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Transcript of Jee Advance-2013 (Paper-i)
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NARAYANA IIT PMT ACADEMYJEE ADVANCE - 2013
PAPER I CODE: 0
1
NARAYANA GROUP OF EDUCATIONAL INSTITUTIONS - INDIA
This booklet contains 33printed pages XYZ Test Booklet Code
PAPER -1 : PHYSIC, CHEMISTY & MATHEMATICS
Please Read the Instructions carefully You are allotted 5 minutes specifically for
this purpose
Important Instructions :
INSTRUCTIONS
A General :
This booklet is your Questions Paper Do not break the seals of this booklet before being instructed to
do so by the invigilators
The question paper CODE is printed on the right hand top corner of this sheet and on the back page(Page No 44) of this booklet
Blank spaces and blank pages are provided in the question paper for your rough work No additionalsheets will be provided for rough work
Blank papers, clipboards, log tables, slide rules, calculators, cameras, cellular phones, pagers andelectronic gadgets are NOT allowed inside the examination hall
Write your name and roll number in the space provided on the back cover of this booklet
Answers to the questions and personal details are to be filled on a two part carbon less paper,
which is provided separately These parts should only be separated at the end of the examination wheninstructed by the invigilator The upper sheet is machine gradable objective Response Sheet (ORS)
which will be retained by the invigilator You will be allowed to take away the bottom sheet at the end
of the examinationUsing a black ball point pen darken the bubbles on the upper original sheet Apply sufficient pressure
so that the impression is created on the bottom duplicate sheet
DO NOT TAMPER WITH/MUTILATE THE ORS OR THE BOOKLET
On breaking the seals of the booklet check that it contains 44 pages and all the 60 questions and
corresponding answers choices are legible Read carefully the instruction printed at the beginning of
each section and TOTAL MARKS 180.
B, Filling the right part of the ORS
The ORS also has a CODE printed on its left and right parts
Check that the CODE printed on the ORS (on both sheets) is the same as that on this booklet and putyour signature affirming that you have verified this
IF THE CODES DO NOT MATCH, ASK FOR A CHANGE OF THE BOOKLET
0
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PART I : PHYSICS
SECTION -1: (Only One option correct Type)
This section contains 10 multiple choice questions Each question has four choices (A), (B), (C) and (D)
out of which ONLY ONE is correct
1 The work done on a particle of mass m by a force,3/ 2 2 2 3/ 22 2
x y K i j
(x y )x y(K being a
constant of appropriate dimensions), when the particle is taken from the point (a,0) to the point(0, a) along a circular path of radius a about the origin in the x-y plane is
(A)2K
a (B)
K
a (C)
K
2a (D) 0
Ans (D)
Sol Direction of Forcey
tanx
F
ds
Force is radially outwards and displacement is tangential Hence work done = 0
2 Two rectangular blocks, having identical dimensions, can be arranged either in configuration I
or in configuration II as shown in the figure One of the blocks has thermal conductivity k and theother 2k The temperature difference between the ends along the x-axis is the same in both the
configurations It takes 9s to transport a certain amount of heat from the hot end to the cold end inthe configuration I The time to transport the same amount of heat in the configuration II is
(A) 20 s (B) 30 s (C) 45 s (D) 60 s
Ans (A)Sol In series, equivalent thermal conductivity
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1 2 1 2
eq 1 2K K K
eq
2 1 1
K K 2K
eq
4KK
3
In parallel
eq 1 2 1 1 2 2K (A A ) K A K A
