time : 3 hrs. max. marks: 180 answers & solutions for jee ... · (d) 50.7 cm ij l u st ku sokyk...
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1
Answers & Solutionsfor
JEE (Advanced)-2018
Time : 3 hrs. Max. Marks: 180
Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005Ph.: 011-47623456 Fax : 011-47623472
PAPER - 2
DATE : 20/05/2018
PART-I : PHYSICS[ kaM-1(vf/ dre vad : 24)
bl [ kaM esa Ng (06) i z' u gSaA
çR;sd i z' u ds l gh mÙkj (mÙkjksa) ds fy, pkj fodYi fn, x, gSaA bu pkj fodYi ksa esa l s , d ; k ,d l s vf/ d fodYi l ghgS(gSa)A
i zR;sd i z'u ds fy, ] i z'u dk(ds) mÙkj nsus gsrq l gh fodYi (fodYi ksa) dks pqusaA
i zR; sd i z'u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk%
i w.kZ vad : +4 ; fn dsoy(l kjs) l gh fodYi (fodYi ksa) dks pquk x; k gSA
vkaf' kd vad : +3 ; fn pkjksa fodYi l gh gSa i jUrq dsoy rhu fodYi ksa dks pquk x; k gSA
vkaf' kd vad : +2 ; fn rhu ; k rhu l s vf/ d fodYi l gh gSa i jUrq dsoy nks fodYi ksa dks pquk x; k gS vkSjpqus gq, nksuksa fodYi l gh fodYi gSaA
vkaf' kd vad : +1 ; fn nks ; k nks l s vf/ d fodYi l gh gSa i jUrq dsoy ,d fodYi dks pquk x; k gS vkSj pqukgqvk fodYi l gh fodYi gSA
' kwU; vad : 0 ; fn fdl h Hkh fodYi dks ugha pquk x; k gS (vFkkZr~ i z' u vuqÙkfjr gS)A
½.kkRed vad : –2 vU; l Hkh i fjfLFkfr ; ksa esaA
mnkgj.k Lo: i % ; fn fdl h i z' u ds fy, dsoy i gyk] rhl jk , oa pkSFkk l gh fodYi gSa vkSj nwl jk fodYi xyr gSa; r ks dsoyl Hkh rhu l gh fodYi ksa dk p; u djus i j gh +4 vad feysasxsA fcuk dksbZ xyr fodYi pqus (mnkgj.k esa nwl jk fodYi )] rhul gh fodYi ksa esa l s fl i QZ nks dks pquus i j (mnkgjar% i gyk rFkk pkSFkk fodYi ) +2 vad feysaxsA fcuk dksbZ xyr fodYi pqus(blmnkgj.k esa nwl jk fodYi )] rhu l gh fodYi ksa esa l s fl i QZ ,d dks pquus i j (i gyk ; k rhl jk ; k pkSFkk fodYi ) +1 vad feysaxsAdksbZ Hkh xyr fodYi pquus i j (bl mnkgj.k esa nwl jk fodYi )] –2 vad feyasxs] pkgs l gh fodYi (fodYi ksa) dks pquk x; k gks; k u pquk x; k gksA
2
JEE (ADVANCED)-2018 (PAPER-2)
1. nzO;eku (mass) m dk ,d d.k 'kq:vkr esa ewy fcUnq (origin) ij fojkekoLFkk esa gSA d.k ij ,d cy yxkus ls og x-v{k ij
pyus yxrk gS vkSj d.k dh xfrt ÅtkZ (kinetic energy) K, le; ds lkFk dKt
dt ds vuqlkj ifjo£rr gksrh gS] tgk¡
,d mfpr foekvksa okyk /ukRed fu;rkad (positive constant) gSA fuEufyf[kr dFkuksa esa ls dkSu lk (ls) lgh gS (gSa)\
(A) d.k ij yxk;k x;k cy fu;r (constant) gS
(B) d.k dh pky le; ds lekuqikfrd (proportional) gS
(C) d.k dh ewy fcUnq ls r; dh x;h nwjh] le; ds lkFk js[kh; rjhds ls (linearly) c<+rh gS
(D) cy laj{kh (conservative) gS
mÙkjmÙkjmÙkjmÙkjmÙkj (A, B, D)
gygygygygy ⇒ 21 dK dvK mv mv
2 dt dt
fn;k gS] ⇒ dK dvt mv t
dt dt
⇒ ∫ ∫v t 2 2
0 0
v tvdv tdt
m 2 m 2
v t
m
dv
adt m
F ma m fu;r
⇒ 2
dS tt S
dt m m 2
2. eku yhft, fd ,d ';ku (viscous) nzo ds ,d cM+s VSad (tank) esa ,d iryh oxkZdkj IysV (thin square plate) rSj jgh
gSA VSad esa nzo dh Å¡pkbZ h, VSad dh pkSM+kbZ ls cgqr de gSA rSjrh gqbZ IysV dks ,d fu;r (constant) osx u0 ls {kSfrt
fn'kk esa [khapk tkrk gSA fuEufyf[kr dFkuksa esa ls dkSu lk (ls) lgh gS (gSa)\
(A) nzo ds }kjk IysV ij yxk;k x;k çfrjks/d cy (resistive force) h ds O;qRØekuqikfrd (inversely poroportional) gS
(B) nzo ds }kjk IysV ij yxk;k x;k çfrjks/d cy IysV ds {ks=kiQy ij fuHkZj ugha djrk gS
(C) VSad dh iQ'kZ (floor) ij yxrk gqvk Li'kZjs[kh; çfrcy (tangential/shear stress) u0 ds lkFk c<+rk gS
(D) IysV ij yxus okys Li'kZjs[kh; çfrcy nzo dh ';kurk (viscosity) ds lkFk js[kh; rjhds ls (linearly) cnyrh gS
mÙkjmÙkjmÙkjmÙkjmÙkj (A, C, D)
gygygygygy u0
Fv
A
Plate
⎛ ⎞ ⎜ ⎟⎝ ⎠
v
dvF A
dz
pw¡fd VSad esa nzo dh Å¡pkbZ h vR;Ur de gS ⎛ ⎞⇒ ⎜ ⎟ ⎝ ⎠
0udv v
dz z h
⎛ ⎞ ⎜ ⎟⎝ ⎠
0v
uF ( )A
h
⎛ ⎞ ⎜ ⎟⎝ ⎠
v v 0 v v
1F ,F u ,F A,F
h
3
JEE (ADVANCED)-2018 (PAPER-2)
3. z-v{k ds l ekUr j ,d vuUr yEckbZ dh i ryh vpkyd (non-conducting) rkj i j ,dl eku js[ kh; vkos' k ?kuRo (uniform
line charge density) gSA] ; g r kj R f=kT; k okys ,d i r ys vpkyd xksyh; dks' k (spherical shell) dks bl çdkjHksnrk gS fd vkdZ (arc) PQ, xksyh; dks' k ds dsUnz O i j 120° dk dks.k cukrh gS] t Sl k fd fp=k esa n' kkZ; k x; k gSA eqDrvkdk' k dk i jkoS| qrkad (permittivity of free space)
0 gSA fuEufyf[ kr dFkuksa esa l s dkSu l k (l s) l gh gS (gSa)\
P
R
O
Q
120°
z
(A) dks' k l s xqt jus okyk oS| qr ÝyDl (electric flux) 0
3R gS
(B) oS| qr {ks=k (electric field) dk z-?kVd (z-component) dks' k ds i "B (surface) ds l Hkh fcUnqvksa i j ' kwU; gS
(C) dks'k l s xqt jus okyk oS| qr ÝyDl (electric flux) 0
2R gS
(D) oS| qr {ks=k (electric field) dks'k ds i "B ds l Hkh fcUnqvksa i j yEcor (normal) gS
mÙkj (A, B)
gy PQ = (2) R sin 60°
3
(2R) 3R2
P
R
O
Q
120°
z
R
Q 3 R i fjc¼
0
q
i fjc¼
0
3 R
rFkk fo| qr {ks=k rkj ds yEcor gS] bl fy, Z-?kVd ' kwU; gksxkA
4. , d r kj dks ,d l edks.k f=kHkqt ds vkdkj esa eksM+ dj f i Qksdl nwjh (focal length) okys ,d vory ni Z.k (concave
mirror) ds l keus j[ kk x; k gS] t Sl k fd fp=k esa n' kkZ; k x; k gSA pkj fodYi fp=kksa esa l s dkSu l k (l s) fp=k eqM+s gq, r kj dsçfr fcEc dk l gh vkdkj xq.kkRed r jhds l s n' kkZrk gS (n' kkZrs gSa)\ (; s fp=k Ldsy (scale) ds vuql kj ugha gSaA)
45°ff
2
4
JEE (ADVANCED)-2018 (PAPER-2)
(A)
> 45°
(B)
(C) 0 < < 45°
(D)
mÙkj (D)
gy fcUnq A dk i zfr fcEc
f/2
B
A
Q
(f – x)
x CPO
1 1 1v f
fv f2
ABAB
I fI 2AB
fAB2
For Height of PQ
1 1 1 1 1 1 f(f x)
vv (f x) f v (f x) f x
PQI f(f x) fPQ x{ (f x)} x
PQf f 2(AB)x
I PQx x f
2(AB)x
PQf
IPQ = 2AB
rFkk 2(AB)x
PQf
5. , d jsfM; ks, fDVo {k; Ja[ kyk (decay chain) esa 23290 Th ukfHkd] 212
82 Pb ukfHkd esa {kf; r gksrk gSA bl {k; i zØe (process)
esa mRl ft Zr gq, (emitted) vkSj – d.kksa dh l a[ ; k Øe' k% N vkSj N gSaA fuEufyf[ kr dFkuksa esa l s dkSu l k (l s) l gh gS (gSa)\
(A) N 5 (B) N 6
(C) N 2 (D) N 4
mÙkj (A, C)
5
JEE (ADVANCED)-2018 (PAPER-2)
gy A A 4Z Z 2X Y
{k;
A AZ Z 1 1X X
{k;
232 21290 82Th Pb
mRl ft Zr -d.kksa dh l a[ ; k
232 2125
4
pw¡fd Z, (90 – 82) rd ?kV t krk gS = 8 dssoy
bl fy, – {k; dh l a[ ; k = 2
6. vuquknh ok; q&LraHk (resonating air column) ds ,d i z; ksx esa èofu dh pky eki us ds fy, 500 Hz dh vkofÙk okys ,dLofj=k f}Hkqt (tuning fork) dk mi ; ksx fd; k t krk gSA vuqukn uyh esa t y dk Lr j cnydj ok; q&LraHk dh yEckbZ cnyht krh gSA nks mÙkjksÙkj (successive) vuqukn] ok; q LrEHk dh yEckbZ 50.7 cm vkSj 83.9 cm i j l qus t krs gSaA fuEufyf[ krdFkuksa esa l s dkSu l k (l s) l gh gS (gSa)\
(A) bl i z; ksx l s fuèkkZfjr èofu dh pky 332 m s–1 gS
(B) bl i z; ksx esa vaR; l a' kksèku (end correction) 0.9 cm gS
(C) èofu r jax dh r jaxnSè; Z (wavelength) 66.4 cm gS
(D) 50.7 cm i j l qus t kus okyk vuqukn] ewy xq.kkofÙk (fundamental harmonic) gS
mÙkj (A, B, C)
gy
(2n – 1) 50.7 e4
2 (n 1) – 1 83.9 e4
83.9 – 50.7 33.2 cm2
= 66.4 cm
v = f = 66.4 × 500 × 10–2 m/s = 332 m/s
For, n = 2, e = – 0.9 cm
[ kaM-2(vf/ dre vad : 24)
bl [ kaM esa vkB (08) i z' u gSaA i zR;sd i z'u dk mÙkj , d l a[ ; kRed eku (NUMERICAL VALUE) gSA
çR; sd i z' u ds mÙkj ds l gh l a[ ; kRed eku(n' keyo vadu esa] n' keyo ds f}rh; LFkku rd : f.Mr@fudfVr ; mnkgj.kr%6.25, 7.00, -0.33, -.30, 30.27, -127.30) dks ekmt + (MOUSE) vkSj vkWu LØhu (ON-SCREEN) opqZvy U; wesfjd dhi SM(VIRTUAL NUMERIC KEYPAD) ds i z; ksx l s mÙkj ds fy, fufnZ"V LFkku i j nt Z djsaA
i zR; sd i z' u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk&
i w.kZ vad : +3 ; fn fl i QZ l gh l a[ ; kRed eku (Numerical value) gh mÙkj Lo: i nt Z fd; k x; k gSA
' kwU; vad : 0 vU; l Hkh i fjfLFkfr ; ksa esaA
6
JEE (ADVANCED)-2018 (PAPER-2)
7. , d Bksl {kSfr t r y (solid horizontal surface) r sy dh ,d i r yh i jr (thin layer) l s <dk (covered) gqvk gSAnzO; eku (mass) m = 0.4 kg dk , d vk; r kdkj xqVdk (rectangular block) bl r y i j fojkekoLFkk esa gSA1.0 N s i fjek.k dk ,d vkosx (impulse) xqVds i j t = 0 l e; i j yxk; k t kr k gS ft l ds i QyLo: i xqVdk x-v{k(x-axis) i j (t) = 0 e–t/ osx l s pyus yxrk gS] t gk¡ 0 , d fLFkj jkf' k gS vkSj = 4 s gSA l e; t = i j] xqVds dkfoLFkki u (displacement)_____ ehVj gSA e–1 = 0.37 ysaA
