × × ω⎢ ⎥⎣ ⎦2 2 21 150 (0.4) 10 50 (0.4) 2 2 6.25 (0.2)

2 2

⇒ ω = =+

40 8 rad/s4 1

(10)

PART–II : CHEMISTRY

[k.M[k.M[k.M[k.M[k.M - 1 : (dsoy ,d lgh fodYi izdkj)(dsoy ,d lgh fodYi izdkj)(dsoy ,d lgh fodYi izdkj)(dsoy ,d lgh fodYi izdkj)(dsoy ,d lgh fodYi izdkj)

bl [k.M esa 10 cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u gSaA izR;sd iz'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls dsoy ,ddsoy ,ddsoy ,ddsoy ,ddsoy ,d lgh gSA

21. fuEufyf[kr ladqy vk;uksa P, Q ,oa R ij fopkj dhft, %

P = [FeF6]3–, Q = [V(H2O)6]2+ vkSj R = [Fe(H2O)6]2+

ladqy vk;uksa dk lgh Øe muds izpØ.k ek=k pqacdh; vk?kw.kZ eku (B.M. esa) ds vuqlkj gS

(A) R < Q < P (B) Q < R < P(C) R < P < Q (D) Q < P < R


ladsrladsrladsrladsrladsr : ladqy vk;uksa P, Q o R esa dsUnzh; /krq vk;u ds bysDVªkWuh; foU;kl fuEu gSa

− += 3 36P [FeF ] ; Fe :

3d

Q = [V(H2O)6]2+; V2+:3d

R = [Fe(H2O)6]2+; Fe2+

3d

izpØ.k ek=k pqacdh; vk?kw.kZ dk lgh Øe Q < R < P gSA

22. ,d Bksl AX esa A+ vk;u ij X– vk;uksa dh O;oLFkk (lgh ekilwpd esa ugha) fp=k esa nh xbZ gSA ;fn X– dk v¼ZO;kl 250pm gS] rc A+ dk v¼ZO;kl gksxk

(A) 104 pm X–

A+(B) 125 pm(C) 183 pm(D) 57 pm


ladsrladsrladsrladsrladsr : /uk;u A+ ½.kk;uksa X– }kjk fufeZr v"ViQydh; fjfDr xzg.k djrs gSaA ,d v"ViQydh; fjfDr esa fcuk fdlh foÑfr dslek;ksftr gksus ds fy, ,d /uk;u ds fy, vf/dre f=kT;k vuqikr 0.414 gksrk gSA ½.kk;u X– dh f=kT;k 250 pm gSA

+

−

=A

X

R0.414

R

+ = × =AR 0.414 250 103.50 104 pm

23. lkekU;r% lYiQkbM v;Ldksa ds :i esa ik, tkus okys /krq gSa

(A) Ag, Cu vkSj Pb (B) Ag, Cu vkSj Sn

(C) Ag, Mg vkSj Pb (D) Al, Cu vkSj Pb


ladsrladsrladsrladsrladsr : flYoj] dkWij vkSj ysM HkwiiZVh esa lkekU;r% Ag2S (flYoj Xykal), CuFeS2 (dkWij ikbjkbVht) vkSj PbS (xSysuk) ds :i esaik, tkrs gSaA

(11)

24. CO2(g), H2O(l) rFkk Xywdksl (Bksl) dh fojpu ekud ,aFkSYiht 25°C ij Øe'k% –400 kJ/eksy] –300 kJ/eksy vkSj –1300kJ/eksy gSA izfr xzke Xywdksl dh 25°C ij ngu ekud ,aFkSYih gS

(A) +2900 kJ (B) –2900 kJ(C) –16.11 kJ (D) +16.11 kJ


ladsrladsrladsrladsrladsr : C6H12O6(s) + 6O2(g) → 6CO2(g) + 6H2O(l)ΔH° = 6 (–400) + 6(–300) – (–1300)ΔH° = –2900 kJ/mol

Δ ° = − = −2900H 16.11 kJ / gm180

25. ,eksfudy H2S ds lkFk vfHkfØ;k djus ij ftl /krq vk;u dk vo{ksi.k lYiQkbM ds :i esa gksrk gS] og gS(A) Fe(III) (B) Al(III)(C) Mg(II) (D) Zn(II)


ladsrladsrladsrladsrladsr : ,eksfudy H2S ds lkFk mipkj ij Zn2+ vk;u ZnS ds :i esa vo{ksfir gks tkrs gSaA Fe3+ vkSj Al3+ vk;u Hkh gkbMªkWDlkbMds :i esa vo{ksfir gksrs gSa ysfdu lYiQkbM ds :i esa ughaA

26. 25°C rkieku ij ,d tyh; foy;u ls esfFkfyu Cyw dk lfØf;r pkjdksy ij vf/'kks"k.k fd;k x;kA bl izØe ds fy;s lghdFku gS

(A) vf/'kks"k.k dks 25°C ij lfØ;.k dh vko';drk gksrh gSA

(B) vf/'kks"k.k izØe esa ,UFkSYih ?kVrh gSA

(C) vf/'kks"k.k rkieku c<+kus ij c<+rk gSA

(D) vf/'kks"k.k vuqRØe.kh; gSA


ladsrladsrladsrladsrladsr : lfØf;r pkjdksy ij esfFkfyu Cyw dk vf/'kks"k.k HkkSfrd vf/'kks"k.k gksrk gS tks m"ek{ksih] cgqijrh; gksrk gS rFkk blesamQtkZ vojks/ ugha gksrkA

27. ,flVksu esa KI ds foy;u dh izR;sd P, Q, R vkSj S ds lkFk vyx&vyx SN2 vfHkfØ;k gksrh gSA bu vfHkfØ;k dh njksa dsifjorZu dk lgh Øe gS

