IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ] · IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ]...
Transcript of IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ] · IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ]...
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com , [email protected]
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IIT-JEE 2013 PAPER-II WITH SOLUTION [CODE - 3 ]
PART - I [PHYSICS]1. nks fi aMksa ft uesa i zR; sd dk nzO; eku M gS] ds chp dh nwj h dks 2L fLFkjj [ kk x; k gSA bu fi aMksa ds dsUnzkasa dks t ksM+us okyh j s[ kk ds e/; fcUnq l s , dm nzO; eku dk d.k yEcor ~ i z{ksfi r fd; k t kr k gSA xq: Rokd"kZ.k fu; r kadG gSA l gh i zFkdu gS ¼gSa½(A) nks fi aMksa ds xq: Rokd"kZ.k {ks=k l s i yk; u ds fy, nzO; eku m dk U; wur e
i zkj afHkd osx L
GM4 gSA
(B) nks fi aMksa ds xq: Rokd"kZ.k {ks=k l s i yk; u ds fy, nzO; eku m dk U; wur e
i zkj afHkd osx L
GM2 gSA
(C) nks fi aMksa ds xq: Rokd"kZ.k {ks=k l s i yk; u ds fy, nzO; eku m dk U; wur e
i zkj afHkd osx LGM2
gSA
(D) nzO; eku m dh Åt kZ l nSo fLFkj j gr h gSASol. B
Åt kZ l aj {k.k l s2GMm GMm GM
L L 2L =
12
mv2 + 2GM
2L
2GMmL
= 12
mv2
v = GM2L
Ei = 12
mv2 Ef = 0.
, d d.k esa fLFkfr t Åt kZ dks ugha fy; k x; k gSAnzO; eku m dh Åt kZ fu; r ugha gSA
2. ?k"kZ.kghu {kSfr t r y i j i M+h gqbZ k cy fLFkj kad dh nzO; eku j fgr fLi zaxds , d fl j s l s m nzO; eku dk d.k t qM+k gqvk gSA bl fLi zax dk nwl j k fl j kc) gSA ; g d.k vi uh l kE; koLFkk l s l e; t = 0 i j i zkj afHkd {kSfr t osxu0 l s xfr eku gks j gk gSA t c d.k dh xfr 0.5 u0 gksr h gS] ; g , d n<+nhokj l s i zR; kLFk l a?kê dj r k gSA bl l a?kê ds ckn(A) t c d.k vi uh l kE; koLFkk esa ykSVr k gS bl dh xfr u0 gksr h gSA(B) t c d.k vi uh l kE; koLFkk l s i gyh ckj xqt j r k gS og l e;
t = km
gSA
(C) t c fLi zax esa l ai hMu vf/kdr e gksr k gS og l e; km
34t
gSA
(D) t c d.k vi uh l kE; koLFkk l s nwl j h ckj xqt j r k gS og l e;
km
35t
gSA
Sol. AD
xfr ' khy ugh gS
m
MPu0
v = u /20
m
MPu0
T/4
T/4 T/6
T/6
u /20
x = A sin tv = A cos t
A2
= A cos t
t = 3
nhokj r d i gqapus esa yxk l e; = T6
e/; fcUnq r d oki l vkus esa yxk l e; = T6 +
T6 =
T3 =
2 m3 k
vf/kdr e fl dqM+us esa yxk l e; = T3 +
T4
= 7T12
= 67
mk
f} r h; l e; = T3 +
T2
= 5T6
l kE; koLFkk fLFkfr esa = 5T6 =
5 m3 k
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3. , d R f=kT; k ds vuar yEcs [ kks[ kys pkyd csyu dh yackbZ esa , d fLFkj/kkj k I cg j gh gSA bl csyu dks 2R f=kT; k dh vuar i fj ukfydk ds vUnjl ek{k j [ kk x; k gSA bl i fj ukfydk esa n yi sVs i zfr bdkbZ yEckbZ esa gS o, d fLFkj /kkj k I gSA , d fcUnq P ds fy, t ks l ek{k l s r nwj h i j gS] dkSul k ¼l s½ i zdFku l gh gS ¼gSa½(A) 0 < r < R esa] pqacdh; {ks=k ' kwU; ugha gSA(B) R < r < 2R esa] pqqacdh; {ks=k dh fn' kk l ek{k dh fn' kk esa gSA(C) R < r < 2R esa] pqacdh; {ks=k r f=kT; k ds oÙk l s Li ' kZj s[ kh; gS ft l dkdsUnz l ek{k i j gSA(D) r > 2R esa] pqacdh; {ks=k ' kwU; ugha gSASol. AD
Bi fj ufydk
= 0 nI
B[ kks[ kys csyu
= ‘' kwU; ‘(vUnj )
R2R
= r2I0
(ckgj )
4. nks okgu] ft uesa i zR; sd dh xfr u gS] , d gh l h/kh {kSfr t l M+d i j , dnwl j s dh vksj vk j gs gSaA ok; q l M+d dh fn' kk esa w ds osx l s cg j gh gSAbuesa l s , d okgu f1 vkofÙk dh l hVh ct kr k gSA nwl j s okgu esa cSBs gq, i zs{kddks l hVh f2 vkofÙk dh l qukbZ nsr h gSA fu' py ok; q esa /ofu dh xfr V gSAl gh i zdFku gS ¼gSa½(A) ; fn ok; q i zs{kd l s L=kksr dh fn' kk esa cgr h gS f2 > f1A(B) ; fn ok; q L=kksr l s i zss{kd dh fn' kk esa cgr h gS f2 > f1 A(C) ; fn ok; q i zs{kd l s L=kksr dh fn' kk esa cgr h gS f2 < f1A(D) ; fn ok; q L=kksr l s i zs{kd dh fn' kk esa cgr h gS f2 < f1ASol. AB
çs{kd l s L=kkssr
2 1v w uf fv w u
f2 > f1
L=kkssr l s çs{kd
2 1v w uf fv w u
f2 > f1
5. O; at d 2d sin= dk mi ; ksx dj r s gq, ge dks eki dj d dk ekut kuuk pkgr s gSaA dk eku 0 o 90° ds chp esa gSA r j ax nS/; Z dk ekugesa i fj ' kq) r % Kkr gS r Fkk ds eki us esa =kqfV] ds l Hkh ekuksa ds fy, l ekugSA t Sl s dk eku 0° l s c<+r k gS r c(A) d esa fuj i s{k =kqfV fLFkj j gr h gSA(B) d esa fuj i s{k =kqfV c<+r h gSA(C) d esa fHkUukRed =kqfV fLFkj ?kVr h gSA(D) d esa fHkUukRed =kqfV ?kVr h gSA
Sol. D2dsin =
d =
sin2
dd = 2
dcoseccos
dcot.ddd ( d fu; r )
ddd
dd =
ddd
va' k
dd = fuj i s{k =kqfV
6. nks vpkyd R1 r Fkk R2 f=kT; k okys xksyksa dks Øe' k% + o -, dl ekuvk; r u vkos' k ?kuRo l s vkosf' kr fd; k x; k gSA bu xksyksa dks fp=k esa n' kkZ,vuql kj bl i zdkj t ksM+ dj j [ kk x; k gS fd os vkaf' kd : i l s vfr NkfnrgSaA vfr Nkfnr {ks=k ds i zR; sd fcanq i j
R1R2
(A) fLFkj oS| qr {ks=k ' kwU; gSA(B) fLFkj oS| qr foHko vpj gSA(C) fLFkj oS| qr {ks=k dk i fj ek.k vpj gSA(D) fLFkj oS| qr {ks=k dh fn' kk , dl eku gSASol. CD
net 1 20
E r r3
O O'
r1 r2
= 0
OO'3
= fu; r
7. fp=k esa fdl h Bksl dh fof' k"V Å"ek /kkfj r k (C) dk r ki eku (T) i jfuHkZj r k dks n' kkZ; k x; k gSA r ki eku esa 0 l s 500 K r d l eku nj l s l ar rof) gksr h gSaA eku dj fd vk; r u esa i fj or Zu mi s{kuh; gS] fuEu i zdFkuesa dkSu l k ¼l s½ r dZl axr l fUudV l gh gS ¼gSa½\
100 200 300 400 500
C
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com , [email protected]
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(A) 0-100 K ds chp vo' kksf"kr Å"ek dh nj r ki eku i j j Sf[ kdvkfJr r k fn[ kk, xhA(B) 0-100 K r d r ki eku dks c<+kus i j vo' kksf"kr Å"ek 400-500 Kr ki eku ds chp c<+kus dh Å"ek dh r qyuk esa de gSA(C) vo' kksf"kr Å"ek dh nj 400-500 K r ki eku ds chp vi fj ofr Zr gSA(D) Å"ek vo' kks"k.k dh nj 200-300 K r ki eku ds chp c<+ j gh gSASol. ABCD
dTdt = fu; r
d = mC.dT A fodYi(B) C, 0 – 100 ds e/; de gS 400 – 500(C) 400 – 500 = C fu; r gSA
(D) dQdt =
MCdTdt C
dQdt
8. , d gkbMªkst u&l eku i j ek.kq ds bysDVªkWu d{k dh f=kT; k 4.5 a0 gS
t gk¡ a0 cksj f=kT; k gSA bl bysDVªkWu dk d{kh; dks.kh; l aosx 2h3
gSA fn; k
gS fd h Iykad fu; r kad o R fj McxZ fu; r kad gSA i j ek.kq ds O; qÙksft r gksusi j mRl ft Zr fofdj .k ds r j axnS/; Z dh l aHkkouk, ¡ gS
(A) R329
(B) R169
(C) R59
(D) R34
Sol. AC
L = nh2
n = 3
R = a0
2nZ
4.5 a0 = a0
9Z
Z = 2.