mÙkj (6.30)
gy m/s0
1v = = 2.5
0.4
–t /ds= 2.5e
dt
s t–t /
0 0
ds = 2.5e dt
t–t /
0
2.5s = e
–1 = –t /2.5 1– e
t = i j] s = 10[1 – e–1] = 10 × 0.63 = 6.30 m
8. , d xsan dks Hkwfe (ground) i j {kSfr t r y (horizontal surface) l s 45º ds dks.k i j i z{ksfi r (projected) fd; k t krkgSA xsan 120 m dh vfèkdre Å¡pkbZ i j i gq¡p dj Hkwfe i j oki l ykSV vkrh gSA Hkwfe l s i gyh ckj Vdjkus ds mi jkar xsan dhxfr t Åt kZ (kinetic energy) vkèkh gks t krh gSA Vdjkus ds rqjar ckn xsan dk osx {kSfr t r y l s 30º dk dks.k cukrk gSAVdjkus ds ckn xsan_____ ehVj dh vfèkdre Å¡pkbZ i j i gq¡prh gSA
mÙkj (30.00)
gy 2 2u sin 45°
= 1202g
2u
= 1204g
; fn i gyh VDdj ds ckn pky v gS] rc
2v= 60
4g pw¡fd u
v2
hvfèkdre = 2 2v sin 30°
2g = 2v
8g = 30 m = 30.00 m
9. , d d.k] ft l dk nzO; eku (mass) 10–3 kg vkSj vkos' k (charge) 1.0 C gS] ' kq#vkr esa fojkekoLFkk esa gSA l e; t = 0 i j
; g d.k , d fo| qr ~ {ks=k (electric field) 0ˆE (t) E sin t I
ds i zHkko esa vkr k gS] t gk¡ 1
0E 1.0 N C gS vkSj
3 110 rad s gSA d.k i j dsoy fo| qr~ cy (electrical force) dk gh i zHkko ekfu; sA rc i jorhZ (subsequent) l e;
i j d.k dh vfèkdre pky_____ m s–1 gSA
mÙkj (2.00)
gy ˆ 3F = qE = 1.0 N sin (10 t) i
3 3Fa = = 10 sin (10 t)
m
7
JEE (ADVANCED)-2018 (PAPER-2)
3 3dv= 10 sin (10 t)
dt
V t
3 3
0 0
dv = 10 sin(10 t)dt 3
33
10v = 1 – cos(10 t)
10
vmax = 2 m/s = 2.00 m/s
10. , d py dqaMyh xSYosuksehVj (moving coil galvanometer) esa 50 i sQjs (turns) gSa vkSj gj i sQjs dk {ks=ki Qy (area)
2 × 10–4 m2 gSA xSYosuksehVj esa mi fLFkr pqEcd l s 0.02 T dk pqEcdh; {ks=k (magnetic field) mRi Uu gksrk gSA fuyacurkj (suspension wire) dk , saBu fu; r kad (torsional constant) 10–4 N m rad–1 gSA xSYosuksehVj esa èkkjk cgus dsl e; ] ; fn dqaMyh 0.2 rad ?kwerh gS rks xSYosuksehVj esa i w.kZ i Sekuk fo{ksi (full scale deflection) gksrk gSA xSYosuksehVj dhdqaMyh dk i zfr jksèk 50 gSA bl xSYosuksehVj dks 0 – 1.0 A dh jsUt (range) esa èkkjk ds eki u djus ; ksX; ,d , sehVj(ammeter) ds : i esa i fjofrZr djuk gSA bl ds fy, , d ' kaV (shunt) i zfr jksèkd dks xSYosuksehVj l s i kÜoZØe (parallel) esal a;ksft r djuk i M+rk gSA bl ' kaV i zfr jksèkd dk eku_____vkse (ohms) gSA
mÙkj (5.56)
gy = BANim = K
–4
m –4
K 10 × 0.2i = =
BAN 0.02 × 2 × 10 × 50
0.2= = 0.1 A
2
0.1 × 50 = 0.9 S 50
S = = 5.56 9
11. , d bLi kr ds r kj] ft l dk O; kl (diameter) 0.5 mm gS vkSj ; ax xq.kkad (Young's modulus) 2 × 1011 Nm–2 gS] l s M
nzO; eku (mass) dk ,d Hkkj yVdk; k t krk gSA Hkkj yVdkus ds ckn r kj dh yackbZ 1.0 m gSA bl r kj ds var esa 10
Hkkxksa okyk ,d ofuZ; j i Sekuk (vernier scale) yxk; k t krk gSA bLi kr ds rkj ds i kl , d vkSj l anHkZ (reference) rkj gSft l i j 1.0 mm vYi rekad (least count) okyk ,d eq[ ; i Sekuk (main scale) yxk gqvk gSA ofuZ; j i Sekus ds 10 Hkkxeq[ ; i Sekus ds 9 Hkkxksa ds cjkcj gSaA ' kq#vkr esa] ofuZ; j i Sekus dk ' kwU; eq[ ; i Sekus ds ' kwU; l s l ai krh (coincident) gSA ; fnbLi kr ds r kj i j yVdk; k x; k Hkkj 1.2 kg l s c<+k;k t krk gS] r ks eq[ ; i Sekus ds Hkkx l s l ai krh gksus okyk ofuZ; j i Sekus dkHkkx_____gSA g = 10 ms–2 vkSj = 3.2 ysaA
mÙkj (3.00)
gyW
LYAL
11 2 –6
1.2 10 4 1
2 10 (0.5) (10 )= 0.3 mm
ofuZ; j dk vYi rekad
91–
10 mm = 0.1 mm
ofuZ; j dk i kB~;kad = 3.
8
JEE (ADVANCED)-2018 (PAPER-2)
12. , di jekf.od vkn' kZ xSl (monatomic ideal gas) ds , d eksy dk vk; r u (volume), #¼ks"e i zl kj (adiabatic
expansion) l s] vi us vkjafHkd eku dk vkB xquk c<+ t krk gSA l koZf=kd xSl fu; r kad (universal gas constant) R dkeku 8.0 J mol–1 K–1 ysaA ; fn xSl dk vkjafHkd rki eku 100 K gks] r ks bl i zfØ; k esa xSl dh vkar fjd Åt kZ (internal
energy) _____t wy (Joule) l s de gks t krh gSA
mÙkj (900.00)
gy 5
n 1,3
T1 V1–1 = T2V2
–1
2–131
2 12
V 1T T 100
V 8
T2 = 25 K
2 13R
U n T – T2
3
1 8 (25 – 100)2
= – 900 J
vkUrfjd Åt kZ esa deh = 900 J
13. ,d çdk'k fo| qr (photoeletric) ç;ksx esa 200 W ' kfDr (power) okyk ,d l ekUr j ,do.khZ çdk' k fdj.k i qat (a parallel
beam of monochromatic light) i w.kZ : i l s vo' kksf"kr djus okys ,d mRl t Zd (perfectly absrobing cathode)
i j fxjrk gSA mRl t Zd ds i nkFkZ dk dk;Z&i Qyu (work function) 6.25 eV gSA çdk'k dh vkofÙk] nsgyh vkofÙk (threshold
frequency) l s FkksM+h gh vf/ d gS] ft l l s mRl £t r gksus okyh çdkf' kd bysDVªkWuksa (photoelectrons) dh xfr t Åt kZ (kinetic
energy) ux.; gSA eku yhft ,s fd çdk'k fo| qr mRl t Zu n{krk (photoelectron emission efficiency) 100% gSA mRl t ZdvkSj l axzkgd (anode) ds chp 500 V dk foHkokUr j (potential difference) yxk; k t krk gSA mRl £t r gksus okys l HkhbysDVªkWu l axzkgd i j vfHkyEc vki fr r (normal incidence) gksdj vo' kksf"kr gks t krs gSaA bysDVªkWuksa dh l axzkgd i j VDdjl s f = n × 10–4 N dk cy yxrk gSA n dk eku ________ gSA bysDVªku dk nzO; eku (mass) me = 9 × 10–31 kg gS vkSj1.0 eV = 1.6 × 10–19 J gSA
mÙkj (24.00)
gy P = 200 J/s,
i zfr l sd.M i QksVkWu dh l a[ ; k (N) = –19
200
(6.25 1.6 10 )
= 2 × 1020
P 2m (KE) 2m(eV)
–31 –192 9 10 1.6 10 500
= 12 × 10–24
F = P × N = 12 × 10–24 × 2 × 1020
= 24 × 10–4N
9
JEE (ADVANCED)-2018 (PAPER-2)
14. , d gkbMªkst u&t Sl k vk; fur (hydrogen-like ionized) i jek.kq dk i jek.kq Øekad (atomic nunber) Z gSA bl i jek.kq esa,d gh bysDVªku gSA bl i jek.kq ds mRl t Zu&Li sDVªe (emission spectrum) esa] n = 2 l s n = 1 l aØe.k (transition) l smRi Uu gksus okys i QksVkWu (photon) dh Åt kZ] n = 3 l s n = 2 l aØe.k (transition) l s mRi Uu gksus okys i QksVkWu (photon)
dh Åt kZ l s 74.8 eV vf/ d gSA gkbMªkst u i jek.kq dh vk; uu Åt kZ (ionization energy) 13.6 eV gSA Z dk eku___________ gSA
mÙkj (3.00)
gy
2 21 1 1 113.6 – Z 74.8 13.6 – Z
1 4 4 9
2 3 513.6 Z – 74.8
4 36 Z2 = 9 Z = 3
[ kaM 3 (vf/ dre vad % 12)
• bl [ kaM esa pkj (04) i z'u gSaA i zR;sd i z' u esa nks (02) l qesyu l wfp;k¡ (matching lists) gSa% l wph–I vkSj l wph–II |
• l wph–I vkSj l wph–II ds rÙoksa ds l qesykuks dks n'kkZrs gq, pkj fodYi fn, x, gSaA bu pkj fodYi ksa esa fl iQZ ,d fodYi gh l gh
l qesyu i znf' kZr djr k gSA
• i zR; sd i z' u ds fy, l gh l qesyu i znf' kZr djr s okys fodYi dks pqusaA
• i zR; sd i z' u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk
i w.kZ vad % +3 ; fn fl i QZ l gh fodYi gh pquk x; k gSA
' kwU; vad % 0 ; fn dksbZ Hkh fodYi ugha pquk x; k gS (vFkkZr~ i z' u vuqÙkfjr gS)A
Í .k vad % –1 vU; l Hkh i fjfLFfr ; ksa esaA
15. fofHkUu vkos' k for j.kksa (charge distributions) l s mRi Uu gksus okys fo| qr {ks=k (electric field) E dk ,d fcUnq P(0, 0, d)
i j eki u fd; k t krk gS vkSj bl fo| qr {ks=k E dh d i j fuHkZjr k vyx&vyx i k; h t krh gSA l wph-I esa E vkSj d ds chp esa vyx&vyx l EcU/ (relations) fn; s x; s gSaA l wph-II fofHkUu çdkj ds vkos' k for j.kksa vkSj muds LFkkuksa dks crkrh gSaA l wph-I ds i Qyuksadk l wph-II l s l acaf/ r vkos' k for j.kksa l s l qesy dhft ,A
l wph-I l wph-II
P. E, d i j fuHkZj ugha djr k gS 1. ewy fcUnq (origin) i j fcUnq vkos'k (point charge) Q
Q. 1
Ed
2. , d y?kq f}/ qzo (small dipole) ft l dk fcUnq vkos' k Q t ks
(0, 0, l) i j gS vkSj –Q t ks (0, 0, –l) i j gSA ekfu; s 2l <<d
R. 2
1E
d3. vuUr (infinite) yEckbZ dk ,dl eku js[ kh; vkos'k ?kuRo
(uniform linear charge density) okyk r kj t ks x-v{kl s l Ei krh (coincident) gS
S. 3
1E
d4. vuUr yEckbZ ds ,dl eku js[ kh; vkos'k ?kuRo okys nks r kj t ks
x-v{k ds l ekUr j gSaA (y = 0, z = l) okys r kj i j + vkos'k?kuRo gS rFkk (y = 0, z = –l) okys r kj i j – vkos'k ?kuRogSA ekfu, 2l <<d
5. ,dl eku vkos'k ?kuRo (uniform surface charge density)
dk vuUr l ery pknj (infinite plane sheet) t ks xy- ryl s l Ei krh gS