H C – Cl3

P

ClO

ClCl

Q R S(A) P > Q > R > S (B) S > P > R > Q(C) P > R > Q > S (D) R > P > S > Q



P

ClO

Cl

Q

Cl

RS

;kSfxd : CH –Cl3 : :

S N2 vfHkfØ;k ds çfrlkisf{kd fØ;k'khyrk,sa

1,00,000 200 79 0.02: : :

(12)

28. fuEu vfHkfØ;k

P + Q R + Sesa P dh 75% vfHkfØ;k dk le; P dh 50% vfHkfØ;k esa fy, x, le; dh rqyuk esa nksxquk gSA Q dh fofHkÂ lkanzrk]vfHkfØ;k le; vuqlkj fp=k esa n'kkZbZ xbZ gSA bl vfHkfØ;k dh leLr dksfV gS

[Q]0

[Q]

le;(A) 2 (B) 3(C) 0 (D) 1


ladsrladsrladsrladsrladsr : P ds lanHkZ esa vfHkfØ;k dksfV ,d gS D;ksafd t3/4] t1/2 dk nqxuk gSA fn;s x;s xzkiQ ls Q ds lanHkZ esa vfHkfØ;k dksfV'kwU; gSA blfy, leLr vfHkfØ;k dksfV ,d gSA

29. lkanz ukbfVªd vEy dk dkiQh le; ckn ihys&Hkwjs jax esa ifjofrZr gksuk fdlds cuus ls gksrk gS\

(A) NO (B) NO2(C) N2O (D) N2O4

mÙkjmÙkjmÙkjmÙkjmÙkj (B)ladsrladsrladsrladsrladsr : lkUnz HNO3 /hjs&/hjs fuEu izdkj vi?kfVr gksrk gS

4HNO3 ⎯→ 4NO2 + 2H2O + O2

NO2 ds fuekZ.k ds dkj.k bldk jax ihyk&Hkwjk gks tkrk gSA

30. ;kSfxd tks tyh; lksfM;e ckbdkcksZusV foy;u }kjk vfHkfØ;k dj CO2 ughaughaughaughaugha nsrk gS] og gS

(A) csUtksbd vEy (B) csUthulYiQksfud vEy

(C) lsfyflfyd vEy (D) dkjcksfyd vEy (iQhukWy)


ladsrladsrladsrladsrladsr : dkcksZfyd vEy (fiQukWy) dkcksZfud vEy ls nqcZy vEy gS vkSj bl dkj.k ;g tyh; NaHCO3 foy;u ds lkFk mipkjij CO2 eqDr ugha djrkA dkcksZfud vEy dh vis{kk csUtksbd vEy] csUthulYiQksfud vEy vkSj lsfyflfyd vEy vf/dvEyh; gksrs gSa vkSj NaHCO3 foy;u ds lkFk CO2 eqDr djsaxsA

[k.M[k.M[k.M[k.M[k.M - 2 : (,d ;k vf/d lgh fodYi izdkj)(,d ;k vf/d lgh fodYi izdkj)(,d ;k vf/d lgh fodYi izdkj)(,d ;k vf/d lgh fodYi izdkj)(,d ;k vf/d lgh fodYi izdkj)

bl [k.M esa 5 cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u gSaA izR;sd iz'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls ,d ,d ,d ,d ,d ;k vf/dvf/dvf/dvf/dvf/d lgh gSaA

31. esfFky ,lhVsV (1M) dh nqcZy vEy (HA, 1M) }kjk ty vi?kVu dh izkjafHkd nj 25°C ij izcy vEy (HX, 1M) dhrqyuk esa 1/100 gSA HA ds Ka dk ewY;kadu gS

(A) 1 × 10–4 (B) 1 × 10–5

(C) 1 × 10–6 (D) 1 × 10–3


ladsrladsrladsrladsrladsr : nqcZy vEy ds lanHkZ esa nj

R1 = K[H+]WA[,LVj]

vkSj izcy vEy ds lanHkZ esa nj

(13)

R2 = K[H+]SA[,LVj]

∴+

+= =WA1

2 SA

[H ]R 1R 100[H ]

∴+

+⎡ ⎤ = = = = α⎣ ⎦SA

WA

[H ] 1H 0.01 M C100 100

α = 0.01nqcZy vEy ds fy, Ka = Cα2

= 1(0.01)2

= 1 × 10–4

32. tert-C;wfVy /uk;u vkSj 2-C;wVhu Øe'k% esa vfrla;qXeu fLFkjrk ftu dkj.kksa ls gksrh gS] os gSa

(A) σ → p (fjDr) vkSj σ → π∗ bysDVªkWu foLFkkuhdj.k

(B) σ → σ* vkSj σ → π bysDVªkWu foLFkkuhdj.k

(C) σ → p (iwfjr) vkSj σ → π bysDVªkWu foLFkkuhdj.k

(D) p (iwfjr) → σ∗ vkSj σ → π∗ bysDVªkWu foLFkkuhdj.k


ladsrladsrladsrladsrladsr : vfrla;qXeu esa r`rh;d C;wfVy dkcZ/uk;u ds fy, σ → p (fjDr) bysDVªkWu foLFkkuhdj.k vkSj 2-C;wVhu ds fy, σ → π*bysDVªkWu foLFkkuhdj.k gksrk gSA

33. mi&lgla;kstd ;kSfxdksa@vk;Ul ds ;qXe lewg esa tks ,d gh izdkj dh leko;ork n'kkZrs gSa] og gSa

(A) [Cr(NH3)5Cl]Cl2 vkSj [Cr(NH3)4Cl2]Cl (B) [Co(NH3)4Cl2]+ vkSj [Pt(NH3)2(H2O)Cl]+