hc
= Rhc × 2 21 2
1 1n n
(l aØe.k 1 3)
1
1 = R
119
=
32R9 1 =
932R
(l aØe.k 1 2)
2
1 = R
114
× 4 2 =
13 R
(l aØe.k 2 3)
3
1 = R
1 14 9
× 4 2 = 13 R
3
1 =
20R36 =
95R
Paragraph for Question 9 and 10, d : {k i Fk ds mPpr e fcUnq l s , d 1 kg nzO; eku ds xqVds dksfoj kekoLFkk l s NksM+k x; k gSA ; g i Fk 40 m f=kT; k dk oÙkh; pki gSA xqVdkvi us i Fk i j fcuk yq<+ds gq, l j dr k gSA bl xqVds i j , d ?k"kZ.k cyr kR{kf.kd osx dh foi j hr fn' kk esa yxr k gSA fp=k esa n' kkZ; s vuql kj ] fcUnqQ r d vkus ds fy, ?k"kZ.k dks vfr Øe dj us ds fy, 150 J dk; Z dj uki M+r k gSA ¼xq: Roh; Roj .k g dk eku = 10 ms-2 yhft , )
1 kg300
R P
O x
R
y
9. t c xqVdk fcUnq Q i j i gqapr k gS] bl dh xfr gS
(A) 5 ms-1 (B) 10 ms-1 (C) 310 ms-1 (D) 20 ms-1
Sol. B
1kg
30º
R = 40m P
R = 40m
mg
60º
mg cos60º
R sin 30º = 20
30º
y
x
xq: Roh; fLFkfr t Åt kZ = 0
Q
wf = Ef – Ei (Ei = 0)
– 150 = 2v200
2
v = 10 m/s
10. fcUnq Q i j xqVds i j yxus okys vfHkyac cy dk i fj ek.k gS(A) 7.5 N (B) 8.6 N (C) 11.5 N (D) 22.5 NSol. A
mg cos 60º + 2mv
R = 10 ×
12
+ 10040 = 7.5 N
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com , [email protected]
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Paragraph for Question 11 and 12, d r ki h; fo| qr l a; a=k 600 kW dh ' kfDr 4000 V i j mRi kfnr dj r kgS] t ks 20 km dh nwj h i j mi HkksDr vksa ds mi ; ksx ds fy, ys t k; h t kr hgSA bl dks ; k r ks mPp /kkj k ogu&{ker k okys dsfcy l s Hkst k t k l dr k gS; k nksuksa fl j ksa i j mPpk; h o vi pk; h VªkUl QkWeZj dk i z; ksxdj fd; k t kl dr k gSA i zR; {k i zs"k.k dk nks"k ; g gS fd bl esa Åt kZ dk {k; cgqr vf/kdgksr k gS t cfd VªkUl QkWeZj ds mi ; ksx ds r j hds esa {k; cgqr de gksr k gSAbl r j hds esa , d mPpk; h VªkUl QkWeZj l a; a=k dh vksj yxk; k t kr k gS ft l l s/kkj k dk eku de gks t k, A mi HkksDr k ds fl j s esa vi pk; h VªkUl QkWeZj dki z; ksx fd; k t kr k gS ft l l s mi HkksDr vksa dks , d fo' ks"k de oksYV i j fo| qr' kfDr nh t k l dsA ; g ekuk t k l dr k gS fd dsfcy ' kq) i zfr j ksf/kr gS r FkkVªkUl QkWeZj vkn' kZ gSa] o mudk ' kfDr xq.kkad , d gSA mfYyf[ kr l eLr /kkj kvksa o oksYVr kvksa dk eki rms eku esa gSA
11. ; fn , sl s dsfcy dk mi ; ksx fd; k t k, ft l dk i zfr j ks/k 4.0 km-1
gS r c i zR; {k i zs"k.k dh fLFkfr esa ' kfDr {k; (% esa) gS(A) 20 (B) 30 (C) 40 (D) 50Sol. B
P = vii = 150 AR = 0.4 /km = 0.4 × 20 = 8 gkfuH = I2 R= 150 × 150 × 8= 180000 J
% ' kfDr {k; = 1000600180000
× 100 = 30 %
12. VªkUl QkWeZj ds i z; ksx dj us okyh fof/k esa] ; g ekusa fd mPpk; hVªkUl QkWeZj ds i zkFkfed o f} r h; d esa yi sVksa dh l a[ ; k dk vuqi kr 1 : 10gSA ; fn fo| qr ' kfDr ] mi HkksDr kvksa dks 200 V i j nh t kr h gS r ks vi pk; hVªkUl QkWeZj esa i zkFkfed o f} r h; d ds y i sVksa dh l a[ ; k dk vuqi kr gS(A) 200 : 1 (B) 150 : 1(C) 100 : 1 (D) 50 : 1Sol. A
mPpk; h
110
4000v
v = 40000 oksYV
vi Ppk; h
200200
40000
200 : 1
Paragraph for Questions 13 and 14x-y r y esa R f=kT; k dh oÙkh; d{kk esa , d Q fcUnq vkos' k dks.kh; xfr
l s i fj Øek dj j gk gSA bl s ywi esa cgr h
2Q
vi fj or hZ /kkj k ds r qY; ekuk
t k l dr k gSaA vc , d , dl eku pqacdh; {ks=k dks /kukRed z-fn' kk esa pkywdj r s gSa ft l dk eku 0 l s B r d , d l Sfd.M esa , dl eku nj l s c<+r kgSA ; g ekfu; s fd bl nkSj ku d{k dh f=kT; k fLFkj j gr h gSA paqcdh; {ks=kds yxkus l s d{k esa , d emf i zsfj r gksr k gSA , d i zsfj r fo| qr {ks=k } kj k bdkbZ/ku vkos' k dks l aoÙk ywi ds pkj ksa vksj ?kqekus esa fd; s x; s dk; Z dh ek=kk dksi zsfj r fo| qr okgd cy (emf) dgk t kr k gSA ; g Kkr gS fd t c , d vkos' k, d d{k esa i fjHkze.k dj r k gS r c ml dk pqacdh; f} /kwzo vk?kw.kZ ml ds dks.kh;l aosx ds vkuqi kfr d gksr k gS ft l dk vkuqi kfr d fLFkj kad gSA13. pqacdh; {ks=k esa i fj or Zu ds nkSj ku d{k esa fdl h fo' ks"k {k.k i j i zsfj rfo| qr {ks=k dk eku gS
(A) 4
BR(B)
2BR
(C) BR (D) 2BR
Sol. B
dE.d l B.dAdt
E (2R) = ddt (B) (R2)
E (2R) = R2 dBdt
E = R dB2 dt =
BR2
14. ft l l e; vUr j ky esa pqacdh; {ks=k esa i fj or Zu gks j gk gS] ml vUr j kyds vUr esa] vkos' k ds d{k l s l acaf/kr pqacdh; f} /kqzo vk?kw.kZ esa i fj or Zu gS
(A) -BQR2 (B) 2
BQR2
(C) 2
BQR2
(D) BQR2
Sol. B or C
E = BR2
F = QE = QBR
2
= FRI
=
2
RQBR2 MR
= QB2M
= T = QB2M
(T)
fn; k gSM = L. bl fy; s M = L
= (I) ()
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com , [email protected]
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= 2MR QB T
2M
bl fy, M = 2R QBT2
T = 1 j [ kus i j
M = ± 2R QB
T
i z' u 15 vkSj 16 ds fy, vuqPNsn
, d ukfHkd AZ X dk nzO; eku (A–Z) U; wVkªWuksa , oa Z i zkWVksuksa ds