eq
3KK
2
Heat Supplied
1 1
1 1
1
K A TQ t
2 2
2 2
2
K A TQ t
2
4K A 3K 2A. 9 t
3 2 2
1 2( Q Q )
6 = 3t2t2 = 2 sec
3 Two non-reactive monatomic ideal gases have their atomic masses in the ratio 2 : 3 The ratio oftheir partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3 The ratio
of their densities is(A) 1 : 4 (B) 1 : 2 (C) 6 : 9 (D) 8 : 9
Ans (D)Sol P1M1= d1RT
P2M2= d2RT
1 1 1 1
2 2 2 2
PM d d 4 2 8
P M d d 3 3 9
4 A particle of mass m is projected from the ground with an initial speed u0at an angle
with the
horizontal At the highest point of its trajectory, it makes a completely inelastic collision with
another identical particle, which was thrown vertically upward from the ground with the same
initial speed u0The angle that the composite system makes with the horizontal immediately afterthe collision is
(A)4
(B)4
(C)4
(D)2
Ans (A)
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Sol
u0 o
2 20
max
u sinH2g
ou cos
ou sin
2 2
2 2 0
0
u sinV u 2g
2g
2 2 2
0V u cos
0v u cos
tan 1
4
5 A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest
Power of the pulse is 30 mW and the speed of light is 83 10
ms-1
The final momentum of the
object is
(A) 170.3 10
kg ms-1
(B) 171.0 10
kg ms-1
(C) 173.0 10
kg ms-1
(D) 179.0 10
kg ms-1
Ans (B)
Sol E = 30 mW 100 ns
= 3000 10-12
J= 3 10-9J
P = E/C9
8
3 10
3 10= 10
-17kg m/s
6 In the Youngs double slit experiment using a monochromatic light of wavelength , the path
difference (in terms of an integer n) corresponding to any point having half the peak intensity is
(A) (2n 1)2
(B) (2n 1)4
(C) (2n 1)8
(D) (2n 1)16
Ans (B)
Sol 200
II cos
2 2
1cos
2 2
(2n 1) / 2
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2. x
x (2n 1) / 4
7 The image of an object, formed by a plano-convex lens at a distance of 8m behind the lens, is
real and is one-third the size of the object The wavelength of light inside the lens is2
3times the
wavelength in free space The radius of the curved surface of the lens is
(A) 1 m (B) 2 m (C) 3 m (D) 6 m
Ans (C)
Sol1
3/ 22 / 3
i
o
h vmh u
1 v 8
3 u u
u = 24
1 1 1( 1)
f R
1 1 1
v u R
R = 3 m
8 One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end ofanother horizontal thin copper wire of length L and radius R When the arrangement is stretched
by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wireis
(A) 025 (B) 050 (C) 200 (D) 400
Ans (C)
SolR
R
2
1
F(2R)
YL
2L
1 2
F LL
R 2Y
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2
2
F
RYL
L
2 2
F LL
R Y
1
2
L2
L
9 A ray of light travelling in the direction1
i 3j2
is incident on a plane mirror After reflection,
it travels along the direction1
i 3j2
The angle of incidence is
(A) 30 (B) 45 (C) 60 (D) 75
Ans (A)
Sol Direction of Normal is1 1
(i 3j) (i 3) j2 2
3j
Angle between incident ray and normal
1(i
3j).( 3j)2cos1
(i 3j 3j2
332
23
o30
10 The diameter of a cylinder is measured using a Vernier calipers with no zero error It is found
that the zero of the Vernier scale lies between 510 cm and 515 cm of the main scale The Vernier
scale has 50 divisions equivalent to 245 cm The 24th
division of the Vernier scale exactlycoincides with one of the main scale divisions The diameter of the cylinder is
(A) 5112 cm (B) 5124 cm (C) 5136 cm (D) 5148 cm
Ans (B)
Sol MSD = 005 cm
2.45VSD 0.049
50cm
LC = 0050049 = 0001 cm
Reading = 510 + 24 0001 cm
= 510 + 0024 cm = 5124 cm
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SECTION -2 : (One or more options correct Type)
The section contains 5 multiple choice questions Each question has four choices (A), (B), (C) and (D)out of which ONE and MORE are correct
11 In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C