(C) [CoBr2Cl2]2– vkSj [PtBr2Cl2]2– (D) [Pt(NH3)3](NO3)Cl vkSj [Pt(NH3)3Cl]Br


ladsrladsrladsrladsrladsr : ladqy vk;uksa dk ;qXe [Co(NH3)4Cl2]+ vkSj [Pt(NH3)2(H2O)Cl]+ T;kferh; leko;ork n'kkZrk gSA ladqyksa dk ;qXe[Pt(NH3)3(NO3)]Cl vkSj [Pt(NH3)3Cl]Br vk;uu leko;ork n'kkZrk gSA fn;s x;s nwljs ;qXeksa esa bl izdkj dh leko;orkugha gSA

34. P, Q, R vkSj S esa ,sjkseSfVd ;kSfxd gS@gSa

Cl

P

Q

AlCl3

NaH

R(NH ) CO4 2 3

100-115 °C

O O

SHClO

(A) P (B) Q(C) R (D) S

mÙkjmÙkjmÙkjmÙkjmÙkj (A, B, C, D)

ladsrladsrladsrladsrladsr : (i) ClAlCl3 AlCl4 (P)

(,jksesfVd)

(ii) NaH(Q) + H2

(,jksesfVd)

Na

(14)

(iii)O O

(NH ) CO4 2 3

100–115 CºR

(NH ) CO4 2 3

Δ2NH + CO + H O3 2 2

O O+ NH3

O O NH2 O NH2OH

. . IMPE

N

H

Δ

–2H O2

N

H

HO OH

IMPEN

H

O OH

H

(R) ,jksesfVd

H

(iv) O HCl OH Cl (S)

(,jksesfVd)

35. csUthu vkSj usÝFkyhu lk/kj.k rkieku ij ,d vkn'kZ foy;u cukrs gSaA bl izØe ds fy;s lgh dFku gS (gSa)

(A) ΔG /ukRed gSA (B) ΔSfudk; /ukRed gSA

(C) ΔSifjos'k = 0 (D) ΔH = 0mÙkjmÙkjmÙkjmÙkjmÙkj (B, C, D)ladsrladsrladsrladsrladsr : csUthu vkSj usÝFkyhu ,d vkn'kZ foy;u cukrs gSaA ,d vkn'kZ foy;u ds fy,] ΔH = 0, ΔSfudk; > 0 vkSj ΔSifjos'k = 0

gksrk gS D;ksafd fudk; vkSj ifjos'k ds eè; mQ"ek mQtkZ dk fofue; ugha gksrkA

[k.M[k.M[k.M[k.M[k.M - 3 : (iw.kk±d eku lgh izdkj)(iw.kk±d eku lgh izdkj)(iw.kk±d eku lgh izdkj)(iw.kk±d eku lgh izdkj)(iw.kk±d eku lgh izdkj)

bl [k.M esa 5 iz'u iz'u iz'u iz'u iz'u gSaA izR;sd iz'u dk mÙkj 0 ls 9 rd (nksuksa 'kkfey) ds chp dk ,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 gSA

36. He vkSj Ne ds ijek.kq nzO;eku Øe'k% 4 vkSj 20 a.m.u. gSA He xSl dh –73ºC ij ns czkWXyh rjax yEckbZ Ne dh 727°C ijns czkWXyh rjax yEckbZ ls "M" xquk gSA 'M' dk eku gS


ladsrladsrladsrladsrladsr : λ=

λHe Ne Ne

Ne He He

M VM V

=

NeNe

Ne

HeHe

He

3RTMM

3RTMM

=

NeNe

Ne

HeHe

He

TMMTMM

= Ne Ne

He He

M TM T

×=

×205 1000

4 200λ = λHe Ne5

(15)

37. EDTA4– ,sfFkyhu Mkb,sehu VsVªk,slhVsV vk;u gSA ladqy vk;u [Co(EDTA)]1– esa N—Co—O vkcU/ dks.kksa dh dqy la[;k gS

mÙkjmÙkjmÙkjmÙkjmÙkj (8)ladsrladsrladsrladsrladsr :

Co

O – C = O

O – C = O

OC CH2 CH2

O

CH2

CH2

N

O NC CH2

OCH2

II

III

IV

I

I

II

vr% cU/ dks.k gSa(1) ∠NI CoOI

(2) ∠NI CoOII

(3) ∠NI CoOIII

(4) ∠NI CoOIV

(5) ∠NII CoOI

(6) ∠NII CoOII

(7) ∠NII CoOIII

(8) ∠NII CoOIV

blfy, iwNs x;s dqy cU/ dks.k 8 gSaA

38. mRikn P esa dkcksZfDlfyd vEy lewgksa dh dqy la[;k gS

O

O O

O

O

1. H O , +Δ3

2. O33. H O2 2

P


ladsrladsrladsrladsrladsr : O

O

O

O

O

H O3+

O

O

O

O

OHΔ∗

O

O

(i) O3

(ii) H O2 2

O

O

HOOCHOOC

(P)

OHOH (–H O, –CO )2 2

vr% vfUre mRikn (P) esa nks dkcksZfDlfyd vEy lewg gSA

∗ xeZ djus ij ty dk ok"ihdj.k gksrk gS vkSj ftl dkj.k vEy dh lkUnzrk c<+rh gS vkSj lkUnz vEyh; ekè;e esafodkcksZfDlyhdj.k o futZyhdj.k gksaxsA

(16)