10
JEE (ADVANCED)-2018 (PAPER-2)
(A) P 5; Q 3, 4; R 1; S 2
(B) P 5; Q 3; R 1, 4; S 2
(C) P 5; Q 3; R 1, 2; S 4
(D) P 4; Q 2, 3; R 1; S 5
mÙkj (B)
gy l wph-II
(1) 2
0
1 QE
4 d
2
1E
d
(2) 30
1 2Q(2 )E
4 d
v{k 3
1E
d
(3)
0
E2 d
1E
d
(4)
2
0 0 0
(2 )E –
2 (d – ) 2 (d ) 2 d 2
1E
d
(5)
0
E2 E, d i j fuHkZj ugha djrk gSA
16. M nzO;eku (mass) okys ,d xzg (planet) ds nks çkÑfrd mi xzg (natural satellites) nks oÙkh; d{kksa esa i fjØe.k (revolve)
dj jgs gSaA mi xzgksa ds chp xq: Rokd"kZ.k cy (gravitational attraction) dh mi s{kk dhft ; sA i gyk mi xzg] ft l dk nzO; eku
m1, d{kh; pky 1, dks.kh; l aosx (angular momentum) L1, xfr t Åt kZ (kinetic energy) K1 vkSj vkorZdky (period
of revolution) T1 gS] R1 f=kT; k okyh d{kk esa LFkkfi r gSA nwl jk mi xzg] ft l dk nzO; eku m2, d{kh; pky v2, dks.kh; l aosx
L2, xfr t Åt kZ K2 vkSj vkorZdky T2 gS] R2 f=kT; k okyh d{kk esa LFkkfi r gSA ; fn m1/m2 = 2 vkSj R1/R2 = 1/4 gks] r ks l wph-
I esa fn, x, vuqi krksa dk l qesy l wph-II esa nh x; h l a[ ; kvksa ds l kFk djsaA
l wph-I l wph-II
P.1
2
v
v 1.18
Q.1
2
L
L 2. 1
R.1
2
K
K 3. 2
S.1
2
T
T 4. 8
(A) P 4; Q 2; R 1; S 3
(B) P 3; Q 2; R 4; S 1
(C) P 2; Q 3; R 1; S 4
(D) P 2; Q 3; R 4; S 1
mÙkj (B)
11
JEE (ADVANCED)-2018 (PAPER-2)
gy 2
2
GMm mv GMv
R RR
ekuk R1 = R R2 = 4R
m2 = m m1 = 2m
l wpht-I
(P) 1 2
2 1
v R 4R2 : 1
v R R
(Q) L = mvR
1 1
2 2
L R(2m)v 1(2) 1: 1
L 4R(m)v 2
m2
v2
R1
m1
v1
M
R2
(R)
21
1
222
1(2m)vK 2 2(4) 8 : 1
1K (m)v2
(S)
3 32 2
1 1
2 2
T R 11: 8
T R 4
17. , di jek.fod vkn' kZ xSl (monatomic ideal gas) dk ,d eksy (one mole), pkj Å"ekxrh; i zØeksa (thermodynamic
processes) l s xqt jr k gS] t Sl k fd uhps PV O; oLFkk fp=k (schematic diagram) eas n' kkZ; k x; k gSA ; gk¡ fn; s x; s i zØeksa
esa ,d l enkch; (isobaric)] , d l evk; r fud (isochoric)] , d l erki h; (isothemal) vkSj , d : ¼ks"e (adiabatic)
gSaA l wph&I esa fn, x, i zØeksa dk l wph&II esa fn, x, l axr dFkuksa l s l qesy djsaA
3P0
P
V0 3V0
I
II
IIIIV
V
P0
l wph-I l wph-II
P. i zØe-I esa 1. xSl }kjk fd; k x; k dk; Z ' kwU; gS
Q. i zØe-II esa 2. xSl dk rki eku ugha cnyrk gS
R. i zØe-III esa 3. xSl vkSj i fjos' k ds chp Å"ek i zokg ugha gksrk gS
S. i zØe-IV esa 4. xSl }kjk fd; k x; k dk;Z 6P0V0 gS
(A) P 4; Q 3; R 1; S 2
(B) P 1; Q 3; R 2; S 4
(C) P 3; Q 4; R 1; S 2
(D) P 3; Q 4; R 2; S 1
mÙkj (C)
12
JEE (ADVANCED)-2018 (PAPER-2)
gy adiabatic Isothermal
dP dPdV dV
l wph-1
(P) i zØe I #¼ks"e Q = 0
(Q) i zØe II l enkch;
W = PV = 3P0 [3V0 – V0] = 6P0V0
(R) i zØe III l evk; r fud W = 0
(S) i zØe (IV) l er ki h; rki = fu; rA
18. uhps nh x; h l wph-I esa] , d d.k ds pkj fofHkUu i Fk] l e; ds fofHkuu i Qyuksa (functions) ds : i esa fn; s x; s gSaA bu i Qyuksaeas vkSj mfpr foekvksa okys / ukRed fu; r kad (positive constants) gSa] t gk¡ | i zR; sd i Fk eas d.k i j yxus okyk
cy ; k r ks ' kwU; gS ; k l aj{kh (conservative) gSA l wph-II eas d.k dh i k¡p HkkSfrd jkf' k; ksa dk fooj.k fn; k x; k gS%p
js[ kh;
l aosx (linear momentum) gS] L ewyfcanq (origin) ds l ki s{k dks.kh; l aosx (angular momentum) gS] K xfr t Åt kZ
(Kinetic energy) gS] U fLFkfr t Åt kZ (potential energy) gS vkSj E dqy mt kZ (total energy) gSA l wph-I ds i zR; sdi Fk dk l wph-II esa fn; s x; s mu jkf' k; ksa l s l qesy dhft , ] t ks ml iFk ds fy, l aj{kh (conserved) gSaA
l wph-I l wph-II
P.
ˆ ˆr(t) t i t j 1.p
Q.
ˆ ˆr(t) cos t i sin t j 2.L
R.
ˆ ˆr(t) (cos t i sin t j) 3. K
S.
2ˆ ˆr(t) t i t j2
4. U
5. E
(A) P 1, 2, 3, 4, 5; Q 2, 5; R 2, 3, 4, 5; S 5
(B) P 1, 2, 3, 4, 5; Q 3, 5; R 2, 3, 4, 5; S 2, 5
(C) P 2, 3, 4; Q 5; R 1, 2, 4; S 2, 5
(D) P 1, 2, 3, 5; Q 2, 5; R 2, 3, 4, 5; S 2, 5
mÙkj (A)
gy t c cy F = 0 rc fLFkfr t Åt kZ U = fu; r
F 0 cy l aj{kh gS dqy Åt kZ E = fu; r
l wph-I
(P) ˆ ˆr(t) t i tj
dr ˆ ˆv i j
dt = fu; r kad p
fu; rkda
2 2|v| constant K = fu; r kad
13
JEE (ADVANCED)-2018 (PAPER-2)
dv
a 0 F 0 U constantdt
E = U + K = fu; r kad
L m(r v) 0
L constant
P 1, 2, 3, 4, 5
(Q) ˆ ˆr(t) cos ti sin tj
dr ˆ ˆv sin t( i) cos tjdt
fu; r kda p
fu; r kda
2 2|v| ( sin t) ( cos t) fu; rkad K fu; rkad
2dv
a r 0dt
E = fu; rkad = K + U
But K fu; rkad U fu; rkad
ˆL m(r v) m (k)
fu; rkda
Q 2, 5
(R) ˆ ˆr(t) (cos ti sin tj)
dr ˆ ˆv [sin t( i) cos tj]dt
fu; rkda p
fu; r kda
|v|
fu; rkda K = fu; rkad
2dva r 0 E , U
dt
fu; rkda fu; rkda
2 ˆL m(r v) m k
fu; rkda
R 2, 3, 4, 5
(S)
2ˆ ˆr(t) t i t j
2
dr ˆ ˆv i tj pdt
fu; r kda fu; r kda
2 2|v| ( t) K
fu; rkda fu; rkda
dv ˆa j 0 E K Udt
fu; rkda
But K fu; rkda
U fu; rkda
21 ˆL m(r v) t k2
fu; r kda
S 5
14
JEE (ADVANCED)-2018 (PAPER-2)
PART-II : CHEMISTRY[ kaM-1(vf/ dre vad : 24)
bl [ kaM esa Ng (06) i z' u gSaA
çR;sd i z' u ds l gh mÙkj (mÙkjksa) ds fy, pkj fodYi fn, x, gSaA bu pkj fodYi ksa esa l s , d ; k ,d l s vf/ d fodYi l ghgS(gSa)A
i zR;sd i z'u ds fy, ] i z'u dk(ds) mÙkj nsus gsrq l gh fodYi (fodYi ksa) dks pqusaA
i zR; sd i z'u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk%
i w.kZ vad : +4 ; fn dsoy(l kjs) l gh fodYi (fodYi ksa) dks pquk x; k gSA
vkaf' kd vad : +3 ; fn pkjksa fodYi l gh gSa i jUrq dsoy rhu fodYi ksa dks pquk x; k gSA
vkaf' kd vad : +2 ; fn rhu ; k rhu l s vf/ d fodYi l gh gSa i jUrq dsoy nks fodYi ksa dks pquk x; k gS vkSjpqus gq, nksuksa fodYi l gh fodYi gSaA
vkaf' kd vad : +1 ; fn nks ; k nks l s vf/ d fodYi l gh gSa i jUrq dsoy ,d fodYi dks pquk x; k gS vkSj pqukgqvk fodYi l gh fodYi gSA
' kwU; vad : 0 ; fn fdl h Hkh fodYi dks ugha pquk x; k gS (vFkkZr~ i z' u vuqÙkfjr gS)A
½.kkRed vad : –2 vU; l Hkh i fjfLFkfr ; ksa esaA
mnkgj.k Lo: i % ; fn fdl h i z' u ds fy, dsoy i gyk] rhl jk , oa pkSFkk l gh fodYi gSa vkSj nwl jk fodYi xyr gSa; r ks dsoyl Hkh rhu l gh fodYi ksa dk p; u djus i j gh +4 vad feysasxsA fcuk dksbZ xyr fodYi pqus (mnkgj.k esa nwl jk fodYi )] rhul gh fodYi ksa esa l s fl i QZ nks dks pquus i j (mnkgjar% i gyk rFkk pkSFkk fodYi ) +2 vad feysaxsA fcuk dksbZ xyr fodYi pqus(blmnkgj.k esa nwl jk fodYi )] rhu l gh fodYi ksa esa l s fl i QZ ,d dks pquus i j (i gyk ; k rhl jk ; k pkSFkk fodYi ) +1 vad feysaxsAdksbZ Hkh xyr fodYi pquus i j (bl mnkgj.k esa nwl jk fodYi )] –2 vad feyasxs] pkgs l gh fodYi (fodYi ksa) dks pquk x; k gks; k u pquk x; k gksA
1. l adqy [Co(en)(NH3)3(H2O)]3+ (en = H2NCH2CH2NH2) ds fo"k; esa l gh fodYi gS (gSa)
(A) bl ds nks T; kferh; l eko; o (geometrical isomers) gksrs gSa
(B) bl ds rhu T; kferh; l eko; o gksaxs ; fn f}narqj (bidentate) 'en' dks nks l k; ukbM fyxUMksa (cyanide ligands) l s cnykt k,
(C) ; g vuqpqEcdh; (paramagnetic) gS
(D) ; g [Co(en)(NH3)4]3+ dh rqyuk esa yach r jax&nSè; Z (wavelength) dk i zdk'k vo' kksf"kr djr k gS
mÙkj (A, B, D)
gy (A) [Co(en)(NH3)3(H2O)]3+
Co
OH2
NH3
NH3
NH3
NH2CH2
H N2CH2
Co
NH3
OH2
NH3
NH3
NH2CH2
NH2CH2
3+ 3+
15
JEE (ADVANCED)-2018 (PAPER-2)
(B) [Co(CN)2(NH
3)3(H
2O)]1+
Co
OH2
NH3
NH3
NH3
NC
NC
NH3
OH2
NH3
NH3
NC
NC
CN
OH2
NH3
CN
H N3
H N3
Co Co
1+ 1+ 1+
(C) Co3+ = [Ar]3d6
en rFkk NH3 dh mifLFkfr esa ;g fuEu pØ.k ladqy cukrk gS
(D) [Co(en)(NH3)
4]3+ e s a e
g rFk k t
2g ds eè; vUrj ky [Co(en)(NH
3)
3(H
2O)]3+ l s vfè kd g SA vr%
[Co(en)(NH3)
4]3+ dh vis{kk [Co(en)(NH
3)
3(H
2O)3+ nh?kZ rjaxnSè;Z vo'kksf"kr djrk gS
2. vyx ls fy, x, Mn2+ vkSj Cu2+ ds ukbVªsV yo.kksa ds foHksnu ds fy, lgh fodYi gS (gSa)
(A) Tokyk ijh{k.k (flame test) esa Mn2+ vfHky{kf.kd (characteristic) gjk jax fn[kkrk gS
(B) vEyh; ekè;e esa H2S izokfgr djus ij dsoy Cu2+ vo{ksi dk cuuk fn[kkrk gS
(C) gYds {kkjdh; ekè;e esa H2S izokfgr djus ij dsoy Mn2+ vo{ksi dk cuuk fn[kkrk gS
(D) Cu2+/Cu dk vip;u foHko (reduction potential) Mn2+/Mn ls mPprj gS (le:i voLFkk ij ekik x;k)
mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)
gygygygygy (A) eSaxuht Tokyk ijh{k.k esa gYdk cSaxuh jax n'kkZrk gS
(B) H S2 2
HClCu CuS
dkyk vo{kis