nzO; ekuksa ds ; ksx l s de gksr k gSA nzO; ekuksa dh deh ds l er qY; Åt kZ dksca/ku Åt kZ dgr s gS A , d nzO; eku M dk Hkkjh ukfHkd m1 , oa m2 æO; ekuksads nks gYds ukfHkdksa esa fo?kfVr gks l dr k gS] ; fn (m1+m2) < M A r Fkkm3 , oa m4 �æO; ekuksa�ds�nks�gYds�ukfHkd�i w.kZ�l ay; u�dj ds�, d�M'æO; eku dk Hkkj h ukfHkd cuk l dr s gSa] ; fn (m3+m4) > M' A dqNi j ek.kqvksa ds æO; eku uhps Vscy es fn; s x; s gS %
11H 1.007825 u 2
1H31H 3.016050 u 4.002603 u2.014102 u 4
2He
6.015123 u 69.925325 u 81.916709 u7.016004 u73Li 70
30Zn 8234Se
151.919803u 208.980388u 209.982876u205.974455u20682 Pb 209
83 Bi 21084 Po
63 Li
15264 Gd
15. l gh i zdFku gSA
(A) ukfHkd 63Li , d , sYQk d.k mRl ft Zr dj l dr k gSA
(B) ukfHkd 21084 Po , d i zksVkWu mRl ft Zr dj l dr k gSA
(C) M~; wwVkWu vkSj , sYQk d.k i w.kZ l ay; u dj l dr s gSA
(D) ukfHkd 7030Zn , oa ukfHkd 82
34Se i w.kZ l ay; u dj l dr s gSASol. C
36Li 6.015123 u
24He 4.002603 u
12H 2.014102 u
36Li 1 2
2 4H + HenzO; eku
denzO; ekuvf/kd
( )l aHko ugh
84210Po 1 83
1 209H + BinzO; eku
denzO; ekuvf/kd
( )l aHko ugh
1 22 4H + He 3
6Li nzO; eku
denzO; ekuvf/kd
( )l aHko
30 3470 82Zn + Se 64
152GdnzO; eku
denzO; ekuvf/kd
( )l aHko ugh
16. t c foj kekoLFkk esa ukfHkd 21084 Po , sYQk&{k; dj r k gS] r c , sYQk d.k
dh xfr t Åt kZ (keV esa) gksr h gSA(A) 5319 (B) 5422(C) 5707 (D) 5818Sol. D
84210Po 82 2
206 4Pb + He M = 0.005818 u( M)c2 = 5.419467 MeV ~ 5420 KeV
K() = 206210 (5420)
= 5316 KeV ~ 5319 KeV
17. , d µ1 vi or Zukad ds l edks.k fi zTe dks µ2 vi or Zukad ds vk; r kdkjCykd esa j [ kk x; k gSA i w.kZ O; oLFkk µ3, vi or Zukad ds ek/; e l s fp=k esan' kkZ; s vuql kj f?kj h gqbZ gSA i zdk' k dh fdj .k ‘e’ vk; r kdkj CykWd i jvfHkycaor vki fr r gksr h gSA µ1,µ2, vkSj µ3, ds ekuksa i j fuHkZj gksr h gqbZi zdk' k dh fdj .k pkj l aHko i Fkksa ‘ef’, ‘eg’, ‘eh’ ; k ‘ei’ eas l s , d ysr h gSA
l wph I esa fn; s x; s i Fkksa dks l wph II dh vi or Zukad dh ' kr ksZ l sl qesfyr dhft , vkSj l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx dj ds mÙkjpqfu; s % l wph I l wph II
(P) e f 1. µ1> 2 µ2
(Q) e g 2. µ2>µ1 , oa µ2>µ3(R) e h 3. µ1=µ2
(S) e i 4. µ2<µ1< 2 µ2 , oa µ2>µ3
Codes :P Q R S
(A) 2 3 1 4(B) 1 2 4 3(C) 4 1 2 3(D) 2 3 4 1
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Sol. D(P) e f : i w.kZ vkUr fj d i j kor Zu ugh gks j gk gS r Fkkfdj .k 1 o 2 l r g ds vfHkyEcor ~ t k j gh gS vkSj fdj .k l r g2 o 3 vfHkyEcor ~ l ss nwj t k j gh gSA2 > 1
(Q) e g : dksbZ >qdko ugh gS 1 = 2
(R) e h : 1 o 2 l r g dh l r g ds vfHkyEcor ~ l ss nwji w.kZ vkUr fj d i j kor Zu ugh gks j gk gS i qu% 2 o 3 ds vfHkyEcor ~
l s nwj t k j gh gSA 2 > 3 vkSj 2 2 > 1 gSA
(S) e i : i w.kZ vkUr fj d i j kor Zu gks j gk gS
sin 45º > 2
1
2 < 1
2
1 > 2 2.
18. l wph I dks l wph II l s l qesfyr dhft , vkSj l wfp; ksa ds uhps fn; sx; s dksM dk i z; ksx dj ds l gh mÙkj pqfu; s %
l wph I l wph II(P) cksYV~l eku fu; r kad 1. [ML2 T–1](Q) ' ; kur k xq.kkad 2. [ML-1 T–1](R) Iykad fu; r kad 3. [MLT-3 K–1](S) Å"ek pkydr k 4. [ML2T–2 K–1]
Codes :P Q R S
(A) 3 1 2 4(B) 3 2 1 4(C) 4 2 1 3(D) 4 1 2 3Sol. C
32
KT = E , k = [ML2 T–2 K–1]
F = 6rv, = [ML–1 T–1]E = h h = [ML2 T–1]
Q = 1 2KA T T t
L
K = [ML T–3 K–1]
19. , d , d&i j ek.kqd vkn' kZ xSl ds , d eksy dks fp=k esa n' kkZ; s PVvkj s[ k ds vuql kj nks pØh; i zØeksa EFGE o EFHE esa yst k; k t kr k gSA l acf) r i zØe ' kq) r % l evk; r fud] l enkch l er ki h; ; k: nks"e gSA
Vl wph I esa fn; s x; s i Fkksa dks l wph II esa fd; s x; s dk; Z ds i fj ek.k
ds l kFk l qesfyr dhft , vkSj l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx l ghmÙkj pqfu; sA l wph I l wph II(P) GE 1. 160 P0V0In2(Q) GH 2. 36 P0V0(R) FH 3. 24 P0V0(S) FG 4. 31 P0V0Codes :
P Q R S(A) 4 3 2 1(B) 4 3 1 2(C) 3 1 2 4(D) 1 3 2 4Sol. A
(32T )0
(32P V )0 0
T0
(8T , 8V )0 0
(P , 32V )0 0
V0 V
P0
32P0
P
G – E l enkch;W = Pv = P0(32 V0 – V0) = 31P0V0 ..........(4)
G – X l enkch;P0(32 V0 – 8V0) = P0 × 24V0
= 24 P0V0 ..........(3)
F – X r1
nR
(Tf – Ti) = WÅ/kks"e
)T8T32(TVP
23
000
00 00
00 T24TVP
23
36 P0V0 ..........(2)
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7
F – G nR (32 T0) ln 32 0
0
vv
nR 32 T0 ln 25
0 00
0
P V5 32T ln2
T
= 160P0V0 ln2 ..........(1)
20. l wph I esa dqN ukfHkdh; i zfØ; k, ¡ nh xbZ gSSA l wph II esa bu i zfØ; kvksads t ud ukfHkd o , d vafr e ukfHkdh; [ kaM fn, x, gSA l wfp; ksa ds uhpsfn; s x, dksM dk i z; ksx dj ds l gh mÙkj pqfu, % l wph I l wph II
(P) , sYQk&{k; 1. 15 158 7O N .....