39. ,d VsVªkisIVkbM esa ,ykuhu ij —COOH xzqi fo|eku gSA blds lEiw.kZ ty vi?kVu }kjk Xykbflu (Gly), oSyhu (Val), isQfuy,sykuhu (Phe) rFkk ,sykfuu (Ala) çkIr gksrs gSaA bl VsVªkisIVkbM dh laHkkfor Üka[kykvksa (çkFkfed lajpukvksa) dh la[;k crk,¡ftuesa —NH2 xzqi fdjsy dsUnz ds lkFk vkcaf/r gSA


ladsrladsrladsrladsrladsr : iz'ukuqlkj C – fljs ij ,sykfuu gS vkSj N – fljs ij fdjsy dkcZu dk vFkZ gS fd bl ij Xykblhu ugha gksuk pkfg,Ablfy, lEHko vuqØe fuEu gS :

Val Phe Gly AlaVal Gly Phe AlaPhe Val Gly AlaPhe Gly Val Ala

blfy, mÙkj (4) gSA

40. eSySehu ij miyC/ bysDVªkWuksa ds ,dkdh ;qXeksa dh dqy la[;k gS


ladsrladsrladsrladsrladsr : eSySehu dh lajpuk fuEu gS

N NH2

N

NH2

H2N

N

vr% eSySehu ds ikl N% ,dkadh bysDVªkWu ;qXe gSaA

PART–III : MATHEMATICS

[k.M[k.M[k.M[k.M[k.M - 1 : (dsoy ,d lgh fodYi izdkjdsoy ,d lgh fodYi izdkjdsoy ,d lgh fodYi izdkjdsoy ,d lgh fodYi izdkjdsoy ,d lgh fodYi izdkj)

bl [k.M esa 10 cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u gSaA izR;sd iz'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls dsoy ,d dsoy ,d dsoy ,d dsoy ,d dsoy ,d lgh gSA

41. a > b > c > 0 ds fy,] (1, 1) rFkk js[kkvksa + + = 0ax by c o + + = 0bx ay c ds izfrPNsn fcUnq ds chp dh nwjh 2 2 lsde gS] rc(A) a + b – c > 0 (B) a – b + c < 0(C) a – b + c > 0 (D) a + b – c < 0


ladsrladsrladsrladsrladsr : + + = 0ax by c ... (1)

+ + = 0bx ay c ... (2)

gy djus ij, cxa b−

=+

rFkk (1) o (2) lsy = x

∴ izfrPNsnh fcUnq y = x ij fLFkr gS

⇒cy

a b−

=+

fn;k gS, 2 2

1 1 2 2c ca b a b

⎛ ⎞ ⎛ ⎞+ + + <⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

(17)

⇒ 2 1 2 2ca b

⎛ ⎞+ <⎜ ⎟+⎝ ⎠

⇒ 2a b ca b+ +

<+

⇒ 2 2a b c a b+ + < +

⇒ 0a b c+ − >

42. varjky π⎡ ⎤

⎢ ⎥⎣ ⎦0,

2 ij oØksa = +sin cosy x x rFkk = −cos siny x x }kjk ifjc¼ {ks=kiQy gS %

(A) −4( 2 1) (B) −2 2( 2 1)

(C) +2( 2 1) (D) +2 2( 2 1)



O π/4 π/2

/2 /4 /2

0 0 /4(sin cos ) (cos sin ) (sin cos )x x dx x x dx x x dx

π π π

π

⎡ ⎤+ − − + −⎢ ⎥

⎢ ⎥⎣ ⎦∫ ∫ ∫

=/2 /2 /4 /4 /2 /2

0 0 0 0 /4 /4cos sin sin cos cos sinx x x x x xπ π π π π π

π π⎡ ⎤− + − + − −⎢ ⎥⎣ ⎦

=1 1 1 1(0 1) (1 0) 1 0 12 2 2 2

⎡ ⎤⎛ ⎞ ⎛ ⎞− − + − − + − − − − −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

= 1 12 2 1 12 2

⎡ ⎤− − + − +⎢ ⎥⎣ ⎦

=2 2 2 2⎡ ⎤− −⎣ ⎦

=4 2 2−

=2 2( 2 1)−

43. ∞ ∞(– , ) esa fcUnqvksa dh la[;k] ftuds fy, − − =2 sin cos 0x x x x , gS %

(A) 6 (B) 4(C) 2 (D) 0


(18)

ladsrladsrladsrladsrladsr : = − −2( ) sin cosf x x x x x

( ) 2 cos sin sinf x x x x x x′ = ⋅ − +

= x(2 – cosx)

f(x) o¼Zeku gksrk gS x > 0

f(x) ßkleku gksrk gS x < 0f(0) = –1

( )f ∞ = ∞

(– )f ∞ = ∞

44.23

1

1 1cot cot 1 2

n

n kk−

= =

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑ ∑ dk eku gS %

(A)2325 (B)

2523

(C) 2324 (D)

2423


ladsrladsrladsrladsrladsr : ( )23

1

1cot 1 ( 1)

nk k−

=+ +∑

=23

1 1

1tan ( 1) tan ( )

nk k− −

=+ −∑

= tan–1(24) – tan–1(1)

vc, ( )1 1cot (tan (24) tan (1)− −−

= 1 24 1cot tan1 24 1

−⎛ ⎞−⎛ ⎞⎜ ⎟⎜ ⎟+ ⋅⎝ ⎠⎝ ⎠

= −⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠1 23cot tan

25 = 2523

45. ,d oØ fcUnq π⎛ ⎞⎜ ⎟⎝ ⎠1,

6 ls xqtjrk gSA ekuk fd izR;sd fcUnq (x, y) ij oØ dh izo.krk ⎛ ⎞+ >⎜ ⎟

⎝ ⎠sec , 0y y x

x x gS] rc oØ dk

lehdj.k gS %

(A) ⎛ ⎞ = +⎜ ⎟⎝ ⎠

1sin log2

y xx

(B) ⎛ ⎞ = +⎜ ⎟⎝ ⎠

cosec log 2y xx

(C) ⎛ ⎞ = +⎜ ⎟⎝ ⎠

2sec log 2y xx

(D) ⎛ ⎞ = +⎜ ⎟⎝ ⎠

2 1cos log2

y xx


ladsrladsrladsrladsrladsr : = + secdy y ydx x x

y = vx j[kus ij

(19)