(C) Cu+2 rFkk Mn+2 nksuksa {kkjh; ekè;e esa H2S ds lkFk vo{ksi cukrs gS
(D) 2Cu /Cu
E 0.34
2Mn /Mn
E 1.18 V
3. ,sfufyu feJ.k vEy (lkUnz HNO3 rFkk lkUnz H
2SO
4 ) ds lkFk 288 K ij vfHkfØ;k djds P (51%), Q (47%) vkSj R(2%)
nsrk gSA fuEufyf[kr vfHkfØ;k vuqØeksa dk (ds) eq[; mRikn (major products(s)) gS (gSa)
1) Ac O, pyridine
2) Br , CH CO H2
2 3 2
3) H O
4) NaNO , HCl/273-278K
5) EtOH,
3
2
+
R
1) Sn/HCl2) Br /H O (excess)
2 2
3) NaNO , HCl/273-278K
4) H PO2
3 2
Smajor product(s)
(A)
BrBr
Br
Br
(B)
Br
Br
Br
Br
(C)
BrBr
Br
(D)
Br
Br
Br Br
mÙkjmÙkjmÙkjmÙkjmÙkj ( D )
16
JEE (ADVANCED)-2018 (PAPER-2)
gy NH2
Conc. HNO 3
& Conc. H SO2 4
NH2
NO2
+
NH2
NO2
+
NH2
NO2
51%(P)
47%(Q)
2%(R)
NH2
NO2 Ac O, Pyridine2
NH – C – CH3
NO2
O
Br2
CH COOH3
NO2
NH – C – CH3
O
Br
H O3+
NO2
NH2
Br
NaNO , HCl2
273-278 K
NO2
N Cl2+ –
Br
EtOH
NO2
Br
(S)
(R)
Sn/HClNO2
Br
(S)
NH2
Br
Br2
H O (excess)2
NH2
Br
Br
Br Br
NaNO , HCl2
Br
Br
Br Br
N Cl2
+ –H PO3 2
Br
Br
Br Br
4. D-Xywdksl dk fi Q' kj i zLrqrhdj.k (Fischer presentation) uhps fn; k x; k gSA
CHO
HHO
HH
OHHOHOH
CH OH2
D-glucose
-L-Xywdksi kbjSuksl (-L-glucopyranose) dh l gh l ajpuk (l ajpuk, ¡) gS (gSa)
(A)
CH OH2
OH
HO OH
H
H
HOH H
OH
(B)
CH OH2
OH
HO OH
H
OH
HH H
OH
17
JEE (ADVANCED)-2018 (PAPER-2)
(C)
CH OH2
O
HO
OH
H
H
HO
H H
H
OH (D)
CH OH2
OHO
OH
H
H
HOH
H
H
OH
mÙkj (D)
gy -L-Xywdksi k; jsukst dh l ajpuk fuEu gS
OH
OH
H
OH
H
CH OH2
H
HO
H
OH5. fLFkj vk; ru ,oa 300 K i j , d i zFke dksfV dh vfHkfØ; k A(g) 2B(g) + C(g) ds fy, ] i zkjaHk (t = 0) vkSj l e; t i j
l ai w.kZ nkc Øe' k% P0 vkSj Pt gSaA ' kq: esa fl i QZ A, [A]0 l kanzrk ds l kFk mi fLFkr gS vkSj A ds vkaf' kd nkc (partial pressure)dksi zkjafHkd ewY; (initial value) ds 1/3 rd i gqapus dk l e; t1/3 gSA l gh fodYi gS (gSa)(eku ys fd ; s l kjh xSl sa vkn' kZ xSl ksa t Sl k O; ogkj djrh gSa)
(A)
Time
In(3
P–P
)0
t
(B)
[A]0
t 1/3
(C)
Time
In(P
– P
)0
t
(D)
[A]0
Rat
e c
on
stan
t
mÙkj (A, D)
gy A 2B + C
P0 – –
P0 – P 2P P
Pt = P0 + 2P
P = t 0P P2
0 0
t 0 0 t0
P 2PKt ln ln
P P 3P PP
2
0 0 tKt ln 2P ln(3P P ) Time
ln (
3P
– P
)0
t
vfHkfØ; k dk osx fu; rakd i zkjfEHkd l aknzrk i j fuHkZj ugh djr k
18
JEE (ADVANCED)-2018 (PAPER-2)
6. vfHkfØ; k A P ds fy, ] [A] vkSj [P] ds l e; ds l kFk r ki eku T1 vkSj T2 i j vkys[ k uhps fn, x, gSa
Time
T1
T2
5
10[A
] / (
mo
l L)
–1
Time
T1
T25
10
[P]
/ (m
ol L
)–1
; fn T2 > T1 ] rks l gh i zdFku gS (gSa)
( H vkSj S dks rki eku fuHkZjrk l s Lora=k ekfu; s vkSj T1 i j ln(K) rFkk T2 i j ln(K) dk vuqi kr 2
1
T
T l s vf/ d gSA ; gk¡
H, S, G vkSj K, Øe' k% ,UFkSYi h] ,UVªkWi h] fxCt (Gibbs) Åt kZ vkSj l kE; koLFkk fLFkjkad gSa)
(A) H 0, S 0 (B) G 0, H 0
(C) G 0, S 0 (D) G 0, S 0
mÙkj (A, C)
gy 1 2
2 1
ln K Tln K T
r ki c<kus i j K ?kVrk gS
H° < 0
xzki Q l s K > 1 G° < 0
1 1 2
2 1
2
– H Sln K T R R T
H Sln K TT R R
1 2 2
2 1 1
( H T S ) T T( H T S ) T T
–H° + T1S° > –H° + T2S°
S° < 0
[ kaM-2(vf/ dre vad : 24)
bl [ kaM esa vkB (08) i z' u gSaA i zR;sd i z'u dk mÙkj , d l a[ ; kRed eku (NUMERICAL VALUE) gSA
çR; sd i z' u ds mÙkj ds l gh l a[ ; kRed eku(n' keyo vadu esa] n' keyo ds f}rh; LFkku rd : f.Mr@fudfVr ; mnkgj.kr%6.25, 7.00, -0.33, -.30, 30.27, -127.30) dks ekmt + (MOUSE) vkSj vkWu LØhu (ON-SCREEN) opqZvy U; wesfjd dhi SM(VIRTUAL NUMERIC KEYPAD) ds i z; ksx l s mÙkj ds fy, fufnZ"V LFkku i j nt Z djsaA
i zR; sd i z' u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk&
i w.kZ vad : +3 ; fn fl i QZ l gh l a[ ; kRed eku (Numerical value) gh mÙkj Lo: i nt Z fd; k x; k gSA
' kwU; vad : 0 vU; l Hkh i fjfLFkfr ; ksa esaA
19
JEE (ADVANCED)-2018 (PAPER-2)
7. uhps fn, x, v.kqvksa esa l s] de l s de ,d l srqca/ (bridging) vkWDl ks l ewg okys ; kSfxdksa dh dqy l a[ ; k____gSA
N2O3, N2O5, P4O6, P4O7, H4P2O5, H5P3O10, H2S2O3, H2S2O5
mÙkj (5.00)
gy
N—NO
N O2 3
O ON N
O
N O2 5
O OO
O
P O4 6
PP O
P O P
O OO O
P O4 7 H P O4 2 5
P O P
PP
O O
O
O O
O
P
O
HOH
O POH
O
H
H P O5 3 10
P
O
OH HO O
P
OH
O
O P
OH
O
OH
H S O2 2 3
S
S
OH HO O
H S O2 2 5
S
O
O HO
S
O
OH
8. mPp rki eku i j gok ds i zokg l s xysuk (galena) (, d v; Ld) dk vkaf' kd vkWDl hdj.k gksrk gSA dqN l e; ckn gok dki zokg can dj fn; k x; k] fdUrq can HkV~Vh dks xje djuk pkyw j[ kk x; k r kfd varoZLrqvksa (contents) dk Lo; a&vi p; u(self-reduction) gksA O2 ds i zfr kg xzg.k i j mRi kfnr Pb dk (kg esa) Hkkj gS____A
(i jek.kq Hkkj g mol–1 esa % O = 16, S = 32, Pb = 207)
mÙkj (6.47)
gy 2 2
2
2PbS 3O 2PbO 2SO
2PbO PbS 3Pb SO
3 eksy O2 l s 3 eksy ysM i zkIr gksrk gS
96 kg vkWDl ht u l s i zkIr ysM = 621 kg
1 kg vkWDl ht u l s i zkIr ysM = 621
6.468 6.47 kg96
9. , d t yh; foy; u esa ?kqfyr MnCl2 dh ek=kk ds eki u ds fy, ] bl s vfHkfØ; k MnCl2 + K2S2O8 + H2O KMnO4 +
H2SO4 + HCl (l ehdj.k l arqfyr ugha gS) ds vuql kj i w.kZr ; k KMnO4 esa i fjofrZr fd; k x; kA l kUnz HCl dh dqN cw¡nsa blfoy; u esa Mkyh x; h vkSj ml s gYds l s xje fd; k x; kA vkxs] i jeSaxusV vk; u dk jax xk;c gksus rd vksDl kfyd vEy (225
mg) dks va'kksa esa Mkyk x; kA i zkjafHkd foy;u esa MnCl2 dh ek=kk (mg esa) _____ gSA
(i jek.kq Hkkj g mol–1 esa : Mn = 55, Cl = 35.5)
mÙkj (126.00)
gy POAC l s,
MnCl2 ds m moles = KMnO4 ds m moles = x((ekuk)
rFkk KMnO4 ds meq = vkWDl sfyd vEy ds meq
225x 5 2
90x = 1
MnCl2 ds m moles = 1
MnCl2 ds mg = (55 + 71) = 126 mg
20
JEE (ADVANCED)-2018 (PAPER-2)
10. fn, x, ; kSfxd X ds fy, / zqo.k ?kw.kZd f=kfoe l eko; oh; ksa (optically active steroisomers) dh l ai w.kZ l a[ ; k ____ gSA
mÙkj (7.00)
gy
HO
HO
OH
OH
dsoy rhu f=kfo; dsUnz mi fLFkr gS
dqy l eko; o = 23 = 8
ysfdu ,d l eko;o i zdkf' kd fuf"Ø; gS
HO
OH
OH
OH
11. fuEufyf[ kr vfHkfØ; k vuqØe esa] ,sl hVksi Q+hukWu ds 10 eksy l s i zkIr D dh cuh ek=kk (xzke esa)_____ gSA
(fn; k x; k gS] i jek.kq Hkkj gmol–1 esa % H = 1, C = 12, N = 14, O = 16, Br = 80. i zR; sd pj.k esa mRi kn dh mi t (%)
dks"Bd esa nh x; h gS)
O
NaOBr
H O3+ A
(60%)
NH , 3 B
(50%)
Br /KOH2 C(50%)
Br (3 equiv)2
AcOHD
(100%)
mÙkj (495.00)
gy
O
NaOBr
H O3+
Acetophenone10 moles
COOH
A(60%)
NH 3
CONH2
B(50%)
Br KOH2
NH2
C(50%)
NH2
D(100%)
Br
Br
Br
D dh yfCèk = 60 50 50
10 1.5 moles100 100 100
D dh ek=kk = 1.5 × 330 = 495.00
21
JEE (ADVANCED)-2018 (PAPER-2)
12. dkWij dk i`"B] dkWij vkWDlkbM ds cuus ls efyu gksrk gSA dkWij dks 1250 K ij xje djrs le; vkWDlkbM cuus ls jksdusds fy, ukbVªkstu xSl dk izokg fd;k x;kA fdUrq ukbVªkstu xSl esa 1 eksy % tyok"i dk vinzO; gSA tyok"i dkWij dkuhps fn, x, vfHkfØ;k ds vuqlkj vkWDlhdj.k djrk gS%
2Cu(s) + H2O(g) Cu
2O(s) + H
2(g)
1250 K ij vkWDlhdj.k jksdus ds fy, H2 dk U;wure vkaf'kd nkc (bar esa) pH
2 pkfg,A In(pH
2) dk eku ____ gSA
(fn;k x;k gS] iw.kZ nkc = 1 bar, R (lkoZtfud xSl fu;rakd) = 8 J K–1mol–1, In(10) = 2.3; Cu(s) vkSj Cu2O(s)
ijLij vfeJ.kh; gSA
Cu(s) vkSj Cu2O(s) ijLij vfeJ.kh; gSA
1250 K: ij% 2Cu(s) + 1
2O
2(g) Cu
2O(s); G– = –78,000 J mol–1
H2(g) +
1
2O
2(g) H
2O(g); G– = –1,78,000 J mol–1; G fxCt ÅtkZ gS)
mÙk jmÙ k jmÙ k jmÙ k jmÙ k j (–14.60)
gygygygygy (i) 2 2
12Cu(s) O (g) Cu O(s), G –78 kJ/mole
2
(ii) 2 2 2
1H (g) O (g) H O(g), G –178 kJ/mole
2
(i) – (ii)
2 2 22Cu(s) H O(g) Cu O(s) H (g), G 100 kJ
G G RT ln K 0
⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠
H5 2
H O2
P10 8 1250 ln 0
P
⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠
H4 52
H O2
P10 ln 10 0
P
H H O2 2ln P – ln P –10
vc, –2H O H O Total2 2
P X P 0.01 1 10
H2ln P 2 ln10 –10
H2ln P 4.6 –10
H2ln P –14.60
13. fuEufyf[kr mRØe.kh; vfHkfØ;k (reversible reaction) ij fopkj djsaA
A(g) + B(g) ��⇀↽��
AB(g)
izrhi vfHkfØ;k (backward reaction) dh lfØ;.k ÅtkZ (activation energy)
vxz vfHkfØ;k (forward reaction)
dh
lfØ;.k ÅtkZ ls 2 RT (J mol–1 esa) vfèkd gSA ;fn vxz vfHkfØ;k dk iwoZ pj?kkrkadh xq.kd (pre-exponetial factor)