(Q) +{k; 2. 238 23492 90U Th .....
(R) fo[ kaMu 3. 185 18483 82Bi Pb .....
(S) i zksVkWu mRl t Zu 4. 239 14094 57Pu La .....
Codes :P Q R S
(A) 4 2 1 3(B) 1 3 2 4(C) 2 1 4 3(D) 4 3 2 1
Sol. C(P) , YQk {k;
42
23490
23892 HeThu (2)
(Q) B+ {k;
01
157
158 NO
(1)
(R) fo[ k.Mu
140239 995794 37Pu La (4)
n 1.5 unstablep
vfLFkj
(S) çksVksu mRl t Zu
PbBi18482
18583 + çksVksu (3)
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8
PART - II [CHEMISTRY]
SECTION – ASINGLE CORRECT
21. Ksp (Ag2CrO4) dk eku 298 K i j 1.1 × 10–10 gSA0.1 M AgNO3 ds foy; u esa Ag2CrO4 dh foys; r k eksy@yhVjesa gSA(A) 1.1 × 10–11 (B) 1.1 × 10–10
(C) 1.1 × 10–12 (D) 1.1 × 10–9
Sol. BAg2CrO4 = 2Ag+
(aq.) + CrO4––
(aq.)—— 0.1 +2s s
0.1 Mksp = [Ag+]2 [CrO4
––]1.1 × 10–12 = 0.12 × (s)s = 1.1 × 10–10 M
22. fuEufyf[ kr vfHkfØ; k ds mRi kn@mRi knksa dks cr k, ¡
OH
CH3
CHCl3OH–
OH
CH3
OHC CHO
P Q
O
H C3 CHCl2
R
H C3 CHCl2
OH OH
CH3
S
CHO
(A) P(eq[ ; ) (B) Q(xkS.k)(C) R(xkS.k) (D) S(eq[ ; )
Sol. B DO–H O
O
CH3 CH3
CH3CH3
CH3
OH CCl2
OC Cl2
H
CCHOCl
Cl-2Cl
OH
OH
Major(S)
O
CH3
CCl2 HOH
O O
C Cl2 CHCl2CH3 CH3
Minor(Q)
23. fuEufyf[ kr vfHkfØ; k dk eq[ ; mRi kn gS
OH
SO H3
t yh; r qY; eku Br (3.0, )2
OH
SO H3
Br Br
P Q R
OH OH
SO H3
S
Br
Br BrOH
BrBr
BrBr Br Br
Br
Sol. B
SO3
SO3
SO3
Br Br
SO3
SO H3
O
O
O
O OH
OH
Br – Br
H
Br
O
BrBr
Br
Br Br
Br Br
Br Br
Br
Br2
– HBr
Br – Br
O
BrO=S=O
– SO 3
O
HOH
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9
Hint:
SO H3
OH
Br
OHBrBr
aq. Br (3.0 mole)2
Bromo desulphonation
24. j kl k; fud vfHkfØ; kvksa (I vkSj II) ds i wj s gksus ds ckn j kl k; fudfeJ.k esa dkcZfud ; kSfxd ¼; kSfxdksa½ dks cr k, ¡A
vfHkfØ; k I :
O
H C3 CH3
Br2(1.0 mol)t yh; NaOH
vfHkfØ; k II :
O
H C3 CH3
Br2(1.0 mol)
CH COOH3
(1.0 mol)
(1.0 mol)
O
H C3 CH Br2
P Q R
O
H C3 CBr3
O
BrH C2 CH Br2
O
BrH C2 CH Br2
S T
O
H C3 ONa
U
CHBr3
(A) vfHkfØ; k I : P and vfHkfØ; k II : P(B) vfHkfØ; k I : U, , l hVksu vkSj vfHkfØ; k II : Q , l hVksu(C) vfHkfØ; k I : T, U, , l hVksu vkSj vfHkfØ; k II : P(D) vfHkfØ; k I : R, , l hVksu vkSj vfHkfØ; k II : S, , l hVksu
Sol. A
Reaction-1
CH –C–CH23
O HOH CH –C–CH23
O
–Br Br—Br (1 me)
CH –C–CH23 –Br
O
Reaction-2
CH –C–CH33
CH –C–OH3
O O
CH –C=CH23
–Br
–Br
Br—Br
–HBr
CH –C–CH23 –Br
CH –C–CH23 –Br
O
O
OH
Enol is Nu
H
25. O3 ds l anHkZ esa l gh oDr O; o gS (gS)(A) O-O vkca/kksa dh yackbZ cj kcj gSA(B) O3 dk r ki h; fo; kst u Å"ek' kks"kh gSA(C) O3 i zfr paqcdh; gSA(D) O3 dh l aj puk cafdr gksr h gSA
Sol. A,C,D
O OO OO O
sp2
vuqukn ds dkj .k l eku cU/k yEckbZ eqM+h gqbZ vkdkj dh r Fkk
i zfr pqEcdh; bl dk fo| Vu m"ek{ksi h gksr k gSA2O3 3O2 + Heat
26. fuEufyf[ kr ukfHkdh; r Rokar j .k8944Be X Be Y
esa (X, Y) gS@gSa(A) (, n) (B) (p, D)(C) (n, D) (D) (, p)
Sol. A,B
(A) 9 0 9 14 0 4 0Be Be r (T)
(B) DBepBe 20
84
11
94 (T)
(C) DBenBe 21
84
10
94 (F)
(D) 9 0 8 14 0 4 1Be Be P (F)
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27. /kkr qvksa ds fu"d"kZ.k esa dkcZu vk/kkfj r vi p; d fof/k dk i z; ksx fduv; Ldksda esa ugha gksr k gS\(A) SnO2 l s fVu(B) Fe2 O3 l s vk; j u(C) Al2O3 l s , syqfefu; e(D) MgCO3. CaCO3 l s eSXuhf' k; e
Sol. C D(A) SnO2 + C Sn + COdkcZu vi p; u fof/k(B)Fe2O3 + CO Fe + CO2dkcZu vi p; u fof/k ysfdu eq[ ; vi pk; d CO. gS
(C) Al2O3 l s , Y; wfefu; e gkWy & gsj ksYV fof/k ds } kj k
vi pf; r gksr k gSA
(D) MgCO3.CaCO3 Hkt Zu MgO.CaO
2(MgO.CaO) +FeSi 1150º 2Mg + Ca2SiO4
+ Fe; g fl fydk r ki h; i zØe dgykr k gSa
28. CaCO3 (Bksl ) ds Å"eh; fo?kVu dh l kE; koLFkk dk v/; ; ufofHkUu voLFkkvksa esa fd; k x; kACaCO3 (s) CaO(s) + CO2(g)bl l kE; koLFkk ds fy; s] l gh i zdFku gS (gSa)(A) H r ki eku i j fuHkZj dj r k gSA(B) l k: E; koLFkk fLFkj kad (K) CaCO3 ds i zkj fEHkd i fj .kke i jfuHkZj ugha dj r k gSA(C) K fu; r r ki eku i j CO2 ds nkc i j fuHkZj dj r k gSA(D) H mRi zsj (vxj gks) ds i zHkko i j fuHkZj ugha dj r k gSA
Sol. ABD(A) H T i j fuHkZj dj r k gSA
2
112
T
TprTT dTCHH (T)
(B) l kE; oLFkk fLFkj kad l kUnzr k i j fuHkZj ugh gSA(T)(C) K fu; r r ki eku i j CO2 ds nkc i j fuHkZj dj r k gSA(F)(D) H mRi zsj d (vxj gks) ds i zHkko i j fuHkZj ugh dj r k gsA(T)
[ k.M-2 : (vuqPNsn i zdkj )bl [ k.M esa fl ) kar ksa] i z; ksxksa vkSj vk¡dMksa vkfn dks n' kkZus okys 4vuqPNsn gSA pkj ksa vuqPNsnksa l s l acaf/kr vkB i z' u gSa] ft uesa l s gjvuqPNsn i j nks i z' u gSA fdl h HKh vuqPNsn esa gj i z' u ds pkj fodYi(A), (B), (C) vkSj (D) gSA ft uesa l s dsoy , d gh l gh gSA
i z' u 29 , oa 30 ds fy; s vuqPNsnnks vdkcZfud yo.kksa ds , d feJ.k dk t yh; foy; u r uq HCI vEy} kj k vi p; u dj , d vo{ksi (P) vkSj , d fQYVªsV (Q) nsr k gSAvo{ksi P xeZ t y esa ?kqyu' khy gSA fQYVªssV (Q) r uq [ kfut vEyh;ek/; e esa H2S } kj k foospu dj vi fj ofr Zr j gr k gSA fdUr q , eksfudyek/; e esa H2S ds l kFk vi {ksi (R) nsr k gSA vo{ksi R ds l kFk t yh;NaOH ek/; e r Fkk H3O2 dh vfHkfØ; k j axhu foy; u (S) nsr h gSA