∴ + = + secdvv x v vdx

⇒ =cos dxv dvx

⇒ sin v = ln x + c

∵ ;g π⎛ ⎞

⎜ ⎟⎝ ⎠1,

6 ls gksdj xqtjrk gS

∴ =12

c

∴ = +1sin ln2

y xx

46. ekuk fd ⎡ ⎤ →⎢ ⎥⎣ ⎦

1: ,12

f ( lHkh okLrfod la[;kvksa dk leqPp;) ,d /ukRed] vpjsrj rFkk vodyuh; iQyu gS ftlds fy;s

f′(x) < 2f(x) rFkk ⎛ ⎞ =⎜ ⎟⎝ ⎠

1 12

f gS] rc ∫1

1/2

( )f x dx dk eku fuEu vUrjky esa gS %

(A) (2e – 1, 2e) (B) (e – 1, 2e – 1)

(C) −⎛ ⎞−⎜ ⎟⎝ ⎠

1 , 12

e e (D) −⎛ ⎞⎜ ⎟⎝ ⎠

10,2

e


ladsrladsrladsrladsrladsr : < 2dy ydx

− −<2 22x xdye yedx

− <2( ) 0xd yedx

⇒ ye–2x ßkleku iQyu gS

D;ksafd, < < 112

x

⇒ e–1 > ye–2x > y(1)e–2

⇒ e2x–1 > y > y(1)e2x–2

⇒ − −> > >∫ ∫ ∫

1 1 12 1 2 2

1/2 1/2 1/2

(1) 0x xe dx ydx y e

∴ −< <∫

1

1/2

102

eydx

(20)

47. ekuk fd = + −ˆ ˆ ˆ3 2PR i j k rFkk = − −ˆ ˆ ˆ3 4SQ i j k ,d lekarj prqHkq Zt PQRS ds fod.kZ fu/kZfjr djrs gSa vkSj

= + +ˆ ˆ ˆ2 3PT i j k ,d vU; lfn'k gSA rc lfn'kksa rFkk,PT PQ PS }kjk fu/kZfjr lekarj "kV~iQyd dk vk;ru gS %

(A) 5 (B) 20(C) 10 (D) 30


ladsrladsrladsrladsrladsr : = + = + − = 1ˆ ˆ ˆ3 2PR a b i j k d S b( )

P

R

Q a( )

= − = − − = 2ˆ ˆ ˆ3 4SQ a b i j k d

⇒ = − −ˆ ˆ ˆ2 3a i j k

⇒ = + +ˆ ˆ ˆ2b i j k

∴ vHkh"V lekUrj "kV~iQyd dk vk;ru = − −2 1 3

1 2 11 2 3

= |2(6 – 2) + 1(3 – 1) – 3(2 – 2)|= 10 ?ku bdkbZ

;k;k;k;k;k

lekUrj prqHkqZt dk {ks=kiQy = ×1 21 | |2

d d

∴ lekUrj "kV~iQyd dk vk;ru = −

− −3 1 2

1 1 3 42 1 2 3

= 10 ?ku bdkbZ

48. ry x + y + z = 3 ij js[kk + += =

−2 1

2 1 3x y z ij fLFkr fcUnqvksa ls yEc Mkys tkrs gSaA yEc&ikn fuEu js[kk ij fLFkr gSa %

(A)− −

= =−

1 25 8 13x y z

(B)− −

= =−

1 22 3 5x y z

(C)− −

= =−

1 24 3 7x y z

(D)− −

= =−

1 22 7 5x y z


ladsrladsrladsrladsrladsr :+ +

= = = λ + + =−

2 1 , 32 1 3

x y z x y z

bl js[kk ij dksbZ fcUnq (2λ – 2, –λ – 1, 3λ) gS

( , , )α β γ

B

A

C

(–2, –1, 0)

;g fcUnq lery (2λ – 2) + (–λ – 1) + (3λ) = 3 dks lUrq"V djsxk

⇒ 4λ – 3 = 3

⇒ λ =32

blfy,] lery dk izfrPNsnh fcUnq −⎛ ⎞ ≡⎜ ⎟⎝ ⎠

5 91, ,2 2

C gSA

vc] js[kk ij fcUnq (–2, –1, 0) gS rFkk AB dk fnd~ vuqikr (fp=k ls) α + β + γ= = =

2 11 1 1

k gS

js[kk AB ij dksbZ lkekU; fcUnq (k – 2, k – 1, k) gS]

;g lehdj.k dks larq"V djsxk blfy,] (k – 2) + (k – 1) + k = 3.⇒ k = 2

(21)

vr%] (α, β, γ) ≡ (0, 1, 2).

blfy,, BC ls gksdj xqtjus okyh js[kk dk lehdj.k

− −= =

−1 2

7 512 2

x y z gS

⇒− −

= =−

1 22 7 5x y z

49. pkj O;fDr Lora=kr;k fdlh ,d leL;k dks izkf;drkvksa 1 3 1 1, , ,2 4 4 8 ds lkFk Bhd gy djrs gSa] rc leL;k ds muesa ls de

ls de ,d O;fDr }kjk Bhd gy fd;s tkus dh izkf;drk gS %

(A)235256 (B)