The switch S1is pressed first to fully charge the capacitor C1and then released The switch S2isthen pressed to charge the capacitor C2After some time, S2 is released and then S3 is pressed
After some time,
(A) the charge on the upper plate of C1is 2CV0
(B) the charge on the upper plate of C1is CV0(C) the charge on the upper plate of C2is 0
(D) the charge on the upper plate of C2isCV0S1 S2 S3
C1 C2
V02V0
Ans B, D
Sol: When s1is closed and then released the charge on upper plate of C 1is +2CVoWhen S2is closed
and then released charge will be distributed equally in C1 and C2 So the charge on the upperplate of C2is +CVoWhen S3is pressed the charge on upper plate of C2will become (CVo)
12 A particle of mass M and positive charge Q, moving with a constant velocity 11u 4ims , enters
a region of uniform static magnetic field normal to the x-y plane The region of the magnetic field
extends form x = 0 to x = L for all values of y After passing through this region, the particle
emerges on the other side after 10 milliseconds with a velocity 12
u 2 3i j ms . The correct
statements (s) is (are)(A) The direction of the magnetic field isz direction
(B) The direction of the magnetic field is +z direction
(C) The magnitude of the magnetic field50 M
3Qunits
(D) The magnitude of the magnetic field isM
3Qunits
Ans A, C
Sol:
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x x x x x x x x
x x x x x x x x
x x x x x x x xx x x x x x x x
x x x x x x x xx x x x x x x x
x x x x x x x x
x x x x x x x xx x x x x x x xx x x x x x x x
x = 0 x = L
30o30
o
x
y
As particle turns toward +ve y axis the direction of field must be alongz axis (as)
[use : F q v B ]
Since2
u 2 3i j
o1tan3
So particle emerges at an angle of 30owith initial direction
Hence Time to turn by 30ois
T T
2 6 12
31 2 mt 10 1012 qB
50 MB
3Q
13 A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the
equation, y(x, t) = (001 m) sin[(628 m1) x] cos [(628 s1)t]
Assuming
= 314, the correct statement (s) is (are)(A) The number of nodes is 5(B) The length of the string is 025 m
(C) The maximum displacement of the midpoint of the string, from its equilibrium position is
001 m
(D) The fundamental frequency is 100 HzAns B, C
Sol: 1 1y x, t 0.01m sin 62.8m xcos 628s t
Here,
2
k 62.8 20 0.1m
And 1628 200 s
1v 10msk
Since string is vibrating in 5th
harmonic so
5 5 0.1L 0.25
2 2
m
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and fundamental frequency
v 10f 20Hz
2L 0.5
The midpoint is the antinode hence A = 001 m14 A solid sphere of radius R and density
is attached to one end of a mass-less spring of forceconstant k The other end of the spring is connected to another solid sphere of radius R and
density 3 The complete arrangement is placed in a liquid of density 2 and is allowed to reach
equilibrium The correct statement (s) is (are)
(A) the net elongation of the spring is34 R g
3k
(B) the net elongation of the spring is3R g
3k
(C) the light sphere is partially submerged
(D) the light sphere is completely submergedAns A, DSol:
A
B
B
34 R 2 g3
kx
34 R g3
If whole system is submerged then upthrust is equal to the weight of the system hence the sphereA must be completely submerged On sphere B net force must be zero as weight of sphere B is
greater than upthrust, hence there must be elongation in spring
So for sphere B 3 34 4
kx R 2 g R 3 g.3 3
34 R g3x
k
15 Two non-conducting solid spheres of radii R and 2 R, having uniform volume charge densities
1
and 2respectively, touch each other The net electric field at a distance 2 R and from the centre
of the smaller sphere, along the line joining the centres of the spheres, is zero The ratio 1
2
can
be
(A)4 (B)32
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(C)32
25 (D) 4
Ans B, D