29. vo{ksi P esa mi fLFkr gSA(A) Pb2+ (B) Hg2
2+
(C) Ag+ (D) Hg2+
Sol. APb2+ + Cr3+ dil. HCl PbCl2 + CrCl3 (Q)
' osr vo{ksi xeZ t y esa
foys; gSa t cfd B.Ms
t y esa vfoys;
30. j axhu foy; u S esa mi fLFkr gSA(A) Fe2(SO4)3 (B) CuSO4(C) ZnSO4 (D) Na2CrO4d
Sol. DCrCl3 + H2S + dil HCl No ppt.CrCl3 + H2S + NH4OH Cr(OH)3(s)
Green (R)
H S 2
NH OH4
2H + OH+
S2– vk; u l kUnzr k c<+r h gSa t ks fd vk; u vo{ksi dkj .k gSA
2H + S+ 2–
NH + OH4+
H O2
Cr(OH)3(s) + NaOH + H O2 2
vkWDl hdkj d Na2CrO4
i hyk foy; u (R)
i z' u 31 , oa 32 ds fy; s vuqPNsn
P vkSj Q , d MkbdkcksZfDl fyd vEy C2H4O4 ds nks l eko; oh gSAnksuksa Br2/H2Odks j axghu dj r s gSA xeZ duus i j P pØh; , ugkbMªbMcukr k gSAr uq {kkj h; KMnO4 } kj k P vkSj Q vyx&vyx vfHfØ; k dj , dvFkok , d l s vf/kd ; kSfxd S, T vkSj U.
COOH COOH COOH
COOH COOH COOH
H H H
H HO H
OH OH OH
OH H OH
S T U
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31. P vkSj Q } kj k cus ; kSfxd Øe' k% gS(A) /kzqo.k ; kSfxd S , d /kzqo.k fuf"Ø; ; qXe (T, U)(B) /kzqo.k fuf"Ø; S , oa /kqzo.k fuf"Ø; ; qXe (T, U)(C) /kzqo.k ?kw.kZd ; qXe (T, U) , oa /kzqo.k ?kw.kZd S(D) /kzqo.k fuf"Ø; ; qXe (T, U) , oa /kqzo.k fuf"Ø; S
Sol. B
H C–OH
HHO–C
O
O
+
H C–OH
H C–O–H
O
O
Br /H O2 2
j axghu gks x; k(Q) (P)
KMnO4Rkuq
fl l 0 gkbMªksDl hyhd j .k KMnO4Rkuq
fl l 0 gkbMªksDl hyhd j .k
COOH COOH COOH
COOH COOH COOH
H H H
H HO H
OH OH OH
OH H OH
S T U
(T) (U) (S) ehl ks ; ksfxd
(i zdkf' kd fu"Øh; ); qXe j sl sfed feJ.k gSA
vr % i zdkf' kd fu"Øh;
32. fuEufyf[ kr vfHkfØ; k vuqØeksa esa V vkSj W
2H /NlQ
+ VAlCl (andhydrous)3
2 4
1.Zn Hg /HCl2.H PO
(A)
O
O
O vkSj
OWV
(B) W
CH OH2
CH OH2V
vkSj
(C)
O
O
O
WV
vkSj
(D) CH OH2
HOH C2
CH OH2V W
vkSj
Sol. A
C – OH
O
H
H
C
OHO
Q
H /Ni2 O
O
O(V)
Fumeric acid
+ AlCl (anhydrous)3
Fridel craft acylation
O
CH2
CH2
CH2
CH2
C – OH
O
O
O
O1.Zn, Hg/HClClemmenson
reduction
COOH
H PO3 4
O(W)
(V)
i z' u 33 , oa 34 ds fy; s vuqPNsn, d xSl ds fuf' pr nzO; eku 'm' dh voLFkk i fj or Zu K l s L M l sN r Fkk oki l K esa fp=k } kj k fn[ kkbZ gS
Pressure
Volume
K L
N M
nkc
vk; r u
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33. Øfed i fj pkyu t ks bu voLFkk i fj or Zuksa esa l gk; d gS] og gS(A) xeZ] BaMk] xeZ] BaMk dj us i j(B) BaMk] xeZ] BaMk] xeZ dj us i j(C) xeZ] BaMk] BaMk] xeZ dj us i j(D) BaMk] xeZ] xeZ] BaMk dj us i j
Sol. C
P
V
N M
LK
K L ; xeZ
L M ; B.Mk
M N ; B.Mk
N K ; xeZ
34. voLFkk i fj or Zuksa dh fLFkfr ; ksa esa l evk; fud çØe ; qXe gS(A) K l s L vkSj L l s M (B) L l s M vkSj N l s K(C) L l s M vkSj M l s N (D) M l s N vkSj N l s K
Sol. BL M & N K : nksuks l evk; r uhd i z+de
i z' u 35 , oa 36 ds fy; s vuqPNsnCl2 xSl r uq vkSj l kanz NaOH ds t yh; foy; u } kj k Øe' k% BaMsvkSj xeZ voLFkk esa vfHkfØ; k dj nks ¼fHkUu½ Dyksjhu ds vkWDl h&vEyds l ksfM; e yo.k] P vkSj Q nsr s gSA Cl2(g) pkj dksy dh mi fLFkfresa SO2 l s vfHkfØ; k dj mRi kn R nsr k gSA R l Qsn QkLQksj l } kj kvfHkfØ; k dj ; kSfxd S nsr k gSA S dh t y&vi ?kVu fØ; k QkLQksj ldk , d vkDl ksvEy T nsr h gSA
35. P and Q, respectively, are the sodium salts of(A) gkbi ksDyksj l vkSj Dyksfj d vEy(B) gkbi ksDyksj l vkSj Dykj l vEy(C) Dyksfj d vkSj i j Dyksfj d vEy(D) Dyksfj d vkSj gkbi ksDyksj l vEy
Sol. ACl2 + NaOH NaOCl + NaCl
(cold & dil.) (P)Cl2 + NaOH NaClO3 + NaCl
(hot & conc.) (Q)NaOCl l ksfM; e gkbi ksDyksj kbV
NaClO3 l ksfM; e Dyksj sV
vr % l EcfU/kr vEy HOCl r Fkk HClO3 gSA
36. R, S vkSj T, Øe' k% gS(A) SO2Cl2, PCl5 vkSj H3PO4
(B) SO2Cl2, PCl3 vkSj H3PO4
(C) SOCl2 PCl3 vkSj H3PO2
(D) SOCl2 PCl5 vkSj H3PO4
Sol. ACl2 + SO2 charcoal SO2Cl2
(R)SO2Cl2 + P4 PCl5 + SO2
(S)PCl5 + H2O H3PO4 (T)
(Oxy acid)
37. l wph I esa fyf[ kr vl ar qfyr vfHkfØ; kvksa esa vi znf' kZr vfHkdeZd @voLFkk, ¡ l wph II eas nh xbZ gSA l wph I dks l wph II l s l qesfyr dhft ,r Fkk l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx dj ds l gh mÙkj pqfu; s&l wph I l wph II
P. PbO2+H2SO4 ? PbSO4+O2 vU; mRi kn 1. NO
Q. Na2S2O3+H2O? NaHSO4+ vU; mRi kn 2. I2
R. N2H4 ? N2 + vU; mRi kn 3. xeZ
S. XeF2 ? Xe + vU; mRi kn 4. Cl2
dksM :P Q R S
(A) 4 2 3 1(B) 3 2 1 4(C) 1 4 2 3(D) 3 4 2 1
Sol. D(S)XeF2 + NO NOF + Xe(Q)Na2S2O3 + Cl2 + H2O NaHSO4+NaCl
+ H2O, sUVhDyksj (Cl2 gVkus i j )
(P)PbO2 + H2SO4 warm PbSO4 + O2 + H2O(R)N2H4 + I2 I– + N2 + H3O
+
bl vfHkfØ; k dk mi ; ksx vk; r uhd fo' ys"k.k esa gkbMªst hu ds
ek=kkRed fo' ys"k.k esa fd; k t kr k gSA38. l wph I eas fn; s x; s j kl k; fud Li kUr j .kksa dks l wph II eas fn, x,
mi ; qDr vfHkdeZdksa ds l kFk l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn,x; s dksM dk i z; ksx dj ds l gh mr j pqfu; s %l wph I
ClP.