21256

(C)3

256 (D)253256


ladsrladsrladsrladsrladsr : = = = =1 3 1 1( ) , ( ) , ( ) , ( )2 4 4 8

P A P B P C P D

P(A ∪ B ∪ C ∪ D) = − ∪ ∪ ∪1 ( )P A B C D

= − ∩ ∩ ∩1 ( )P A B C D

= −1 ( ) ( ) ( ) ( )P A P B P C P D

= − ⋅ ⋅ ⋅1 1 3 712 4 4 8

= −211

256 =

235256

50. ekuk fd lfEeJ la[;k,¡ α rFkk α1

Øe'k% o`Ùk − + − =2 2 20 0( ) ( )x x y y r rFkk − + − =2 2 2

0 0( ) ( ) 4x x y y r ij fLFkr gSaA

;fn = +0 0 0z x iy lehdj.k = +2 2

02 2,z r dks lUrq"V djrk gS] rc α =

(A)12 (B) 1

2

(C)17 (D)

13


ladsrladsrladsrladsrladsr : α, − + − =2 2 20 0( ) ( )x x y y r ij fLFkr gS ... (1) ( , )x y0 0

r

21 α=

α α , − + − =2 2 2

0 0( ) ( ) 4x x y y r ij fLFkr gS ... (2)

ekuk α, α1 + iβ1 gS

(22)

α , 2 2 2 2 20 0 0 0( ) ( ) 2( )x y x y xx yy r+ + + − + = ij fLFkr gS

2 2 20 1 0 1 02( )z x y rα + − α + β = ... (3)

lehdj.k (2) ls, 2 21 0 1 002 2

( )1 2 4x yz rα + β+ − =

α α

⇒ 2 2 220 1 0 1 01 2( ) 4z x y r+ α − α + β = α ... (4)

(3) – (4) ls] ( )2 2 2 2201 (1 ) 1 4z rα − + − α = − α

( )2 2 220( 1)(1 ) 1 4z rα − − = − α ...(5)

vc, ( )2202 1r z= −

2

20

21

rz

=−

(5) esa izfrLFkkfir djus ij,2 21 2(1 4 )α − = − − α

⇒ 2 21 2 8α − = − + α

⇒ 27 1α =

⇒17

α =

[k.M[k.M[k.M[k.M[k.M - 2 : (,d ;k vf/d lgh fodYi izdkj,d ;k vf/d lgh fodYi izdkj,d ;k vf/d lgh fodYi izdkj,d ;k vf/d lgh fodYi izdkj,d ;k vf/d lgh fodYi izdkj)

bl [k.M esa 5 cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u cgqfodYi iz'u gSaA izR;sd iz'u esa pkj fodYi (A), (B), (C) vkSj (D) gSa] ftuesa ls ,d,d,d,d,d ;k vf/dvf/dvf/dvf/dvf/d lgh gSaA.

51. ,d js[kk l ] tks ewyfcUnq ls xqtjrh gS] js[kkvksa

+ + − + + +1ˆ ˆ ˆ: (3 ) ( 1 2 ) (4 2 ) , l t i t j t k −∞ < < ∞t

+ + + + +2ˆ ˆ ˆ: (3 2 ) (3 2 ) (2 ) ,l s i s j s k −∞ < < ∞s

ij yEcor~ gSA rc] l2 ij fLFkr fcUnq (fcUnqvksa) ds funsZ'kkad] tks js[kkvksa l rFkk l1 ds izfrPNsn fcUnq ls 17 dh nwjh ij gS

(gSa)] fuEu gS (gSa) %

(A)⎛ ⎞⎜ ⎟⎝ ⎠

7 7 5, ,3 3 3 (B) (–1, –1, 0)

(C) (1, 1, 1) (D)⎛ ⎞⎜ ⎟⎝ ⎠

7 7 8, ,9 9 9


ladsrladsrladsrladsrladsr : = = = λx y za b c (js[kk l dk lehdj.k)

js[kk l1 dk lehdj.k, 3 1 41 2 2

x y z t− + −= = =

js[kk l2 dk lehdj.k, 3 3 22 2 1

x y z s− − −= = =

(23)

js[kk l dk fnd~ vuqikr

ˆ ˆ ˆ1 2 22 2 1

i j k

}kjk fn;k x;k gS

= ˆ ˆ ˆ2 3 2i j k− + −

js[kk l dk lehdj.k 2 3 2x y z

= = = λ− − gS

l rFkk l1 dk izfrPNsnh fcUnq,

2 3 t− λ = + ...(1)

3 2 1tλ = − ...(2)t dk eku j[kus ij, 3 2( 2 3) 1λ = − λ − − gS

3 4 6 1λ = − λ − −

7 7λ = −

1λ = −

izfrPNsnh fcUnq (2, –3, 2) gSa

blfy,, 2 2 2(3 2 2) (3 2 3) (2 2) 17s s s+ − + + + + + − =

⇒ 2 2 24 4 1 36 24 4 17s s s s s+ + + + + + =

⇒ 29 28 20 0s s+ + =

⇒102, 9

s = − −

i.e., izfrPNsnh fcUnq (–1, –1, 0) rFkk ⎛ ⎞⎜ ⎟⎝ ⎠

7 7 8, ,9 9 9 gSa

52. ekuk fd f(x) = x sin πx, x > 0 ] rc lHkh /u&iw.kk±dksa n ds fy, f ′(x) fuEu ij 'kwU; gksrk gS %

(A) varjky ⎛ ⎞+⎜ ⎟⎝ ⎠

1,2

n n esa ,dek=k ,d fcUnq ij (B) varjky ⎛ ⎞+ +⎜ ⎟⎝ ⎠

1 , 12

n n esa ,dek=k ,d fcUnq ij

(C) varjky (n, n + 1) esa ,dek=k ,d fcUnq ij (D) varjky (n, n + 1) esa nks fcUnqvksa ij

mÙkjmÙkjmÙkjmÙkjmÙkj (B, C)ladsrladsrladsrladsrladsr : f(x) = x sinπx

f ′(x) = sin πx + πx cos πx = 0⇒ – tan πx = πx

0 1 32

2 52

12

Li"Vr% f ′(x) dk 1 , 12

n n⎛ ⎞+ +⎜ ⎟⎝ ⎠

esa ,d ewy gS] blh rjg f ′(x) dk (n, n + 1) esa ,d ewy gSA

(24)