izrhi vfHkfØ;k ds iwoZ pj?kkrkadh xq.kd ls 4 xq.kk gS] rks 300 K ij vfHkfØ;k ds G (J mol–1 esa) dk fujis{k (absolute)
eku ____ gSA
(fn;k x;k gS] In(2) = 0.7, 300 K ij RT = 2500 J mol–1, G fxCt ÅtkZ gS)
mÙk jmÙ k jmÙ k jmÙ k jmÙ k j (8500.00)
22
JEE (ADVANCED)-2018 (PAPER-2)
gygygygygy A(g) + B(g)� AB(g)
b f
a aE –E 2RT
f
b
A4
A
f
b
KK
K
af–E /RT
f fK A e
ab–E /RT
b bK A e
a ab f(E –E )/RTf f
b b
K Ae
K A
K = 4e2RT/RT
K = 4e2
G° = – RT In K
= – RT (2 + ln 4)
= – 2500 (2 + 2 × 0.7)
= – 8500 J mol–1
fujis{k eku 8500.00 gS
14. ,d oS|qrjklk;fud lsy% A(s) | An+(aq, 2 M) || B2n+(aq, 1 M) | B(s) ij fopkj dhft,A 300 K ij lsy vfHkfØ;k ds
H dk ewY; mlds G ls nqxuk gSA ;fn lsy dk emf 'kwU; gS] rks 300 K ij lsy vfHkfØ;k esa B ds izfr eksy cuus ds
fy;s S (J K–1 mol–1 esa) dk eku _____ gSA
(fn;k x;k gS] In(2) = 0.7, R ( lkoZtfud xSl fu;rkad) = 8.3 J K–1mol–1 | H, S vkSj G, Øe'k% ,UFkSYih] ,UVªkih vkSj fxCt
(Gibbs) ÅtkZ gSaA)
mÙk jmÙ k jmÙ k jmÙ k jmÙ k j (–11.62)
gy n –
2n –
2n n
A A ne
B 2ne B
2A B 2A B
H° = 2G°, Ecell
= 0
G° = H° – TS°
G° = TS° G
ST
n 2
2n
– RTlnK [A ]S – R ln
T [B ]
2
2– 8.3 ln
1
S° = – 11.62 JK–1 mol–1
23
JEE (ADVANCED)-2018 (PAPER-2)
[ kaM 3 (vf/ dre vad % 12)• bl [ kaM esa pkj (04) i z'u gSaA i zR;sd i z' u esa nks (02) l qesyu l wfp;k¡ (matching lists) gSa% l wph–I vkSj l wph–II |• l wph–I vkSj l wph–II ds rÙoksa ds l qesykuks dks n'kkZrs gq, pkj fodYi fn, x, gSaA bu pkj fodYi ksa esa fl iQZ ,d fodYi gh l gh
l qesyu i znf' kZr djr k gSA• i zR; sd i z' u ds fy, l gh l qesyu i znf' kZr djus okys fodYi dks pqusaA• i zR; sd i z' u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk
i w.kZ vad %+3 ; fn fl i QZ l gh fodYi gh pquk x; k gSA' kwU; vad %0 ; fn dksbZ Hkh fodYi ugha pquk x; k gS (vFkkZr~ i z' u vuqÙkfjr gS)AÍ .kkRed vad %–1 vU; l Hkh i fjfLFfr ; ksa esaA
15. l wph-I (List-I) ds i zR; sd l adj d{kd (hybrid orbitals) ds l sV dks l wph-II(List-II) esa fn, x, l adqy (l adqyksa) ds l kFkl qesy djsaA
l wph-I l wph-II
P. dsp2 1. [FeF6]4–
Q. sp3 2. [Ti(H2O)
3Cl
3
R. sp3d2 3. [Cr(NH3)
6]3+
S. d2sp3 4. [FeCl4]2–
5. Ni(CO)4
6. [Ni(CN)4]2–
l gh fodYi gS(A) P 5; Q 4,6; R 2,3; S 1
(B) P 5,6; Q 4; R 3, 4; S 1,2
(C) P 6; Q 4,5; R 1; S 2,3
(D) P 4,6; Q 5,6; R 1,2; S 3
mÙkj (C)
gy 1. [FeF6]4–
Fe+2
4s 4p3d
mPp pØ.k l adqy gS D;ksafd F– , d nqcZy {ks=k fyxs.M gSA vr% l adj.k sp3d2 gS
2. [Ti(H2O)3Cl3]
Ti3+
d sp2 3
d sp2 3
3. [Cr(NH3)6]3+
Cr3+
d sp2 3
d sp2 3
4. [FeCl4]2–
Fe+2 : 3d6, Cl– nqcZy {ks=k fyxs.M gSA
sp3
3d 4s 4p
24
JEE (ADVANCED)-2018 (PAPER-2)
5. Ni(CO)4
Ni0 – 3d84s2, CO i zcy {ks=k fyxs.M gSA
Ni
sp3
6. [Ni(CN)4]2–
Ni+2
dsp2
CN– i zcy {ks=k fyxs.M gSA
16. l wph-I (List-I) dh vfHkfØ;kvksa ds eq[ ; mRi kn dk l wph-II(List-II) esa fn, x, , d ; k vusd l q; ksX; vfHkdkjdksa ds l kFk vfHkfØ;kdjus i j bfPNr mRi kn X cuk;k t k l drk gSA
(fn; k x; k] vfHkxkeh vfHkofÙk(migratory aptitude) dk Øe% ,sfjy > , sfYdy > gkbMªkst u)
O
Ph
Ph
OH
Me
X
l wph-I l wph-II
P.
PhHO
PhOH
Me
Me
H SO2 4+1. I
2, NaOH
Q.
PhH N2
PhOH
H
Me
HNO2+2. [Ag(NH
3)
2]OH
R.
PhHO
MeOH
Ph
Me
H SO2 4+3. i sQgfyax foy; u
S.
PhBr
PhOH
H
Me
AgNO3+4. HCHO, NaOH
5. NaOBr
25
JEE (ADVANCED)-2018 (PAPER-2)
l gh fodYi gS
(A) P 1; Q 2,3; R 1,4; S 2,4
(B) P 1,5; Q 3,4; R 4,5; S 3
(C) P 1,5; Q 3,4; R 5; S 2,4
(D) P 1,5; Q 2,3; R 1,5; S 2,3
mÙkj (D)
gy (P)Me
MeHOPhHO
Ph
+ H SO2 4
O
MePhMe
PhI /NaOH2
O
PhMe
PhC – OH
NaOBr
O
PhMe
PhC – OH
P 1, 5
(Q)
H
MeH N2
Ph
+ HNO2
O
HPhMe
Ph[Ag(NH ) ]OH3 2
PhMe
PhCOOH
Fehling
Ph
OH
SolutionPh
Me
PhCOOH
Q 2, 3
(R)Ph
MeHO
Ph
+ H SO2 4
O
MeMePh
Ph
Me
OH
NaOBr I /NaOH2
PhMePh
COOHPh
MePh
COOH
R 1, 5
(S)Ph
MePh
Ph
+ AgNO3
MePh
Ph
OH
Fehlingsolution
[Ag(NH ) ]OH3 2
Me
Ph
Ph
COOH
Br
CHO
Me
Ph
Ph
COOH
S 2, 3
26
JEE (ADVANCED)-2018 (PAPER-2)
17. l wph-I (List-I) esa vfHkfØ;k; sa gSa vkSj l wph-II (List-II) esa eq[ ; mRi kn gSaA
l wph-I l wph-II
P.ONa Br
+ 1.OH
Q.OMe
HBr+ 2.Br
R.Br
NaOMe+ 3.OMe
S.ONa
MeBr+ 4.
5.O
l wph-I dh i zR; sd vfHkfØ; k dk l wph-II ds ,d ; k vusd mRi knksa ds l kFk l qesy djsa vkSj l gh fodYi pqusaA
(A) P 1,5; Q 2; R 3; S 4
(B) P 1,4; Q 2; R 4; S 3
(C) P 1,4; Q 1,2; R 3,4; S 4
(D) P 4,5; Q 4; R 4; S 3,4
mÙkj (B)
gy (P) ONa + Br + OH
3° gSykbM ds l kFk i zHkkoh : i l s foyksi u gksrk gSA
(Q) OMe + HBr Br + MeOH
(R) Br + NaOMe
(S) ONa + MeBr OMe
P 1, 4; Q 2; R 4; S 3.
27
JEE (ADVANCED)-2018 (PAPER-2)
18. l wph-I (List-I) esa vyx vyx t yh; foy; uksa dk t y ds l kFk ruqdj.k djus ds i zØe fn, x, gSA foy; u ds ruqdj.k l s[H+] i j gq, i zHkko l wph-II(List-II) esa fn, x, gSaA
(è; ku nsa] nqcZy vEy vkSj nqcZy {kkj dh fo; kst u ek=kk ()(degree of dissociation)<<1 gS; yo.k ds t y&vi ?kVu dhek=kk(degree of hydrolysis of salt)<<1 gS; [H+], H+ vk; uksa dh l kanzrk dks fu: fi r djr k gS)
l wph-I l wph-II
P. (0.1 M NaOH dk 10 mL + 0.1 M , fl fVd 1. ruqdj.k djus i j [H+] ds eku esa dksbZ cnyko ugha gksrkgS
vEy dk 20 mL) dk 60 mL rd ruqdj.k
Q. (0.1 M NaOH dk 20 mL + 0.1 M , fl fVd 2. ruqdj.k djus i j [H+] dk eku cnydj bl ds i zkjafHkdvEy dk 20 mL) dk 80 mL rd ruqdj.k eku dk vkèkk gksrk gS
R. (0.1 M HCI dk 20 mL + 0.1 M veksfu;k 3. ruqdj.k djus i j [H+] dk eku cnydj bl ds i zkjafHkdfoy; u dk 20 mL) dk 80 mL rd ruqdj.k eku dk nks xq.kk gksrk gS
S. 10 ml Ni(OH)2 dk l ar Ir foy;u (saturated 4. ruqdj.k djus i j [H+] dk eku cnydj bl ds i zkjafHkd
solution) t ks vkfèkD; Bksl Ni(OH)2 ds l kFk eku dk 1
2 xq.kk gksrk gS
l kE;koLFkk esa gS] ml dk 20 mL rd ruqdj.k 5. ruqdj.k djus [H+] dk eku cnydj bl ds i zkjafHkd eku
fd;k x; k (Bksl Ni(OH)2 ruqdj.k ds i ' pkr dk 2 xq.kk gksrk gSA
Hkh mi fLFkr gSA)
l wph-I esa fn, x, i zR;sd i zØe dks l wph-II esa fn, x, ,d ; k vusd i zHkko (i zHkkoksa) ds l kFk l qesy djsaA l gh fodYi gS
(A) P 4; Q 2; R 3; S 1
(B) P 4; Q 3; R 2; S 3
(C) P 1; Q 4; R 5; S 3
(D) P 1; Q 5; R 4; S 1
mÙkj (D)
gy (P) 3 old
20 0.1– 10 0.1 1CH COOH
30 30
–
3old
1CH COO
30
[yo.k] = [vEy] okyk ci Qj
ruqrk i j pH i fjofrZr ugha gksrh (P) (1)
(Q) [ ] CH COO 3
–old
=
20 0.1 2
40 40
[ ] CH COO 3
–new
= 2
80
28
JEE (ADVANCED)-2018 (PAPER-2)
CH COO 3
– + H O2
c
–3
xxCH COOH OH
– 2 – 22old new
h[OH ] [OH ]x
Kc 2/40 2/80
[OH ] – 2
new =
– 2old[OH ]
2
–
– oldnew
[OH ][OH ]
2
new old[H ] 2[H ]
(Q) (5)
(R) 4 old
20 0.1 2[NH ]
40 40
4 new2
[NH ]80
4 2 4yc y
NH H O NH OH H
2 22old new[H ] [H ]y
Khc 2/40 2/80
22 old
new[H ]
[H ]2
oldnew
[H ][H ]
2
(R) (4)
(S) , d l ar Ir foy;u ds fy; s – 3 sp[OH ] 2K
foy;u dk vk; ru dqN Hkh gks] [H+] fu; r jgrk gSA
(S) (1)
END OF CHEMISTRY
29
JEE (ADVANCED)-2018 (PAPER-2)
MATHEMATICS[ kaM-1(vf/ dre vad : 24)
bl [ kaM esa Ng (06) i z' u gSaA
çR;sd i z' u ds l gh mÙkj (mÙkjksa) ds fy, pkj fodYi fn, x, gSaA bu pkj fodYi ksa esa l s , d ; k ,d l s vf/ d fodYi l ghgS(gSa)A
i zR;sd i z'u ds fy, ] i z'u dk(ds) mÙkj nsus gsrq l gh fodYi (fodYi ksa) dks pqusaA
i zR; sd i z'u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk%
i w.kZ vad : +4 ; fn dsoy(l kjs) l gh fodYi (fodYi ksa) dks pquk x; k gSA
vkaf' kd vad : +3 ; fn pkjksa fodYi l gh gSa i jUrq dsoy rhu fodYi ksa dks pquk x; k gSA
vkaf' kd vad : +2 ; fn rhu ; k rhu l s vf/ d fodYi l gh gSa i jUrq dsoy nks fodYi ksa dks pquk x; k gS vkSjpqus gq, nksuksa fodYi l gh fodYi gSaA
vkaf' kd vad : +1 ; fn nks ; k nks l s vf/ d fodYi l gh gSa i jUrq dsoy ,d fodYi dks pquk x; k gS vkSj pqukgqvk fodYi l gh fodYi gSA
' kwU; vad : 0 ; fn fdl h Hkh fodYi dks ugha pquk x; k gS (vFkkZr~ i z' u vuqÙkfjr gS)A
½.kkRed vad : –2 vU; l Hkh i fjfLFkfr ; ksa esaA