Sol: If 1and 2both are both positive or both negative then field will be zero at a point in betweenthe line joining the centre
ie,3
1 2
2
o o
R
3 32RR
1
2
4
Case II: If one of the charge density is negative and other is positive the point will lies on the left
side of smaller sphere33
1 2
2 2
o o
2RR0
3 32R 5R
1
2
32
25
SECTION-3: (Integer value correct Type)
The section contains 5 questions The answer to each question is a single digit integer, ranging from 0 to
9 (both inclusive)
Integer
16 A bob of mass m, suspended by a string of length l1, is given a minimum velocity reqired tocomplete a full circle in the vertical plane At the highest point, it collides elastically with anotherbob of mass m suspended by a string of length l2, which is initially at rest Both the strings are
mass-less and inextensible If the second bob, after collision acquires the minimum speed
required to complete a full circle in the vertical plane, the ratio 1
2
l
lis
Ans 5
Sol: The speed of the mass attached to string of length l1 at the highest point will be 1gl which
should be equal to the critical speed of mass attached to the string of length l 2at the bottom-mostpoint ie
1 25gl
gl
or 1
2
5l
l
17 A particle of mass 02 kg is moving in one dimension under a force that delivers a constant power
05 W to the particle If the initial speed (in ms1
) of the particle is zero, the speed (in ms1
) after 5s is
Ans 5
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Sol: w k
21pt mv2
1v 5ms
18 The work functions of Silver and Sodium are 46 and 23 eV, respectively The ratio of the slope of
the stopping potential versus frequency plot for Silver to that of Sodium is
Ans 1
Sol: maxkE hf W and max skE ev
slope of either graph ish
ewhich is constant
ratio of slopes = 1
sv
f
19 A freshly prepared sample of a radioisotope of half-life 1386 s has activity 103disintegrations
per second Given that ln 2 = 0693, the fraction of the initial number of nuclei (expressed in
nearest integer percentage) that will decay in the first 80 s after preparation of the sample is
Ans 4
Sol: Required fraction =
t
o
o
N 1 e
1 e 0.04N
% fraction 4%
20 A uniform circular disc of mass 50 kg and radius 04 m is rotating with an angular velocity of 10
rad s1
about its own axis, which is vertical Two uniform circular rings, each of mass 625 kg and
radius 02 m, are gently placed symmetrically on the disc in such a manner that they are touching
each other along the axis of the disc and are horizontal Assume that the friction is large enoughsuch that the rings are at rest relative to the disc and the system rotates about the original axis
The new angular velocity (in rad s1
) of the system is
Ans 8
Sol: From conservation of angular momentum
2 22 21 1 R R 50 R 10 50 R 2 6.25 6.25
2 2 2 2
8rad/s
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PART II : CHEMISTRY
SECTION1 (Only One option correct Type)
This section contains 10 multiple choice questions Each question has four choices (A) (B), (C) and (D)
out of the which ONLY ONEis correct
21 Consider the following complex ions, P, Q and R
P = [FeF6]3-
, Q = [V(H2O)6]2+
and R = [Fe(H2O)6]2+
The correct order of the complex ions, according to their spinonly magnetic moment values (in
BM) is(A) R < Q < P (B) Q < R < P (C) R < P < Q (D) Q < P < R
Ans (B)
[FeF6]
3
, Fe
3+
has 5 unpaired electron= 35 BM
[V(H2O)6]2+
, V2+
has 3 unpaired electrons
= 15 BM
[Fe(H2O)6]2+
, Fe2+
has 4 unpaired electrons
= 24 BM
P > R > Q
22 The arrangement of X ions around A+ion in solid AX is given in the figure (not drawn to scale)
If the radius of X is 250 pm, the radius of A+is
(A) 104 pm (B) 125 pm (C) 183 pm (D) 57 pmAns (A)
cation in octahedral voids
r0.414
r (without distortion)
r0.414