53. ekuk fd

+

== −∑

( 1)4 2 2

1( 1)

k kn

nk

S k , rc Sn fuEu eku ys ldrk gS %

(A) 1056 (B) 1088(C) 1120 (D) 1332

mÙkjmÙkjmÙkjmÙkjmÙkj (A, D)

ladsrladsrladsrladsrladsr :+

== −∑

( 1)4 2 2

1( 1)

k kn

nk

S k

2 2 2 2 2 2 2 2 2 2–1 – 2 3 4 5 6 ........ (4 3) (4 2) (4 1) (4 )nS n n n n= + + − − + − − − − + − +

Sn = 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2(3 1 ) (4 2 ) (7 5 ) (8 6 ) (11 9 ) (12 10 )...... (4 1) (4 3) (4 ) (4 2)n n n n

2− + − + − + − + − + − ++ − − − + − −

Sn = 2(1 3) 2(4 2) 2(7 5) 2(8 6) ...... 2(4 1 4 3) 2(4 4 2)n n n n+ + + + + + + + + − + − + + −

Sn = 2 4 (4 1)2[1 2 3 ..... 4 ]

2n nn ⋅ +

+ + + + =

A ls, 4n(4n + 1) = 10564n2 + n = 2644n2 + n – 264 = 0n = 8

B ls, 4n(4n + 1) = 1088 (laHko ugha gS)C ls, 4n(4n + 1) = 1120 (laHko ugha gS)D ls, 4n(4n + 1) = 1332

n = 954. 3 × 3 vkO;wgksa M rFkk N ds fy, fuEu esa ls dkSu izdFku lR; ugha ugha ugha ugha ugha gS (gSa) \

(A) M ds lefer ;k fo"ke&lefer gksus ds vuqlkj NTMN lefer ;k fo"ke&lefer gSA(B) lHkh lefer vkO;wgksa M rFkk N ds fy, MN – NM fo"ke&lefer gS(C) lHkh lefer vkO;wgksa M rFkk N ds fy, MN lefer gS(D) lHkh O;qRØe.kh; vkO;wgksa M rFkk N ds fy, (adj M) (adj N) = adj(MN)

mÙkjmÙkjmÙkjmÙkjmÙkj (C, D)ladsrladsrladsrladsrladsr : (NTMN )T = – NTMT(NT )T

NTMTN(A) ;fn M fo"ke lefer gS] (NTMN)T = –NTMN,

vr% fo"ke leferA

;fn M lefer gS] (MTMN)T = NTMN,vr% lefer

∴ fodYi (A) lR; gSA(B) (MN – NM)T = (MN)T – (NM)T

= NTMT – MTNT

= –(MTMT – NTMT) = –(MN – NM)∴ fo"ke lefer] fodYi (B) lR; gSA

(C) (MN)T = NTMT

leferk rFkk fo"ke leferrk M rFkk N dh izdfr ij fuHkZj gS, fodYi (C) xyr gSA(D) adj(MM) = adj(N) adjM,

fodYi (D) xyr gSA

(25)

55. ,d fuf'pr ifjeki dh vk;rkdkj pknj dks] ftldh Hkqtkvksa dh yEckbZ;k¡ 8 : 15 ds vuqikr esa gSa] lHkh pkjksa fdukjksa lsleku {ks=kiQy ds oxZ fudky dj ,d [kqyh vk;rkdkj isVh esa ifjofrZr fd;k tkrk gSA ;fn fudkys x;s oxks± dk dqy {ks=kiQy100 gS] rc ifj.kkeh isVh dk vk;ru egÙke gSA rc vk;rkdkj pknj dh Hkqtkvkssa dh yEckb;k¡ fuEu gSa %

(A) 24 (B) 32(C) 45 (D) 60

mÙkjmÙkjmÙkjmÙkjmÙkj (A, C)

ladsrladsrladsrladsrladsr : (8 2 )(15 2 )V x x x= λ − λ −

= 3 2 24 46 120x x x− λ + λx 15λ

8λ2 212 92 120 0dV x x

dx= − λ + λ = , at x = 5

⇒ 260 230 150 0λ − λ + =

26 23 15 0λ − λ + =

(6 5)( 3) 0λ − λ − =

λ = 3 ds fy, Hkqtkvksa dh yEckbZ;k¡ 45, 24 gSaA

[k.M[k.M[k.M[k.M[k.M - 3 : (iw.kk±d eku lgh izdkjiw.kk±d eku lgh izdkjiw.kk±d eku lgh izdkjiw.kk±d eku lgh izdkjiw.kk±d eku lgh izdkj)

bl [k.M esa 5 iz'uiz'uiz'uiz'uiz'u gSaA izR;sd iz'u dk mÙkj 0 ls 9 rd ( nksuksa 'kkfey) ds chp dk ,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 gSaA

56. vkB lfn'kksa dk leqPp; V = { }+ + ∈ −ˆ ˆ ˆ : , , { 1,1}ai bj ck a b c yhft;sA V ls rhu vleryh; lfn'k 2p izdkj ls pqus tk ldrs

gSaA rc p dk eku gS %


ladsrladsrladsrladsrladsr : fp=k esa dqy 8 lfn'k n'kkZ;s x, gSaA

lfn'kksa dh dqy la[;k = 8C3 = 56

leryksa dh la[;k = 2 × (6 × 2) = 2456 – 24 = 32 25

57. rhu Lora=k ?kVukvksa E1, E2 rFkk E3 esa ls dsoy E1 ds ?kVus dh izkf;drk α gS] dsoy E2 ds ?kVus dh izkf;drk β gS rFkkdsoy E3 ds ?kVus dh izkf;drk γ gSA ekuk fd ?kVukvksa E1, E2 ;k E3 esa ls fdlh ds Hkh u ?kVus dh izkf;drk p ] lehdj.kksa

