(1)
Answers & Solutionsforforforforfor
JEE (Advanced)-2013
Time : 3 hrs. Max. Marks: 180
DATE : 02/06/2013
PAPER - 1 (Code - 0)
CODE
0
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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
mÙkjmÙkjmÙkjmÙkjmÙkj (D)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (B)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (C)
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°
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (5)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (1)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (4)
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) %
mÙkjmÙkjmÙkjmÙkjmÙkj (8)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (B)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (C)
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)
mÙkjmÙkjmÙkjmÙkjmÙkj (D)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (B)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (B)
ladsrladsrladsrladsrladsr :
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 fofHk 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
mÙkjmÙkjmÙkjmÙkjmÙkj (D)
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)
mÙkjmÙkjmÙkjmÙkjmÙkj (D)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (5)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (2)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (4)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (6)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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)
mÙkjmÙkjmÙkjmÙkjmÙkj (B)
ladsrladsrladsrladsrladsr :
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
mÙkjmÙkjmÙkjmÙkjmÙkj (C)
(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
mÙkjmÙkjmÙkjmÙkjmÙkj (B)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (D)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (C)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (D)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (A)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (C)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)
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 %
mÙkjmÙkjmÙkjmÙkjmÙkj (5)
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
mÙkjmÙkjmÙkjmÙkjmÙkj (6)
(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 =
mÙkjmÙkjmÙkjmÙkjmÙkj (5)
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
=
mÙkjmÙkjmÙkjmÙkjmÙkj (9)
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