ONaQ. OEt
R.OH
S.
OH
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l wph II1. (i) Hg(OAc)2; (ii) NaBH22. NaOEt3. Et-Br4. (i) BH3; (ii) H2O2/NaOH
dksM :P Q R S
(A) 2 3 1 4(B) 3 2 1 4(C) 2 3 4 1(D) 3 2 4 1
Sol. A
(P) NaOEtCl + EtOH + NaCl
(Q) O Na + Et–Br O – Et
(R) Hg(OAc)2NaBH4
OH (Oxymercuration)
demurcuration
(S) (i) BH3
(ii) H O /NaOH2 2
OH (Hydroboration
oxidation)
39. X ds t yh; foy; u esa Øe' k% Y dk t yh; foy; u /khj s & /khj s Mkykt kr k gS t Sl s l wph I es fn[ kk; k x; k gSA bu vfHkfØ; kvksa l s mRi Uupkydr k dh fHkUur k l wph II esa nh xbZ gSA l wph I dks l wph II l sl wesfyr dhft , r Fkk l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksx dj dsl gh mr j pqfu; s &l wph IP. (C2H5)3N + CH3COOH
X YQ.KI (0.1M) + AgNO3(0.01M)
X YR. CH3COOH + KOH
X YS. NaOH + HI
X Yl wph II1. pkydr k ?kVr h gS vkSj r Ri ' pkr ~ c<r h gSA2. pkydr k ?kVr h gS vkSj r Ri ' pkr ~ vf/kd i fj ofr Zr ugha gksr h gSA3. pkydr k c<r h gS vksj r Ri ' pkr ~ vf/kd i fj ofr Zr ugha gksr hA4. pkydr k vf/kd i fj ofr Zr ugah gksr h vkSj i Ri ' pkr ~! c<r h gSA
Sol. A(P) (C2H5)3N + CH3COOH salt WB WA
Eq. Point
G
Vol. of WB added
Ans = 3
(Q) KI + AgNO3 AgI + KNO3
Eq. Point
G
Vol. of KI added
Ans = 4
(R) CH3COOH + KOH CH3COOk + H2O WA SB
Eq. Point
G
Vol. of WA added
Ans = 2
(S) NaOH + HI NaI + H2O SB SA
Eq. Point
G
Vol. of SB added
Ans = 1
40. ekud vi pk; d foHko 25ºC i j fuEufyf[ kr gSAEº (Fe3+, Fe2+) = + 0.77 V;Eº (Fe2+, Fe) = 0.44 VEº (Cu2+, Cu) = +0.34 VEº (Cu+, Cu) = +0.52 VEº [O2(g) + 2H2O+ 4e– 2H2O] = + 1.23 V;Eº [O2(g) + 2H2O+ 4e– 4OH–] = + 0.40 VEº (Cr3+, Cr) = –0.74 V;Eº (Cr2–, Cr) = 0.91 Vl wph I esa fn; s x; s j sMkDl ; qXeksa dks l wph II esa fn; s x; s vi pk; dfoHko l s l wesfyr dhft , r Fkk l wfp; ksa ds uhps fn; s x; s dksM dk i z; ksxdj ds l gh mÙkj pqfu; s &
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l wph IP. Eº (Fe3+, Fe)Q. Eº (4H2O 4H+ + 4OH–)R. Eº (Cu2+ + Cu 2Cu+)S. Eº (Cr3+, Cr2–)l wph II1. –0.18 V2. –0.4 V3. –0.04 V4. –0.83 V
Sol. D
(P) (1) e– + Fe+3 Fe+2, 01E = + 0.77 V
(2) 2e– + Fe+2 Fe, 02E = – 0.44 V
__________________________ (3) 3e– + Fe+3 Fe(s) __________________________(3) = (1) + (2)
02
01
03 GGG
)FE2()FE1(FE3 02
01
03
3)44.0(277.0
3E2E
E02
010
3
= – 0.037= – 0.04VP = 3
(Q) (1) O2(g) + 2H2O + e– OH– V40.0E0
(2) 2H2O O2 + 4H+ + 4e– –1.23 V __________________________ (3) 4H2O 4H+ + 4OH–
__________________________(3) = (1) + (2)
02
01
03 GGG
0 0 03 1 24FE ( 4FE ) ( 4FE )
0 0 03 1 2E E E
= 0.40 + (–1.23)= – 0.83 VQ = 4
(R) (1) Cu+2 + 2e– Cu 01E = + 0.34 V
(2) Cu+ + e– Cu 02E = + 0.52 V
(3) Cu+2 + e– Cu+ 03E
(3) = (1) – (2)0 0 03 1 2G G G
0 0 03 1 21FE 2FE ( 1FE )
0 0 03 1 2E 2E E
= 2 × 0.34 – (0.52)= 0.16 V
(1) Cu+2 + e– Cu+ 01E = + 0.16 V
(2) Cu Cu+ + e– 02E = – 0.52 V
(3) Cu+2 + Cu 2Cu+ 03E = ?
(3) = (1) + (2)0 0 03 1 2G G G
0 0 03 1 21FE ( 1FE ) ( 1FE )
0 0 03 1 2E E E = + 0.16 + (–0.52) = – 0.36 V
(S) (1) Cr+3 + 3e– Cr 01E 0.74V
(2) Cr+2 + 2e– Cr 02E 0.91V
__________________________
(3) Cr+3 + e– Cr+2 03E ?
__________________________(3) = (1) – (2)
0 0 03 1 2G G G
)FE2()FE3(FE1 02
01
03
)91.0(2)74.0(3E2E3E 02
01
03
= – 0.4 VS = 2
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15
PART - III [MATHEMATICS]
SECTION – ASingle Correct
41. a R (l Hkh okLr fod l a[ ; kvksa dk l eqPp; ), a – 1, ds fy,
nlim
)]nna(...)2na()1na[()1n()n...21(
1a
aaa
= 601
r c a =
(A) 5 (B) 7 (C) 215
(D) 217
Sol. B,D
nlim
a 1 a a 11 2
a 1 2
1 .n a n a n .....1(a 1)
1 601n(n 1) .n a
2
nlim
1 22
a
1 a a .... 1a 1 n n1 6011 n1 a
n 2
11a 1
1 60a2
(a + 1) (2a + 1) = 1202a2 + 3a – 119 = 02a2 + 17a – 14a – 119 = 0
(a – 7) (2a + 17) = 0
a = 7, 172
42. x – v{k dks ewyfcUnq l s 3 nwj h i j Li ' kZ dj us okyk (okys) r Fkk
y – v{k i j 72 var %[ k.M cukus okyk (okys) oÙk gS (gSa)(A) x2 + y2 – 6x + 8y + 9 = 0(B) x2 + y2 – 6x + 7y + 9 = 0(C) x2 + y2 – 6x – 8y + 9 = 0(D) x2 + y2 – 6x – 7y + 9 = 0
Sol. A,C
2 Cf2 = 72f2 – C = 7 .....(i)r Fkk oÙk (3, 0) l s
; k (–3, 0) .....(ii)r Fkk f=kT; k = 4
43. nks j s[ kk, ¡ L1 : x = 5, 2z
3y
r Fkk L2 : x =
,
2
z1
y l er yh; gSaA r c dk eku gks l dr k gS
(A) 1 (B) 2 (C) 3 (D) 4Sol. A,D
05x
= 3
y =
z2
0x
= 1
y
= 2
z
005210
230
= 0
a = 4, 1
44. f=kHkqt PQR, esa P cgÙke dks.k gS r Fkk cos P = 31
A bl ds
vfr fjDr f=kHkqt dk var %oÙk Hkqt kvksa PQ, QR r Fkk RP dks Øe' k%N, L r Fkk M i j bl r j g Li ' kZ dj r k gS fd PN, QL r Fkk RMdh yEckb; k¡ Øekxr l e i w.kZl a[ ; k, ¡ gSA r c f=kHkqt dh Hkqt k(Hkqt kvksa) dh l EHkkfor yEckbZ (yEckb; k¡) gS (gSa)(A) 16 (B) 18 (C) 24 (D) 22
Sol. B,D
cos P = PR.PQ.2QRPRPQ 222
31
= 1m3m2
1m2
2
P 2m (2m+2) R
(2m+2)
(2m+4)
Q
(2m+4)
2m
m = 4 r Fkk m = –1bl fy, Hkqt kvksa dh yEckbZ = 10, 20, 22
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45. ekuk fd w =2
i3 r Fkk P = {wn : n = 1, 2, 3, .....}.