mnkgj.k Lo: i % ; fn fdl h i z' u ds fy, dsoy i gyk] rhl jk , oa pkSFkk l gh fodYi gSa vkSj nwl jk fodYi xyr gSa; r ks dsoyl Hkh rhu l gh fodYi ksa dk p; u djus i j gh +4 vad feysasxsA fcuk dksbZ xyr fodYi pqus (mnkgj.k esa nwl jk fodYi )] rhul gh fodYi ksa esa l s fl i QZ nks dks pquus i j (mnkgjar% i gyk rFkk pkSFkk fodYi ) +2 vad feysaxsA fcuk dksbZ xyr fodYi pqus(blmnkgj.k esa nwl jk fodYi )] rhu l gh fodYi ksa esa l s fl i QZ ,d dks pquus i j (i gyk ; k rhl jk ; k pkSFkk fodYi ) +1 vad feysaxsAdksbZ Hkh xyr fodYi pquus i j (bl mnkgj.k esa nwl jk fodYi )] –2 vad feyasxs] pkgs l gh fodYi (fodYi ksa) dks pquk x; k gks; k u pquk x; k gksA
1. fdl h Hkh / ukRed i w.kk±d (positive integer) n ds fy, , fn : (0, ) ,
n 1
n j 11
f (x) tan1 (x j)(x j 1)
l Hkh x (0, ) ds fy, ] ds }kjk i fjHkkf"kr gSA(; gk¡ i zfrykse f=kdks.kferh; i Qyu (inverse trigonometric function)
tan–1x,
,
2 2 esa eku / kj.k djr k gSA) rc fuEufyf[ kr esa l s dkSul k (l s) dFku l R; gS (gSa)\
(A) 5 2j 1 jtan (f (0)) 55
(B) 10 2j 1 j j(1 f (0))sec (f (0)) 10
(C) fdl h Hkh fu; r (fixed) / ukRed i w.kk±d n ds fy, ]
nx
1lim tan(f (x))
n
(D) fdl h Hkh fu; r (fixed) / ukRed i w.kk±d n ds fy, ]
2n
xlim sec (f (x)) 1
mÙkj (D)
30
JEE (ADVANCED)-2018 (PAPER-2)
gygygygygy ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
1 1 1
n
nf (x) tan (n x) tan (x) tan
1 (n x)x
⇒ 1
n nf (0) tan (n) tan(f (0)) n
⇒
2
n n2 2 2 2
1 1 1 nf (x) f (0) 1
1 (n x) 1 x 1 n 1 n
(A)
∑5
2 2 2 2 2 2j
j 1
5 6 11tan (f (0)) 1 2 3 4 5 55
6
(B)
⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦
∑ ∑210 10
2 2j j 2
j 1 j 1
j(1 f (0))sec (f (0)) 1 1 j 10
1 j
(C)
nx x
nlim tan(f (x)) lim 0
1 (n x)x
(D)
2
nx
lim (1 tan (f (x)) 1 0 1
2. ekuk fd T, fcanqvksa P(-2, 7) vkSj Q(2, -5) ls xqtjus okyh js[kk (line) gSA ekuk fd F1 mu lHkh o`Ùk ;qXeksa (pairs
of circles) (S1, S
2) dk leqPp; (set) gS fd js[kk T, S
1 ds fcanq P ij vkSj S
2 ds fcanq Q ij Li'khZ (tangent)
gS rFkk o`Ùk S1 o S
2 ,d nwljs dks fcanq] ekuk fd M, ij Li'kZ djrs gSaA tc ;qXe (S
1, S
2), F
1 esa fopfjr (varies)
djrk gS rks ekuk fd leqPp; (set) E1, fcanq M ds fcanqiFk (locus) dks n'kkZrk gSA ekuk fd F
2 mu ljy js[kk&[k.Mksa
(straight line segments) dk leqPp; gS] tks fcanq R(1, 1) ls xqtjrha gSa rFkk E1 ds nks fHkUu fcanqvksa ds ;qXe
(pair of distinct points) dks tksM+rha gSaA ekuk fd E2, leqPp; F
2 ds js[kk[k.Mksa ds eè; fcanqvksa dk leqPp; gSA rc
fuEufyf[kr esa ls dkSulk (ls) dFku lR; lR; lR; lR; lR; gS (gSa)\
(A) fcanq (–2, 7) leqPp; E1 esa fLFkr gS
(B) fcanq ⎛ ⎞⎜ ⎟⎝ ⎠
4 7,
5 5 leqPp; E
2 esa fLFkr ugha ugha ugha ugha ugha gS
(C) fcanq ⎛ ⎞⎜ ⎟⎝ ⎠
1,1
2 leqPp; E
2 esa fLFkr gS
(D) fcanq ⎛ ⎞⎜ ⎟⎝ ⎠
30,
2 leqPp; E
1 esa fLFkr ugha ugha ugha ugha ugha gS
mÙkjmÙkjmÙkjmÙkjmÙkj (B, D)
gygygygygyP
Q
T
M
31
JEE (ADVANCED)-2018 (PAPER-2)
PMQ = 90°
5 7
12 2
M dk fcanqiFk
x2 + y2 – 2y – 39 = 0 ...(1)
oÙk dh ml thok dk lehdj.k ftldk eè; fcnq (, ) gS
S1 = T
x + y – (y + ) – 39 = 2 + 2 – 2 – 39
R(1, 1)
( , )
x + y – 2y – 39 = 02 2
;g (1, 1) ls xqtjrh gS
2 + 2 – 2 – + 1 = 0
fcUnqiFk : x2 + y2 – x – 2y + 1 = 0 ...(2)
fodYi (A) xyr gS ;|fi ;g lehdj.k (1) dks larq"V djrk gS vU;Fkk js[kk T f}rh; o`Ùk dks nks fcanqvksa ij Li'kZ djsxhA fcanq
(4/5, 7/5) lehdj.k (2) dks larq"V djrs gS ysfdu iqu% bl fLFkfr esa thok dk ,d fljk (–2, 7) gksxk tks E1 esa lfEefyr
ugha gSA vr% ⎛ ⎞⎜ ⎟⎝ ⎠
4 7,
5 5, E
2 esa fLFkr ugha gSaA (1/2, 1), lehdj.k (2) dks larq"V ugha djrk gS] vr% ;g E
2 esa fLFkr ugha gSaA
(0, 3/2), lehdj.k (1) dks larq"V ugha djrk gS] vr% E1 esa fLFkr ugha gSA
3. ekuk fd S mu lHkh LrEHk vkO;wgksa (column matrices)
1
2
3
b
b
b
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
dk leqPp; (set) gS ftuds fy, b1, b
2, b
3 � vkSj
okLrfod pjksa (real variables) okys lehdj.k fudk; (system of equations)
–x + 2y + 5z = b1
2x – 4y + 3z = b2
x – 2y + 2z = b3
dk de ls de ,d gy (solution) gSA rc fuEufyf[kr okLrfod pjksa okys fudk;ksa esa ls fdl (dkSuls) fudk; (fudk;ksa)
dk Hkh izR;sd 1
2
3
b
b
b
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
S ds fy, de ls de ,d gy gS\
(A) x + 2y + 3z = b1, 4y + 5z = b
2 and x + 2y + 6z = b
3
(A) x + 2y + 3z = b1, 4y + 5z = b
2 vkSj x + 2y + 6z = b
3
(B) x + y + 3z = b1, 5x + 2y + 6z = b
2 vkSj –2x – y – 3z = b
3
(C) –x + 2y – 5z = b1, 2x – 4y + 10z = b
2 vkSj x – 2y + 5z = b
3
(D) x + 2y + 5z = b1, 2x + 3z = b
2 vkSj x + 4y – 5z = b
3
mÙkjmÙkjmÙkjmÙkjmÙkj (A, D)
Sol. nh x;h lehdj.kksa ds fudk; dks fuEukuqlkj fy[kk tk ldrk gS
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⇒ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
1
2
3
1 2 5 x b
2 4 3 y b AX B
1 2 2 z b
32
JEE (ADVANCED)-2018 (PAPER-2)
⎡ ⎤
⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
∼ ∼
11 1
2 1 2 3 1 2 3
3 1 31 3
b1 2 5 b 1 2 5 b 1 2 5
1 13A |B 2 4 3 b 0 0 6 b b b 0 0 0 b b b
7 71 2 2 b 0 0 7 b b 0 0 7
b b
bl fudk; ds gy ds fy,, 1 3
2
b 13bb 0
7 7
b1 + 7b
2 – 13b
3 = 0 (b
1, b
2, b
3) = (–7K
2 + 13K
3, K
2, K
3) tgk¡ K
2, K
3 R...(i)
(A) 0 (b1, b
2, b
3) dk dksbZ laHkkfor leqPp; ,d gy nsrk gSA
(i) ds ekuksa dk çR;sd leqPp; fodYi (A) ds fudk; dk de ls de ,d gy çnku djrk gS
(B )
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
∼ ∼
1 1 1
2 1 2 1 2 3
3 1 3 1 3
1 1 3 b 1 1 3 b 1 1 3 b
5 2 6 b 3 0 0 2b b 0 0 0 b b 3b
2 1 3 b 1 0 0 b b 1 0 0 b b
bl fudk; ds gy ds fy,] b1 + b
2 + 3b
3 = 0
(b1, b
2, b
3) = (–K
2 –3K
3, K
2, K
3), K
2, K
3 R ...(ii)
Li"Vr% (i) }kjk çnf'kZr leqPp;] (ii) esa fLFkr ugha gS
bl fudk; esa (i) ds (b1, b
2, b
3) ds çR;sd leqPp; ds fy, dksbZ gy ugha gSA
(C) lehdj.ksa lekUrj leryksa ;k le:i leryksa dh gSA gy ds fy,] bu leÙkyksa dk le:i gksuk vko';d gS] ftlds fy,(b
1, b
2, b
3) = (–K
3, 2K
3, K
3) ...(iii)
(b1, b
2, b
3) ds ,sls çR;sd eku (i) esa fLFkr gS ysfdu (i) esa (b
1, b
2, b
3) dk çR;sd eku (iii) esa mifLFkr ugha gS
(i) esa çR;sd (b1, b
2, b
3), fodYi (C) ds lehdj.k fudk; dk gy çnku ugha djrk gSA
(D) 0 (i) ds ekuksa dk çR;sd leqPP; fodYi (D) ds fudk; dk de ls de ,d gy çnku djrk gSA
4. ,slh nks ljy js[kkvksa (straight lines) ij fopkj dhft,] ftuesa ls izR;sd] o`Ùk (circle) 2 2 1x y
2 vkSj ijoy;
(parabola) y2 = 4x nksuksa ij gh Li'khZ (tangent) gSA eku fd ;s js[kk,a fcanq Q ij izfrPNsn (intersect) djrh gSA ,d ,sls
nh?kZo`Ùk (ellipse) ij fopkj dhft, ftldk dsanz (centre) ewyfcanw (origin) 0(0,0) ij gS vkSj ftldk v/Z&nh?kkZ{k (semi-
major axis) OQ gSA ;fn bl nh?kZo`Ùk ds y?kq v{k (minor axis) dh yEckbZ 2 gS] rc fuEufyf[kr esa ls dkSulk (ls) dFku
lR; lR; lR; lR; lR; gS (gSa)\
(A) nh?kZo`Ùk dh mRdsUnzrk (eccentricity)
1
2 gS vkSj ukfHkyEc thok (latus rectum) dh yEckbZ 1 gS
(B) nh?kZoÙk dh mRdsUnzrk 1
2 gS vkSj ukfHkyEc thok dh yEckbZ
1
2 gS
(C) js[kkvksa 1x
2 o x = 1 ds chp nh?kZo`Ùk }kjk ifjc¼ (bounded) {ks=k (region) dk {ks=kiQy (area) 1
( 2)4 2
gS
(D) js[kkvksa 1x
2 o x = 1 ds chp nh?kZoÙk }kjk ifjc¼ {ks=k dk {ks=kiQy 1
( 2)16
gS
33
JEE (ADVANCED)-2018 (PAPER-2)
mÙkj (A, C)
gy 1
y mxm
, 2 2 1x y
2 i j Li ' kZ js[ kk Hkh gSA
2
11m
m 121 m
mHk; fu"B Li ' kZ js[ kk; sa y = x + 1 o y = –x – 1 Q(–1, 0)
nh?kZoÙk dk l ehdj.k
2 2
2 2
x y1
1 1
2
b2 = a2(1 – e2)
21 11(1 e ) e
2 2
ukfHkyEc dh yEckbZ =
2
122b 2 1
a 1
y
xQ(–1, 0) (0, 0)
x = 11
x2
O
21y 1 x
2
1 2
1
2
1A 2 1 x dx
2
12 1
1
2
1 12 x 1 x sin x
2 2
2
4 2 oxZ bdkbZ
5. ekuk fd s, t, r ' kwU; sÙkj (non-zero) l fEeJ l a[ ; k; sa (complex numbers) gSa vkSj L l ehdj.k (equation) sz tz r 0 ds
gyksa (solutions) z = x + iy (x, y ,i 1) dk l eqPp; gS] t gk¡ z x iy A rc fuEufyf[ kr esa l s dkSul k (l s) dFku
l R; gS (gSa)\
(A) ; fn L esa Bhd ,d vo; o (element) gS] r c |s| |t|
(B) ; fn |s| = |t|, r c L esa vuUr (infinitely many) vo; o gSa
(C) L {z : |z – 1 + i| = 5} esa vo;oksa dh vf/ dre l a[ ; k 2 gS
(D) ; fn L esa ,d l s T;knk vo; o gS] r c L esa vuUr vo; o gSa
mÙkj (A, C, D)
34
JEE (ADVANCED)-2018 (PAPER-2)
gy sz t z r 0 ...(i)
(i) dk l a; qXeh ysus i j
s z t z r 0 ...(ii)
l ehdj.k (i) o (ii) esa l s z dk foyksi u djus i j
(ss z t s z sr) (t s z t t z t r) 0
2 2z(| s | | t | ) tr rs