250
r+= 0414
250 = 1035 104 (approx)
23 Sulfide ores are common for the metals
(A) Ag, Cu and Pb (B) Ag, Cu and Sn (C) Ag, Mg and Pb (D) Al, Cu and Pb
Ans (A)Common sulphide ore are given by
Ag Ag2S : Silver glance
Cu CuS : Copper glance
Pb PbS : Galena
24 The standard enthalpies of formation of CO2(g), H2O(l) and glucose (s) at 250C are -400 kJ/mol,
-300 kJ/mol and -1300 kJ/mol, respectively The standard enthalpy of combustion per gram of
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glucoses at 250C is
(A) +2900 kJ (B) -2900 kJ (C) -1611 kJ (D) +1611 kJAns (C)
I C + O2 CO2 H = 400 kJ/mol
II H2 + 21
O2
H2O H = 300 kJ/mol
III 6C + 6H2+ 3O2 C6H12O6 H = 1300 kJ/mol
C6H12O6 + 9O2 6CO2 + 6H2O
H = (400 6) + (300 6)(1300)
H =2900 kJ / mole
H of per gram of Glucose = 2900
16.11kJ / gram180
25 Upon treatment with ammoniacal H2S the metal ion that precipitates as a sulfide is
(A) Fe(III) (B) Al (III) (C) Mg(II) (D) Zn (II)
Ans (D)On reaction with H2S in presence of NH4Cl/NH4OH only Zn reacts and gives ZnS (white)
Zn2+
+ H2S ZnS (white ppt)
(IV group)
26 Methylene blue, from its aqueous solution, is adsorbed on activated charcoal at 250C For this
process, the correct statement is(A) The adsorption requires activation at 25
0C
(B) The adsorption is accompanied by a decreases in enthalpy
(C) The adsorption increases with increases of temperate
(D) The adsorption is irreversibleAns (B)
Adsorption of phenolphthalein is exothermic at 250C
27 KI in acetone, undergoes SN2 reaction with each of P, Q, R and S The rate of the reaction vary
as
H3C - Cl ClCl
Cl
O
P Q RS
(A) P > Q > R > S (B) S > P > R > Q (C) P > R > Q > S (D) R > P > S > Q
Ans (B)
S > P > R > Q
SN2
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Note :Methyl halide has lower rate of SN2 then allyl halide according to the data given in Jerry
March (4th
Edition, Page No 339, Book NameAdvance Organic Chemistry)
Average relative SN2 rates for some alkyl substratesReactant Relative rate
Methyl 30
Isopropyl 0025
Allyl 40
But considering the different options given in question the most appropriate order
S > P > R > Q
SN2
28 In the reaction
P + Q R + S
the time taken for 75% reaction of P is twice the time taken for 50% reaction of P theconcentration of Q varies with reaction time as shown in the figure The overall order of the
reaction is
timetime
[Q]
[Q]0
(A) 2 (B) 3 (C) 0 (D) 1
Ans (D)
In the reaction
P + Q R + Stime taken for 75% of composition is two times of the decomposition of 50% of P then the order
of reaction with respect to P is 1
for
P C0 C0/2 C0/4
t t
according to graph concentration of Q completely decomposed in given time then order of
reaction with respect to Q is 0
overall order of reaction = 1 + 0 = 1
29 Concentrated nitric acid, upon long standing, turns yellow-brown due to the formation of
(A) NO (B) NO2 (C) N2O (D) N2O4Ans (B)
On long standing concentrated HNO3 gives brown colored gas
HNO3(con) NO2(g) + H2O + O2(brown coloured)
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30 The compound that does NOT liberate CO2, on treatment with aqueous sodium bicarbonatesolution is
(A) Benzoic acid (B) Benzenesulphonic acid(C) Salicylic acid (D) Carbolic acid (Phenol)
Ans (D)
Phenol does not reacts with NaHCO3Because phenol is less acidic then HCO3 Thus according
to acid-base concept PhO can not abstract the hydrogen from HCO3 and do not give CO2gas
COOH SO3H COOH
OH
OH
SECTION2 : (One or more options correct Type)
This section contains 5 multiple choice questions Each question has four choices (A), (B), (C) and
(D) out of which ONE or MORE are correct
31 The initial rate of hydrolysis of methyl acetate (1 M) by a weak acid (HA ,1M) is 1/100th
of that