bl d s v fr fj Dr H1 =
21zRe:Cz r Fkk H2
=
21zRe:Cz , t gk¡ C l fEeJ l a[ ; kvksa dk l eqPp;
gSA ; fn z1 P H1, z2 P H2 r Fkk O ewyfcUnq i znf' kZrdj r k gS, r c z1 Oz2 =
(A)2
(B)6
(C)32
(D)65
Sol. C,D
½/6
/3
–½
z1z1z '2
z "2
z "'2
H1
H2
O
P = , 2, 3 ...............
1 2
1 2
z o z ' 2 /3z oz " 5 /6
46. ; fn 3x = 4x – 1, r c x =
(A) 12log22log2
3
3
(B) 3log22
2
(C) 3log11
4 (D) 13log23log2
2
2
Sol. A,B,C3x = 4x–1
x = (x – 1) log34 x (1 – log34) = –log34
x = 12log22log2
3
3 = 3log2
2
2 = 3log11
4
47. bdkbZ dk , d l fEeJ ?kuewy yhft ; s, t gk¡ 1 r Fkk
P = [pij] , d n × n vkO; wg yhft ; s, t gk¡ pij = i+jA r c
P2 0, t c n =(A) 57 (B) 55 (C) 58 (D) 56
Sol. B,C,DLi "Vr % n, 3 dk xq.kt gksuk pkfg,
48. Qyu f(x) = 2 |x| + |x + 2| – ||x + 2 |– 2|x|| dk
, d LFkkuh; U; wur e ; k , d LFkkuh; vf/kdr e ft u x ds eku i j
gS] og gSa
(A) – 2 (B) 32
(C) 2 (D) 32
Sol. A,B
f(x) = (2|x| + |x + 2|) – |x|2|2x|
f(x) =
|x|2|2x|if|2x|2|x|2|2x|if|x|4
y=2|x|
y=|x+2|
2–2 O32
x
y
So f(x) =
2x,2x2x0,x2
0x32,x2
3/2x2,2x2x,2x
f'(x) =
2x12x02
0x3/223/2x21
2x1
Li "Vr % U; wur e r Fkk vf/kdr e x = –2, 32
, 0 i zkIr gksr k gS
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i z' u 49 r Fkk 50 ds fy; s vuqPNsn
ekuk fd f : [0, 1] R (l Hkh okLr fod l a[ ; kvksa dk l eqPp; ), d Qyu gSA eku yhft ; s Qyu f nks ckj vodyuh; gS,f (0) = f (1) = 0 r Fkk f" (x) – 2f' (x) + f(x) ex,x [0, 1] dks l ar q"V dj r k gSA
49. fuEu esa l s dkSu 0 < x < 1 ds fy, l R; gS ?
(A) 0 < f(x) < (B) 21)x(f
21
(C) 1)x(f41
(D) – < f (x) < 0
Sol. Df(x) = f(x) e–x
g''(x) 1g(1) = g(0) = 0
dsoy l aHko vkys[ k gks l dr k gS
1/4 1
g'(x) < 0 f(x) < 0 – < f(x) < 0
50. ; fn Qyu e–x f(x) var j ky [0, 1] esa vi uk U; wur e eku
x =41
i j ysr k gS, r c fuEu esa l s dkSu l R; gS ?
(A) f '(x) < f(x), 41
< x < 43
(B) f '(x) > f(x), 0 < x < 41
(C) f ' (x) < f (x), 0 < x < 41
(D) f ' (x) < f (x), 43
< x < 1
Sol. C
f(x),
41,0 esa âkl eku gS
g(x) = e–x f(x)g'(x) = e–x f'(x) – e–x f(x) < 0
x
41,0
e–x (f'(x) – f(x)) < 0
x
41,0
i z' u 51 , oa 52 ds fy; s vuqPNsn
ekuk fd PQ i j oy; y2 = 4ax dh , d ukHkh; t hok gSA fcUnqvksa
P r Fkk Q i j i j oy; dh Li ' kZ j s[ kk, ¡ , d fcUnq i j feyr h gSa t ks
fd j s[ kk y = 2x + a, a > 0 i j fLFkr gSA
51. t hok PQ dh yEckbZ gS :(A) 7a (B) 5a (C) 2a (D) 3a
Sol. B
(–a,–a)
P(at ,2at)2
(a,0)
y =4ax2
Q
O
yEckbZ = 4a cosec2 y (–a) = 2a (x – a)COC: y = –2x + 2atan = –2
2
1
5 sin = 52
cosec = 25
yEckbZ = 45
× 4a
52. ; fn t hok PQ, y2 = 4ax ds ' kh"kZ i j dks.k var fj r dj r h gS
r c tan =
(A) 732
(B) 732
(C) 532
(D) 532
Sol. D
(–a,–a)
P(at ,2at )1 12
(a,0)
y =4ax2
Q
O
12
1 ta2,
ta
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MOP = 1t2
MOQ = –2t1
– 2 = aat
at22
1
1
t1 = 2
15
tan = 41
t2t2
11
tan = –32
(t1 + 1/t1)
t1 dk eku j [ kus i j (D)
i z' u 53 , oa 54 ds fy; s vuqPNsnekuk fd S = S1 S2 S3, t gk¡S1 = {z C : |z| < 4},
S2 =
0
i31i31zIm:Cz r Fkk
S3 = {z C : Re z > 0}.
53. S dk {ks=kQy =
(A) 10
3 (B) 20
3
(C) 16
3 (D) 32
3Sol. B
S1 = {z C: |z| < 4}
S2 =
0
i31i31zlm:Cz
S3 = {z C: Re z > 0}
(4,0)
x+y = 03
Nk; kafdr Hkkx dk {ks=kQy = 203
oxZ bdkbZ
54.Z S
min |1–3i – z| =
(A) 2 – 3
2(B)
2 32
(C) 3 – 3
2(D)
3 32
Sol. C
z Smin
|1 – 3i – z| =
(1, –3) l s j s[ kk 0yx3 dh yEcor ~ nwj h 2
33
bdkbZ
i z' u 55 , oa 56 ds fy; s vuqPNsn, d i sVh B1 esa 1 l Qsn xsan, 3 yky xsans r Fkk 2 dkyh xsans gSaA
, d nwl j h i sVh B2 esa 2 l Qsn xsans, 3 yky xsans r Fkk 4 dkyh xsansa gSaA , dr hl j h i sVh B3 esa 3 l Qsn xasnsa, 4 yky xsansa r Fkk 5 dkyh xsansa gSaA
55. ; fn i sfV; ksa B1, B2 r Fkk B3 esa i zR; sd l s 1 xsan fudkyh t kr h gS]r c fudkyh xbZ l Hkh 3 xsanksa ds , d gh j ax ds gksus dh i zkf; dr k gS :
(A) 82648 (B)
90648
(C) 558648 (D)
566648
Sol. A; fn B1, B2 r Fkk B3 esa l s , d ckWy fudkyh t kr h gS .......