(α – 2β)p = αβ rFkk (β – 3γ)p = 2βγ dks lUrq"V djrh gSA lHkh izkf;drk,¡ vUrjky (0, 1) esa fLFkr ekuh tkrh gSa] rc

d s ?kVus dh ikz f;drkds ?kVu s dh ikz f;drk

=1

3

EE


(26)

ladsrladsrladsrladsrladsr : = α1 2 3( ) ( ) ( )P E P E P E ...(i)= β1 2 3( ) ( ) ( )P E P E P E ...(ii)= γ1 2 3( ) ( ) ( )P E P E P E ...(iii)

= ρ1 2 3( ) ( ) ( )P E P E P E ...(iv)(i) dks (iv) ls foHkkftr djus ij,

α α α − β= = =

αβρ βα − β

1

1

( ) 2( )

2

P EP E

⇒1

1

( )( ) 2

P EP E

β=α − β

⇒1

1

1 ( )( ) 2P E

P E− β

=α − β

⇒1

1 1( ) 2P E

β− =

α − β

⇒1

1( ) 2P E

α − β=α − β

⇒ 12( )P E α − β

=α −β …(v)

vc, 2

2 3αβ βγ

=α − β β − γ

⇒2

2 3α γ

=α − β β − γ

⇒ αβ – 3γα = 2γα – 4βγ⇒ αβ = 5γα – 4βγ

γ = 5 4αβα − β …(vi)

(iii) dks (iv) ls foHkkftr djus ij]

γ γ γ α − β= = =

αβρ αβα − β

3

3

( ) ( 2 )( )

2

P EP E

3

3

( ) 25 4( )

P EP E

α − β=

α − β

3

3

( ) 5 4( ) 2

P EP E

α − β=

α − β

3

3

1 ( ) 5 4( ) 2P E

P E− α − β

=α − β

3

1 6 6 6( )( ) 2 2P E

α − β α − β= =

α − β α − β

P(E3) = 2

6( )α − βα −β

1

3

2( )

2( )6( )

P EP E

α − βα − β=α − βα −β

= 6

(27)

58. (1 + x)n+5 ds rhu Øekxr inksa ds xq.kkad 5 : 10 : 14. ds vuqikr esa gSaA rc] n =

mÙkjmÙkjmÙkjmÙkjmÙkj (6)ladsrladsrladsrladsrladsr : ekuk Øekxr in tr+2, tr + 1, tr gSa

vr%, 1 105

r

r

tt+ =

⇒( 5) ( 1) 1 2

1n r

r+ − + +

=+

⇒ n – 3r + 3 = 0 …(i)

blh izdkj, 2

1

1410

r

r

tt+

+=

⇒ 5n – 12r + 6 = 0 …(ii)

gy djus ij, 6n =

59. ,d xM~Mh esa n dkMZ gSa tks la[;kvksa 1 ls n }kjk fpfUgr gSaA nks Øekxr la[;kvksa okys dkMZ xM~Mh ls fudky fn;s tkrs gSa vkSjvof'k"V dkMks± dh la[;kvksa dk ;ksx 1224 gSA ;fn fudkys x, dkMks± dh fpfUgr la[;kvksa esa ls y?kqrj la[;k k gS] rck – 20 =


ladsrladsrladsrladsrladsr : n dk og y?kqÙke eku ftlds fy,

( 1) 12242

n n +>

n(n + 1) > 2448⇒ n > 49n = 50 ds fy,

( 1)2

n n +⇒ 1275

vr%, k + (k + 1) = 1275 – 1224 = 51k = 25k – 20 = 5

60. fcUnq (h, 0) ls xqtjus okyh ,d ÅèokZ/j js[kk nh?kZo`Ùk + =2 2

14 3x y dks fcUnqvksa P rFkk Q ij dkVrh gSA ekuk fd fcUnqvksa P

rF k k Q ij n h? k Z o ` Ù k dh Li' k Z j s [ k k, ¡ fcUn q R ij feyrh g S a A ; fn Δ(h) = f = k H k qt PQR dk { k s = ki Qy]

Δ1 = ≤ ≤

Δ1/2 1max ( )

hh vkSj Δ2 =

≤ ≤Δ

1/2 1min ( )

hh gS] rc Δ − Δ1 2

8 85

=


ladsrladsrladsrladsrladsr : ≡2 2

+ = 14 3x yS

ekuk P rFkk Q ] (h, β) rFkk (h, –β) gSa

blfy,, R ⎛ ⎞⎜ ⎟⎝ ⎠

4 , 0h gS

vc, Δ = ⎛ ⎞β ⎜ ⎟⎝ ⎠

1 4 × 2 × –2

hh

R

Q(h, – )β

h, ( 0)

P h, ( )β

(28)

= ⎛ ⎞⎜ ⎟⎝ ⎠

2 43 1 – × –4h h

h

= ( )2 3/24 –32

hh

vr%, Δd

dh < 0

i.e. Δ ßkleku gS

i.e. Δ1 = ⎛ ⎞Δ⎜ ⎟⎝ ⎠

12 = 15 45

8

rFkk Δ2 = Δ(1) = 92

vc 1 –5Δ Δ =2

8 8 9

Documents

Regd. Office : Aakash Tower, Plot No.-4, Sec-11, MLU ... · Regd. Office : Aakash Tower, Plot No.-4, Sec-11, MLU, Dwarka, New Delhi-110075 Ph.: 011-47623456 Fax : 011-47623472 funsZ'k@INSTRUCTIONS