of a strong acid (HX, 1 M), at 25oC The aK of HA is
(A) 1 10-4
(B) 1 10-5
(C) 1 10-6
(D) 1 10-3
Ans (A)
1 1r H
2 2r H
1 1
22
Hr
r H
1
a2
H 1100
H
K C
Ka C = 10-4
Ka= 1 10-4
32 The hyperconjugative stabilities of tert-butyl cation and 2-butene, respectively, are due to
(A) p (empty) and * electron delocalisations
(B) * and electron delocalisations
(C) p (filled) and electron delocalisations
(D) p (filled) * and * electron delocalisations
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Ans (A)
t-butyl carbocation is stabilized by p overlapping
2-butene is stabilized by * overlapping
33 The pair(s) of coordination complexes/ions exhibiting the same kind of isomerism is(are)
(A) 3 25Cr NH Cl Cl and 3 24Cr NH Cl Cl
(B)3 24
Co NH Cl and 3 22Pt NH H O Cl
(C)2
2 2CoBr Cl and2
2 2PtBr Cl
(D) 3 33Pt NH NO Cl and 3 3Pt NH Cl Br
Ans (B), (D)
(A) 3 25Cr NH Cl Cl does not exhibit isomerism, 3 24Cr NH Cl Cl exhibits geometrical
isomerism
(B)3 24
Co NH Cl and 3 22Pt NH H O Cl
Both can show geometrical isomerism
(C)2
2 2CoBr Cl - tetrahedral so it does not exhibit isomerism
2
2 2PtBr Cl - square planar so it exhibits geometrical isomerism
(D) 3 33Pt NH NO Cl and 3 3Pt NH Cl Br
Both can show isonisation isomerism
34 Among P, Q, R and S, the aromatic compound(s) is/are(A) P (B) Q (C) R (D) SCl
3AlCl
P
NaH Q
O O
4 32NH CO
o100 115 C
R
OHCl
S
Ans (A), (B), (C), (D)
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Cl
3AlCl4
AlCl
2 electronaromatic
H
NaH
Na
2H
6 electronaromatic
O
O
OH
O
OO
OH
O
H
enolizationO
H
H
baseO
H
6 electronaromatic
O
H
OH
6 electronaromatic
35 Benzene and naphthalene form an ideal solution at room temperature For this process, the true
statement(s) is(are)
(A) G is positive (B) systemS is positive
(C) surroundingsS 0 (D) H 0
Ans (B), (C), (D)
system system systemG H T S
system surroundingH T S So as solution is ideal
systemH 0
Sosurrounding
S 0
As mixing is spontaneous so systemG 0
system systemH T S 0
So assystemH = 0 , systemS must be greater than zero
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SECTION3 : (Integer value correct Type)
This section contains 5 questions The answer to each question is a single digit integer, ranging from 0 to9 (both inclusive)
36 The atomic masses of He and Ne are 4 and 20 amu, respectively The value of the de Broglie
wavelength of He gas at 73oC is M times that of the de Broglie wavelength of Ne at 727
oC M
is
Ans 5He : m1= 4, T1= 200 K
Ne : m2= 20, T2= 1000 K
He
Ne Ne
h
m
T
v
M
So HeHe
He He
mh
m T
NeNe
Ne Ne
mh
m T
He Ne Ne
Ne He He
T m
1000 2025 5
T m 4 200
374
EDTA is ethylenediaminetetraacetate ion The total number of N Co O bond angles in1
Co EDTA complex ion is
Ans 8
38 The total number of carboxylic acid groups in the product P is
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O
O
O O
O
31. H O ,
2. O3
3. H O2 2
P
Ans 2
O
O
O O
O
O
O
3H O
O
O O
O
OH
OH
keto acid
3 2 2O / H O
O
OO
O
OH
OH
39 A tetrapeptide has
COOH group on alanine This produces glycine (Gly), valine (Val), phenyl
alanine (Phe) and alanine (Ala), on complete hydrolysis For this tetrapeptide, the number of
possible sequences (primary structures) with NH2group attached to a chiral center isAns 4
3 2CH CH CH CO H
CH3
NH2
Valine (V)
3 2CH CH CO H
NH2
Alanine (A)
2 2H C CH CO H
NH2
Phenylalanine (P)
2 2 2H N CH CO H Glycine (G)
As (A) is at the end so options possible with NH2at beginning are(i) VPGA, (ii) VGPA, (iii) PVGA and (iv) PGVA
40 The total number of lone-pairs of electrons in melamine isAns 6
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N
N
N
NH2NH2
NH2 Melamine
Total number of lone pairs = 6
PART III : MATHEMATICS
SECTION1 (Only One option correct Type)