(A) ; fn | s | | t | , r c z dk vf}rh; eku gS
(B) ; fn | s | | t | vkSj r t r s 0 , r c z ds vuUr eku gSa
; fn | s | | t | rFkk r t r s 0 , rc z dk dksbZ eku ugha gS
L fjDr l eqPp; ; k vuUr l eqPp; gks l drk gS
(C) z dk fcanqi Fk l Hkh fLFkfr ; ksa esa fjDr l eqPp; ; k , dy l eqPp; ; k ,d js[ kk gSA ; g fn, x, oÙk dks vf/ d l s vf/ d nksfcanqvksa i j çfrPNsn djsxkA
(D) ; fn L esa ,d l s vf/ d vo;o gSa] r c L ds vuUr vo;o gSaA
6. ekuk fd f : (0, ) , d , sl k f}vodyuh; (twice differentiable) i Qyu (function) gS fd
2
t x
f(x)sint f(t)sinxlim sin x
t x l Hkh x (0, ) ds fy, A
; fn
f ,
6 12 r c fuEufyf[ kr esa l s dkSul k (l s) dFku l R; gS (gSa)\
(A)
f
4 4 2
(B) 4
2xf(x) x
6 l Hkh x (0, ) ds fy,
(C) , d , sl s (0, ) vfLrRo (existence) gS ft l ds fy, f() = 0
(D)
f f 0
2 2
mÙkj (B, C, D)
gy
2
t x
f(x) sint f(t) sinxlim sin x
t x
L.H. fu;e ,oa N.L. çes; dk ç; ksx djus i j,
f(x) cosx – f(x) sinx = sin2x
1
f(x) x csinx
...(i)
35
JEE (ADVANCED)-2018 (PAPER-2)
f c 0 f(x) x sinx6 12
(A)
f
4 4 2
(B) pw¡fd 3x
sinx x6
4
2 xx sinx x
6
4
2 xf(x) x
6
(C) [0, ] esa f(x) l r r gS rFkk (0, ) esa vodyuh; gS rFkk f(0) = f() = 0
f() = 0, (0, )
(D) f(x) = –x cosx – sinx, f(x) = x sinx – 2 cosx
f f 02 2 2 2
[ kaM-2(vf/ dre vad : 24)
bl [ kaM esa vkB (08) i z' u gSaA i zR;sd i z'u dk mÙkj , d l a[ ; kRed eku (NUMERICAL VALUE) gSA
çR; sd i z' u ds mÙkj ds l gh l a[ ; kRed eku(n' keyo vadu esa] n' keyo ds f}rh; LFkku rd : f.Mr@fudfVr ; mnkgj.kr%6.25, 7.00, -0.33, -.30, 30.27, -127.30) dks ekmt + (MOUSE) vkSj vkWu LØhu (ON-SCREEN) opqZvy U; wesfjd dhi SM(VIRTUAL NUMERIC KEYPAD) ds i z; ksx l s mÙkj ds fy, fufnZ"V LFkku i j nt Z djsaA
i zR; sd i z' u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk&
i w.kZ vad : +3 ; fn fl i QZ l gh l a[ ; kRed eku (Numerical value) gh mÙkj Lo: i nt Z fd; k x; k gSA
' kwU; vad : 0 vU; l Hkh i fjfLFkfr ; ksa esaA
7. l ekdy (integral)
12
10 2 6 4
1 3dx
(x 1) (1 x)
dk eku gS ____A
mÙkj (2.00)
gy
1/2
1/42 60
1 3I dx
(x 1) (1 x)
1/2
1/460 8
1 3dx
1 x(x 1)
1 x
1/2
3/20 2
1 3dx
1 x(1 x)
1 x
2
1 x 2dxt dt
1 x (1 x)j[ kus i j
1/31/3 1/2
3/21
1
1 1 3 1 tI (1 3) (1 3)( 3 1) 2
12 2t2
36
JEE (ADVANCED)-2018 (PAPER-2)
8. ekuk fd P, 3 × 3 dksfV (order) dk ,d , sl k vkO; wg (matrix) gS fd P dh l Hkh i zfof"V; k¡ (entries) l eqPp; (set) {–1,
0, 1} esa l s gSA rc P ds l kjf.kd (determinant) dk vf/ dre l aHkkfor eku (maximum possible value) gS ____ A
mÙkj (4.00)
gy
11 12 13
21 22 23
31 32 33
a a a
P a a a
a a a
|P| dh vf/ dre l aHkkouk 6 gks l drh gS ; fn
21 22 22 23 21 23
31 32 32 33 31 33
a a a a a a2
a a a a a a
i jUrq] eSfVªDl 21 22
31 32
a a
a a , 2 (ekuk) ds : i esa l eqPp; gS rFkk 22 23
32 33
a a
a a , 2 ; k –2 ds : i esa l eqPp; gS] 21 23
31 33
a a
a a
Lor% ' kwU; eku ysrk gSA
vr%, |P| 6. vxyh l aHkkouk 4 gSA
1 1 1
P 1 1 1
1 1 1
, sl h ,d l aHkkouk gSA
9. ekuk fd l eqPp; (set) X esa Bhd 5 vo; o (elements) gS vkSj l eqPp; Y esa Bhd 7 vo; o gSaA ; fn X l s Y esa ,dSdh
i Qyuksa (one-one functions) dh l a[ ; k gS vkSj Y l s X esa vkPNknd (onto) i Qyuksa dh l a[ ; k gS] rc 1
( )5!
dk eku
gS ____ A
mÙkj (119.00)
gy = X l s Y esa ,dSdh i Qyuksa dh l a[ ; k
= Y l s X esa vkPNknd i Qyuksa dh l a[ ; k
= Y ds 7 vo; oksa esa l s 5 l ewg cukb, rFkk bu l ewgksa dk X ds 5 vo; oksa esa Øe i fjorZu dhft ,
4 2 3
7 75 5
3. 1 . 4 2 1 2. 3
3
1 7 7 75 3. 4 5. 22 3
= 7 [5 + 15 – 3] = 119
10. ekuk fd f : ,d l sl k vodyuh; i Qyu (differentiable function) gS ft l ds fy; s f(0) = 0A ; fn y = f(x), vody
l ehdj.k (differential equation) dy
(2 5y)(5y – 2)dx
dks l arq"V djr k gS] rc x –lim f(x) dk eku gS___A
mÙkj (0.40)
37
JEE (ADVANCED)-2018 (PAPER-2)
gy dy
(2 5y)(5y 2)dx
dydx
(5y 2)(5y 2)
1 1 1dy dx
4 5y 2 5y 2
1 5y 2ln x C
20 5y 2
5y 2
f(0) 0 ln x5y 2
5y 2
x ln5y 2
5y 2 20 y
5y 2 5
x
2lim f(x) 0.40
5
11. ekuk fd f : , d , sl k vodyuh; i Qyu (differentiable function) gS ft l ds fy; s f(0) = 1, vkSj t ks l Hkhx, y ds fy, l ehdj.k f(x + y) = f(x)f '(y) + f '(x)f(y) dks l arq"V djr k gSA rc loge(f(4)) dk eku gS _____A
mÙkj (2.00)
gy
f(0) = 1, f : R R
f(x + y) = f(x) f(y) + f(x) f(y) x, y R
x = y = 0 j[ kus i j f(0) = 2f(0) f(0) f(0) = 1/2
y = 0 j[ kus i j ,
f(x) = f(x) f(0) + f(x) f(0)
f(x) = 12
f(x) + f(x)
2f (x) 1
=f(x) ln(f(x)) =
x+ C
2
f(0) = 1 C = 0
ln(f(x)) =x2
ln(f(4)) = 4
= 22
38
JEE (ADVANCED)-2018 (PAPER-2)
12. ekuk fd P çFke v"Bka' k (first octant) esa ,d fcanq gS] ft l dk l ery (plane) x + y = 3 esa çfr fcEc (image) Q (vFkkZrjs[ kk[ k.M PQ l er y x + y = 3 ds yEcor gS vkSj PQ dk eè; fcanq l ery x + y = 3 esa fLFkr gS) z-v{k (axis) i j fLFkrgSA ekuk fd P dh x-v{k l s nwjh 5 gSA ; fn P dk xy-l er y esa çfr fcEc R gS] r c PR dh yEckbZ gS ______A
mÙkj (8.00)
gy P = (,, ), Q = (0, 0, K) pw¡fd ;g z-v{k i j fLFkr gS
PQ dk eè; fcanq vFkkZr~, , ,
K2 2 2
fuEu dks l arq"V djrk gS
x + y = 3+ = 6 ...(1)
PQ ds fnd~ vuqi kr = (– 0, – 0, – K) = (p, p, 0)
= ,oa = K ...(2)
(1) , oa (2) = = 3
P = (, , K) = (3, 3, K)
x-v{k l s P dh nwjh = 5
2 + 2 = 25 2 + K2 = 25 32 + K2 = 25|K| = 4
PR dh yEckbZ = 2|K| = 8
13. çFke v"Bkax (first octant) esa ,d , sl s ?ku (cube) i j fopkj dhft ; s] ft l dh Hkqt kvksa (sides) OP, OQ vkSj OR yEckbZ 1
gS vkSj t ks Øe' k% x-v{k (axis), y-v{k vkSj z-v{k ds vuqfn' k (along) gSa] t gk¡ O(0, 0, 0) ewyfcanq (origin) gSA ekuk fd
?ku dk dsaæ (centre)
1 1 1S , ,
2 2 2 gS] vkSj ' kh"kZ (vertex) T ewyfcanq O ds l Eeq[ k (Opposite) okyk og ' kh"kZ gS fd
fcanq S fod.kZ (diagonal) OT i j fLFkr gSA ; fn
p SP, q SQ, r SR t ST, | (p q) (r t) | vkjS rc dk ekugS_______A
mÙkj (0.50)
gy 1 1 1 1ˆ ˆˆ ˆ ˆ ˆp SP i j k = (i j k)
2 2 2 2
Y(0,1,0)Q
T (1,1,1)
O
S1 1 1
, ,2 2 2
P(1,0,0)X
ZR(0, 0, 1)
1 ˆˆ ˆq SQ ( i + j k)
2
1 ˆˆ ˆr SR ( i j + k)
2
1 ˆˆ ˆt ST (i + j + k)
2
ˆˆ ˆi j k
1 1 ˆ ˆp q 1 1 1 (i + j)4 2
1 1 1
ˆ ˆˆ ˆ ˆ ˆi j k i j k 1 1 1 ˆ ˆr × t 1 1 1 1 1 1 ( i + j)4 4 2
1 1 1 0 0 2
1 1 1ˆ ˆˆ ˆ ˆ ˆ(p q) (r t) (i + j) ( i + j) 2k k
4 4 2
1
(p q) (r t) 0.502
39
JEE (ADVANCED)-2018 (PAPER-2)
14. ekuk fd X = (10C1)2 + 2(10C2)2 + 3(10C3)2 + ... + 10(10C10)2, t gk¡ 10Cr, r{1, 2, ...,10}, f}i n xq.kkadksa (binomial
coefficients) dks n'kkZrs gSaA rc 1
X1430
dk eku gS____A
mÙkj (646.00)
gy (1 + x)10 = 10C0 + 10C1x + 10C2x2 + ... + 10C10x10
nksuksa vksj x ds l ki s{k vodyu djus i j
10(1 + x)9 = 10C1 + 2·10C2x + 310C3x2 + ...+ 1010C10x9 ...(1)
(1 + x)10 = 10C0x10 + 10C1x9 + ... + 10C10 ...(2)
10(1 + x)9(1 + x)10 esa x9 dk xq.kkad l eku gS
pw¡fd (10C1)2 + 2(10C2)2 + 3(10C3)2 + ... + 10(10C10)2
= 10(1 + x)19 esa x9 dk xq.kkad = 10 × 19C9 = X
1X
1430 =
1 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11× 10 ×
1430 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 = 646
[ kaM 3 (vf/ dre vad % 12)
• bl [ kaM esa pkj (04) i z'u gSaA i zR;sd i z' u esa nks (02) l qesyu l wfp;k¡ (matching lists) gSa% l wph–I vkSj l wph–II |
• l wph–I vkSj l wph–II ds rÙoksa ds l qesykuks dks n'kkZrs gq, pkj fodYi fn, x, gSaA bu pkj fodYi ksa esa fl iQZ ,d fodYi gh l ghl qesyu i znf' kZr djr k gSA
• i zR; sd i z' u ds fy, l gh l qesyu i znf' kZr djus okys fodYi dks pqusaA
• i zR; sd i z' u ds mÙkj dk ewY; kadu fuEu vadu ; kst uk ds vuql kj gksxk
i w.kZ vad %+3 ; fn fl i QZ l gh fodYi gh pquk x; k gSA
' kwU; vad %0 ; fn dksbZ Hkh fodYi ugha pquk x; k gS (vFkkZr~ i z' u vuqÙkfjr gS)A
Í .kkRed vad %–1 vU; l Hkh i fjfLFfr ; ksa esaA
15. ekuk fd
1
xE x : x 1 0
x – 1vkSj
vkSj
–1
2 1 e
xE x E : sin log , (real number)
x –1,d okLrfod l a[ Õkk gS
(; gk¡ çfr ykse f=kdks.kferh; i Qyu (inverse trigonometric function) sin–1x, – ,
2 2 esa eku / kj.k djr k gS)
ekuk fd i Qyu
1 e
xf : E , f(x) log
x – 1 ds }kjk i fjHkkf"kr gS vkSj i Qyu
–12 2
xg : E , g(x) sin log
x – 1
ds }kjk i fjHkkf"kr gSA
40
JEE (ADVANCED)-2018 (PAPER-2)