B1
1 W3 R2 B
B2
2 W3 R4 B
B3
3 W4 R5 B
WWW BBB RRR
= 1 2 3 3 3 46 9 12 6 9 12 +
2 4 56 9 12
6 36 406 8 12
= 82
6 9 12
56. ; fn 2 xsansa , d ; knfPNd p; fur i sVh l s (fcuk i zfr LFkki u ds)fudkyh t kr h gSa r Fkk , d xsan l Qsn o nwl j h xsan yky fudyr h gS, r cbu 2 xsanksa dh i sVh B2 l s fudyus dh i zkf; dr k gS :
(A) 116181 (B)
126181
(C) 65
181 (D) 55181
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Sol. D
212
29
26
29
C43
C32
C31
C3x2
112
61
51
61
=
18155
57. l wph I dks l wph II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn,x, dksM dk i z; ksx dj ds l gh mÙkj pqfu; s :
l wph I l wph II
P.
1 /22–1 –14
2 –1 –1
1 cos(tan y) y sin(tan y) yy cot (sin y) tan(sin y)
1.
1 52 3
dk eku gS
Q. ; fn cos x+cos y+cos z = 0 = sin x 2. 2
+ sin y + sin z r c cos 2
yx dk l EHkkfor
eku gS
R. ; fn cos
– x4 cos 2x + sin x 3.
12
sin 2x sec x=cos x sin 2x sec x+
cos
x4 cos 2x dk l EHkkfor eku gS
S. ; fn cot –1 2 –1sin 1– x sin tan x 6 , 4. 1
x 0. r c x dk l EHkkfor eku gSCodes :
P Q R S(A) 4 3 1 2(B) 4 3 2 1(C) 3 4 2 1(D) 3 4 1 2
Sol. B(P)
2/1
42
11
11
2 y)ytan(sin)ycot(sin)ysin(tany)ycos(tan
y1
tan–1 y = y = tan i zfr LFkkfi r dj us i j
1 /222
2 42
y 1 y1 1 y y1y
1 /2
2 4 42
1 y (1 y ) yy
= 1
(Q)oxZ dj ds ; ksx dj us i j3 + 2 (cos (x – y) + cos (y – z) + cos (z – x))=0cos (x – y) + cos (y – z) + cos (z – x) = –3/2
cos 2
yx cos
2zx
cos 2
xz =
81
....(i)
; fn cos 2
)yx( = 1
x = y
cos2 2
zy =
81
(vl aHko)
bl fy, cos 2
)yx( =
21
(R) cos 2x 2 sin 4
sin x = sec x sin 2x (cos x – sin x)
cos x sin x = 2
x = 4
sec x = 2
(S) 1
58. , d j s[ kk L:y = mx + 3, y–v{k ds fcUnq E(0, 3) r Fkki j oy; ds pki y2 = 16x, 0 y 6 ds fcUnq F(x0,y0) i j feyr h gSAi j oy; dh fcUnq F(x0,y0) i j Li ' khZ y-v{k dks fcUnq G(0,y1) i j dkVr hgSA j s[ kk L dh i zo.kr k m , sl h pquh t kr h gS fd f=kHkqt EFG ds {ks=kQy dk, d LFkkuh; vf/kdr e gSAl wph I dks l wph II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn, dksM dki z; ksx dj ds l gh mÙkj pqfu; s :
l wph I l wph II
P. m = 1. 12
Q. EFG dk egÙke {ks=kQy gS 2. 4R. y0= 3. 2S. y1= 4. 1
P Q R S(A) 4 1 2 3(B) 3 4 1 2(C) 1 3 2 4(D) 1 3 4 2
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Sol. A
(0,3)E
(0,y )1
F(x ,40 x0 ) (x , y )0 0
Tf = yy0 = 8(x + x0)Tf, G l s xqt j r h gSy1 y0 = 8 (0 + x0)y1 y0 = 8 x0
x0 = 8yy 01
vc, 16y 2
0 = 8yy 01
2yy 0
1
= 21
(3 – y1) . x0
= 21
(3 – y1). 8yy 01
= 161
(3 – y1). y1. (2y1)
= 18
(3 – y1). y12
dyd
= 18
(6y1 – 3y12) = 0
y1 0 y1 = 2y1 = 2 & y0 = 4
vc m = 0x3y
0
0
= 1
m = 1
EFG = 21
. (3 – 2). 1 = 21
D = 21
P – 4 (m = 1)
Q – 1 (A = 21
)
R – 2 (y0 = 4)S – 3 (y1 = 2)
59. l wph-I dks l wph-II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn,x, dksM dk i z; ksx dj ds l gh mÙkj pqfu; s :
l wph-I l wph-II
P. l fn' kksa
a,b r Fkk c } kj k fu/kkZfj r l ekUr j 1. 100
"kV~Qyd dk vk; r u 2 gSA r c l fn' kksa
2(a b),3(b c) r Fkk
(c a) } kj k fu/kkZfj r
l ekUr j "kV~Qyd dk vk; r u gS
Q. l fn' kksa
a,b r Fkk c } kj k fu/kkZfj r l ekUr j 2. 30
"kV~Qyd dk vk; r u 5 gSA r c l fn' kksa
3(a b),(b c) r Fkk
2(c a) } kj k fu/kkZfj r
l ekUr j "kV~Qyd dk vk; r u gSR. , d f=kHkqt dk {ks=kQy] ft l dh l ayXu 3. 24
Hkqt k, ¡ l fn' kksa a r Fkk b } kj k fu/kkZfj r
gS] 20 gSA r c l fn' kksa
(2a 3b) r Fkk
(a– b)
} kj k fu/kkZfj r l ayXu Hkqt kvksa okys f=kHkqt dk {ks=kQy gSS. , d l ekUr j pr qHkqZt dk {ks=kQy ft l dh l ayXu 4. 60
Hkqt k, ¡ l fn' kksa a r Fkk b } kj k fu/kkZfj r gS, 30 gSA
r c l fn' kksa
(a b) r Fkk a } kj k fu/kkZfj r l ayXu
Hkqt kvksa okys l ekUr j pr qHkqZt dk {ks=kQy gSCodes :
P Q R S(A) 4 2 3 1(B) 2 3 1 4(C) 3 4 1 2(D) 1 4 3 2
Sol. C
(P) ]accb3ba2[
= 6 ]accbba[
= 6 ]cba[ 2 = 24
P 3
(Q) [3(a b) b c 2c a]
= 6 [a b b c c a]
= 6 × 2 ]cba[
= 60Q 4
(R) 21
ba
= 20, ba
= 40
21
)ba()b3a2(
= 25
|ba|
= 100
R 1
(S) a)ba(
= ab
= 30
S 2
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60. j s[ kk, ¡ 1x –1 y z 3L :
2 –1 1
,
2x – 4 y 3 z 3L :
1 1 2
r Fkk l er y P1:7x + y + 2z = 3,
P2 : 3x + 5y – 6z = 4 yhft ; sA ekuk fd ax + by + cz = d,j s[ kkvska L1 r Fkk L2 ds i zfr PNsn fcUnq l s xqt j us okyk r Fkk l er y P1 o P2
ds yEcor , l er y dk l ehdj .k gSAl wph-I dks l wph-II l s l qesfyr dhft , r Fkk l wfp; ksa ds uhps fn, x, dksMdk i z; ksx dj ds l gh mÙkj pqfu; s :
l wph-I l wph-IIP. a = 1. 13Q. b = 2. – 3R. c = 3. 1S. d = 4. – 2
Codes :P Q R S
(A) 3 2 4 1(B) 1 3 4 2(C) 3 2 1 4(D) 2 4 1 3
Sol. AL1 i j dksbZ fcUnq ( 2 + 1, –, – 3)L2 i j dksbZ fcUnq ( + 4, – 3, 2 – 3)bl fy, = 1 r Fkk = 2bl fy, i zfr PNsn fcUnq (5, –2, –1) gS
P1 r Fkk P2 ds nksuksa ds yEcor ~ l fn' k = 653217kji
= k32j48i16 , vr % r y dk l ehdj .k
–16 (x – 5) + 48 (y + 2) + 32 (7 + 1) = 0x – 3y – 2z = 13a = 1b = –3c = –2d = 13