This section contains 10 multiple choice questions Each question has four choices (A) (B), (C) and (D)
out of the which ONLY ONEis correct
41 For 0,a b c the distance between 1,1 and the point of intersection of the lines
0ax by c and 0bx ay c
is less than 2 2. then
(A) 0a b c (B) 0a b c (C) 0a b c (D) 0a b c 41 (A)
Point of intersection ,c c
a b a b
Distance of (1, 1) from point of
intersection
2 2
1 1c c
a b a b
2 a b c
a b
Acc to question
2 2 2a b c
a b
0a b c
42 The area enclosed by the curves sin cosy x x and cos siny x x over the interval 0,2
is
(A) 4 2 1 (B) 2 2 2 1 (C) 2 2 1 (D) 2 2 2 1
42 (B)
Required Area/4 /2
0 /4
2sin 2cosx dx x dx
/4 /2
0 /42 cos 2 sinx x
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1 12 1 2 1
2 2
4 2 2 2 2 2 1
43 The number of points in , for which2 sin cos 0,x x x x is
(A) 6 (B) 4 (C) 2 (D) 0
43 (C)
Let2 sin cos
' 2 sin cos sin
' 2 cos
f x x x x x
f x x x x x x
f x x x
When 0 ' 0x f x f x is sI function
When 0 ' 0x f x f x is sd function
At x = 0 ' 0f x
0 1f
When x f x
f x will be zero at exactly two points
44 The value of cot23
1
1 1
cot cot 1 2n
n k
k is
(A)23
25 (B)
25
23 (C)
23
24 (D)
24
23
44 (B)23
1
1 1
cot cot 1 2n
n k
k
1 1
2. 12 2 1
2
n n
k k
n nk k n n
231 2
1
cot 1n
n n
1 2cot 1rT r r
1
2
1tan
1 r r
1 1
tan1 1
r r
r r
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1 1tan 1 tanrT r r
231
1
tan 24
4
r
n
T
1
11 1
2524cot tan 2414 23
124
45 A curve passes through the point 1, .6
Let the slope of the curve at each point ,x y be
sec , 0.y y
xx x
Then the equation of the curve is
(A)
1
sin log 2
y
xx (B) cos log 2
y
ec xx
(C)2
sec log 2y
xx
(D)2 1
cos log2
yx
x
45 (A)
Given
secdy y y
dx x x (i)
Let y vx
dy dvv x
dx dx
(ii)
Substituting (ii) in (i)
secdv
v x v vdx
cos
dxv dx
x
At 1,6 6
x y v at 1x
/6 1
cos
v xdx
v dvx
/6 1sin logv xv x
sin sin log6
v x
1sin log
2
yx
x
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46 Let1
: ,12
f (the set of all real numbers) be a positive, non-constant and differentiable
function such that ' 2f x f x and1
1.2f Then the value of
1
1/2
f x dx lies in the
interval
(A) 2 1, 2e e (B) 1, 2 1e e (C)1
, 12
ee (D)
10,
2
e
46 (D)
' 2f x
f x
2 2' 2 0
x xe f x e f x
Or 2 0xd
e f x
dx
2xe f x is sd nf in1
,12
2 1xe f x e
2 10 xf x e
11 2 1
1/2 1/2
02
xef x dx
1
1/2
10
2
ef x dx
47 Let 3 2PR i j kand 3 4SQ i j k determine diagonals of a parallelogram PQRS and
2 3PT i j kbe another vector Then the volume of the parallelepiped determined by the
vector ,PT PQ and PS is
(A) 5 (B) 20 (C) 10 (D) 30
47 (C)S R
P Q
a
b
3 2a b i j k 2 3c i j k
3 4a b i j k
10 10 10a b a b i j k
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2 10 10 10a b i j k
5a b i j k
Now volume of parallelepiped
. 5 1 2 3c a b c a b
10
48 Perpendiculars are drawn from points on the line2 1
2 1 3
x y zto the plane 3.x y z The
feet of perpendiculars lie on the line
(A)1 2
5 8 13
x y z (B)
1 2
2 3 5
x y z
(C)1 2
4 3 7
x y z (D)
1 2
2 7 5
x y z
48 (D)
The given line is2 1
2 1 3
x y zr
General point on the line is
2 2, 1, 3P
r r r
Now the direction ratio of PQ
are 2 2, 1, 3r r r
which are proportional to (1, 1, 1)
2 2 1 31 1 1
r r r k
2 2
1
3
k r
k r
k r
Point , , lies on pane 3x y z so
6 42 2 1 3 3
3
rk r k r k r k
Now6 4 3
2 2
3 2
rr r
6 4 3 31
3 7
rr r and
6 4r 3 63r r
3 5
3 3 3 3 6 1 2
2 7 5 2 7 5
x y z
P
A
(-2,-1, 0)
Q
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49 Four person independently solve a certain problem correctly with probabilities1 3 1 1
, , , .2 4 4 8
Then
the probability that the problem is solved correctly by at least one of them is
(A) 235
256 (B) 21
256 (C) 3
256 (D) 253
256
49 (A)
Prob that the problem is solved correctly = 1prob that problem is not solved correctly bynone of them
1 1 3 71 . . .