l wph-I l wph-II
P. f dk i fj l j (range) gS 1.
1 e, ,
1 e e 1
Q. g dk i fj l j esa l ekfgr (contained) gS 2. (0, 1)
R. f ds çkUr (donain) esa l ekfgr gS 3.
1 1,
2 2
S. g dk çkUr gS 4. ( , 0) (0, )
5.
e,
e 1
6.
1 e( , 0) ,
2 e 1
fn, gq, fodYi ksa esa l s l gh fodYi gS:
(A) P 4; Q 2; R 1; S 1
(B) P 3; Q 3; R 6; S 5
(C) P 4; Q 2; R 1; S 6
(D) P 4; Q 3; R 6; S 5
mÙkj (A)
gy f(x) ds çk¡r ds fy, ;
x
0 x 0 x 1x 1
Õkk
f
1 eD : ( , 0) (1, ) , ,
1 e e 1
f(x) ds i jkl ds fy, ;
ekuk
y
e y
x ey log x
x 1 e 1
vc x < 0 ; k x > 1
y
y
e0
e 1;k
y
y
e1
e 1
0 < ey < 1 ;k y
10
e 1
– < y < 0 ;k ey > 0
f(x) dk i jkl ( , 0) (0, ) gS
g(x) ds çk¡r ds fy, ;
e
x1 log 1
x 1rFkk
x
0x 1
41
JEE (ADVANCED)-2018 (PAPER-2)
1 xe
e x 1rFkk x < 0 or x > 1
x(e 1) 10
x 1rFkk
(e 1)x e
0x 1
;k x < 0 ;k x > 1
⎛ ⎤ ⎡ ⎞ ⎜ ⎟⎥ ⎢ ⎝ ⎦ ⎣ ⎠
1 ex , ,
e 1 e 1
g(x) ds ijkl ds fy,;
⎛ ⎞ ⎜ ⎟⎝ ⎠e
x1 log 0
x 1 ;k ⎛ ⎞ ⎜ ⎟⎝ ⎠
x0 log 1
x 1
g(x) 0 0 g(x)2 2
Õkk
g(x) dk ijkl = ⎡ ⎞ ⎛ ⎤ ⎟ ⎜⎢ ⎥
⎣ ⎠ ⎝ ⎦, 0 0,
2 2
R 1, P 4, S 1, Q 2
16. ,d gkbZ Ldwy (high school) esa] 6 ckydksa M1, M
2, M
3, M
4, M
5, M
6 vkSj 5 ckfydkvksa G
1, G
2, G
3, G
4, G
5 ds lewg (group)
esa ls ,d lfefr (committee) cukbZ tkuh gSSA
(i) ekuk fd 1 lfefr dks bl çdkj ls cukus ds rjhdksa (ways) dh dqy la[;k gS fd lfefr esa 5 lnL; gSa] ftuesa ls Bhd
(exactly) 3 ckyd vkSj 2 ckfydk,a gSaA
(ii) ekuk fd 2 lfefr dks bl çdkj ls cukusa ds rjhdksa dh dqy la[;k gS fd lfefr esa de ls de (at least) 2 lnL; gSa]
vkSj ckydksa vkSj ckfydkvksa dh la[;k cjkcj (equal) gSA
(iii)ekuk fd 3 lfefr dks bl çdkj ls cukusa ds rjhdksa dh dqy la[;k gS fd lfefr esa 5 lnL; gSa] ftuesa ls de ls de 2
ckfydk,a gSA
(iv) ekuk fd 4 lfefr dks bl çdkj ls cukusa ds rjhdksa dh dqy la[;k gS fd lfefr esa 4 lnL; gSa] ftuesa ls de ls de 2
ckfydk,a gSa vkSj M1 o G
1 lfefr esa ,d lkFk ugha gSaA
lwphlwphlwphlwphlwph-I lwphlwphlwphlwphlwph-II
P. 1 dk eku gS 1. 136
Q. 2 dk eku gS 2. 189
R. 3 dk eku gS 3. 192
S. 4 dk eku gS 4. 200
5. 381
6. 461
fn, fodYiksa esa ls lgh fodYi gS:
(A) P 4; Q 6; R 2; S 1
(B) P 1; Q 4; R 2; S 3
(C) P 4; Q 6; R 5; S 2
(D) P 4; Q 2; R 3; S 1
mÙkjmÙkjmÙkjmÙkjmÙkj (C)
42
JEE (ADVANCED)-2018 (PAPER-2)
gygygygygy
(i) 1 = 6C
3 × 6C
2 = 20 × 10 = 200
(ii) 2 = 6C
1 × 5C
1 + 6C
2 × 5C
2 + 6C
3 × 5C
3 + 6C
4 × 5C
4 + 6C
5 × 5C
5
= 30 + 150 + 200 + 75 + 6 = 461
(iii) 3 = 5C
2 × 6C
3 + 5C
3 × 6C
2 + 5C
4 × 6C
1 + 5C
5
200 + 150 + 30 + 1 = 381
(iv) 4 = (5C
2 × 6C
3 – 4C
1 × 5C
1) + (5C
3 × 6C
1 – 4C
2 × 5C
0)
= (150 – 20) + (60 – 6) + 5
= 130 + 60 – 1 = 190 – 1 = 189
17. ekuk fd 2 2
2 2
x yH: 1
a b, tgk¡ a > b > 0, xy-lery (plane) esa ,d ,slk vfrijoy; (hyperbola) gS ftldk la;qXeh v{k
(conjugate axis) LM mlds ,d 'kh"kZ (vertex) N ij 60º dk dks.k (angle) varfjr (subtend) djrk gSA ekuk fd
f=kHkqt (triangle) LMN dk {ks=kiQy (area) 4 3 gSA
lwphlwphlwphlwphlwph-I lwphlwphlwphlwphlwph-II
P. H ds la;qXeh v{k dh yEckbZ gS 1. 8
Q. H dh mRdsUærk (eccentricity) gS 2.4
3
R. H dh ukfHk;ksa (foci) ds chp dh nwjh gS 3.2
3
S. H ds ukfHkyEc thok (latus rectum) dh 4. 4
yEckbZ gS
fn, gq, fodYiksa esa lgh foDyi gS :
(A) P 4; Q 2; R 1; S 3
(B) P 4; Q 3; R 1; S 2
(C) P 4; Q 1; R 3; S 2
(D) P 3; Q 4; R 2; S 1
mÙkj (B)
Sol.30°
L
b
M
O a
y
xN
btan30
a
a
b3
...(i)
43
JEE (ADVANCED)-2018 (PAPER-2)
vc OLN dk {ks=ki Qy = 1
ab2
1
ab 2 32
ab 4 3 ...(ii)
(i) rFkk (ii) l s a 2 3, b = 2
vc 2
2
b 4 2e 1 1
12a 3
ukfHk; ksa ds eè; nwjh = 2ae
2
2 2 3 83
ukfHkyEc dh yEckbZ = 22b 2 4 4
a 2 3 3
18. ekuk fd i Qyu f1 : , f
2: – ,
2 2 , f
3 :
2– 1, e – 2 rFkk f4 : bl çdkj i fjHkkf"kr gSa fd
(i) f1(x) = 2– x1 – e ,sin
(ii) f2(x) =
–1
sinx
tan xx 0
1 x 0
Õkfn
Õkfn, t gk¡ çfrykse f=kdks.kferh; i Qyu (inverse trigonometric function) tan–1x,
– ,2 2 esa
eku / kj.k djrk gS]
(iii) f3(x ) = [sin(log
e(x + 2))]; t gk¡ t , [t ], t l s NksVk t ds cjkcj egÙke i w.kk±d (greatest integer) dks n'kkZrk gS
(iv) f4(x) =
2 1x sin x 0x
0 x 0
Õkfn
Õkfn
l wph-I l wph-II
P. i Qyu f1
1. x = 0 i j l arr (continuous) ugha gS
Q. i Qyu f2
2. x = 0 i j l ar r gS vkSj x = 0 i j vodyuh;
(differentiable) ugha gS
R. i Qyu f3
3. x = 0 i j vodyuh; gS vkSj x = 0 i j bl dk vodyt(derivative) l arr ugha gS
S. i Qyu f4
4. x = 0 i j vodyuh; gS vkSj x = 0 i j bl dk vodytl ar r gS
fn, gq, fodYi ksa esa l s l gh fodYi gS :(A) P 2; Q 3; R 1; S 4(B) P 4; Q 1; R 2; S 3(C) P 4; Q 2; R 1; S 3
(D) P 2; Q 1; R 4; S 3
mÙkj (D)
44
JEE (ADVANCED)-2018 (PAPER-2)
gy ( i ) f1 : R R
2x1f (x) sin 1 e
= 2x
1sin 1
e
f1(x) l oZ=k l r r~ gS] i jUrq f1(0) fo| eku ugha gS] i QyLo: i x = 0 i j f1(x) vodyuh; i Qyu ugha gS f1(x) vodyuh;i Qyu ugha gSA
(ii) 2f : , R2 2
1
2
1
sinx; x 0
tan xf (x) 1 ; x 0
sinx; x 0
tan x
x = 0 i j LHL dk eku –1 gS
x = 0 i j RHL dk eku 1 gS
x = 0 i j f2(x) vl r r~ gS,
(iii) 23f : 1, e 2 R
3 ef (x) sin(log (x 2)
21 x e 2
21 x 2 e
e0 log (x 2)2
e0 sin(log (x 2)) 1
[sin(loge(x + 2)] = 0
f3(x) = 0
x = 0 i j f3(x) l r r~ ,oa vodyuh; gS
(iv) f4 : R R
2
4
1x sin ; x 0
f (x) x
0 ; x 0
45
JEE (ADVANCED)-2018 (PAPER-2)
24
x 0 x 0
1lim f (x) lim x sin 0
x
= x = 0 i j f4(x) dk eku
x = 0 i j f4(x) l r r~ gS
vc, h 0 h 0
1sin
f(0 h) f(0) hf (0 ) lim lim 01hh
h 0 h 0
1sin
f(0 h) f(0) hf (0 ) lim lim 01hh
x = 0 i j f(x) vodyuh; gS
vc, 24 2
1 1 1f (x) 2 sin x . cos
x xx
= 1 1
2 sin cosx x
fujar j gS
x = 0 i j vl r r~ i Qyu gS