2 4 4 8
211
256
235
256
50 Let complex numbers
and1
lie on circles2 2 2
0 0x x y y r and
2 2 2
0 0 4 ,x x y y r respectively If 0 0 0z x iy satisfies the equation2 2
02 2,z r then
(A)1
2 (B)
1
2 (C)
1
7 (D)
1
3
50 (c)
0z r (i)
0
12z r
01 2z r (ii)
(ii)/(i) gives
0
0
12
z
z
2
0 0 0 01 1 4z z z z
Or2 2 2 2 2
0 0 0 0 0 01 4 .z z z z z z (A)
Or2 2 4 2 2 2
0 0 0 01 4 4 1 4z z z z (B)
From (i)2 2 2
0 0 0z z z r
22 2
0 0 12
rz z r
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22
12
r (C)
From (B) & (C)2 2
2 4 21 1 4 4 1
2 2
r r
22 2
1 1 42
r
1
7
SECTION2 : (One or more options correct Type)
This section contains 5 multiple choice questionsEach question has four choices (A), (B), (C) and (D)
out of which ONE or MOREare correct
51 A line lpassing through the origin is perpendicular to the lines
1 : 3 t i 1 2t j 4 2t k, t
2 : 3 2s i 3 2s j 2 s k, s
Then, the coordinate(s) of the point(s) on2
at a distance of
17 from the point of intersection
of and 1 is (are)
(A)7 7 5
, ,3 3 3
(B) 1, 1,0 (C) 1,1,1 (D)7 7 8
, ,9 9 9
Key (B, D)
Sol Given equation of line 1 and 2 are
x 3 y 1 z 4
1 2 2 (1)
andx 3 y 3 z 2
2 2 1 (2)
dr of line are 2,3, 2
Equation of line
is
x y z
2 3 2 (3)
Any point on (3) is 2k,3k, 2k
To find point of intersection of (1) and (3)
We have
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3k 1 2k 42k 3
2 2
4k 6 3k 1 2k 4
7k 7 k 1
Point of intersection of (1) and (3) is P 2, 3,2
Any point on line 2 is Q 3 2S,3 2S,2 S
PQ
17 2 2 21 2S 6 2S S 17
29S 28S 20 029S 10S 18S 20 0
9S 10 S 2 0
10S ,S 2
9
Q point is7 7 8
, ,9 9 9
or 1, 1,0
52 Let f x x sin x, x 0 Then for all natural numbers n,f ' x vanishes at
(A) a unique point in the interval1
n, n2
(B) a unique point in the interval1
n , n 1
2
(C) a unique point in the interval n, n 1
(D) two points in the interval n, n 1
Key (B, C)
Sol
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y x
y
x
1x 2
3
x 2
5x
2
2,01, 00,0
f ' x x cos x. sin x , x > 0
f ' x x cos sin x
Now f ' x 0
xcos x sin x 0 x tan x
Now taking
y tan x, and y x
From graph,
A unique point in the interval 1n , n 12
and a unique in the internal n, n 1
n N
53 Let
k k 14n 2
2
n
k 1
S 1 k Then nS can take value(s)
(A) 1056 (B) 1088 (C) 1120 (D) 1332
Key (A, D)
Sol
k k 14n 2
2
n
k 1
S 1 k
2 2 2 2 2 2
nS 1 2 3 4 5 6 .....4n terms
2 2 2 2
nS 1 5 9 ..... 4n 3 22 2 22 6 10 ...... 4n 2
22 2 23 7 11 ..... 4n 1 22 2 24 8 12 ...... 4n
n n n n n n2 2
n
r 1 r 1 r 1 r 1 r 1 r 1
S 16 r 24 r 9 1 16 r 16 r 4 1
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n n n2
r 1 r 1 r 1
16 r 8 r 1 216 r
n n
r 1 r 1
32 r 12 1
n n 132 12n
2= 4n 4n 1
Therefore the required solutions are 1056, 1332
54 For 3 3 matrices M and N, which of the following statement(s) is (are) NOT correct?
(A)T
N M N is symmetric or skew symmetric, according as M is symmetric or skew symmetric
(B) MN NM is skew symmetric for all symmetric matrices M and N(C) MN is symmetric for all symmetric matrices M and N
(D) adjM adjN adj MN for all invertible matrices M and N
Sol (C, D)(A)
T T TTT T TN MN N MN MN N
T TN M N TN MN (If M is symmetric TM M
)TN MN (If M is skew-symmetric TM M
)
So option A is correct
(B)T T T
MN NM MN NM
T
T T T T
T
N NN M M N
M M
NM MN
MN NM
So option B is correct
(C)T T TMN N M NM (M and N are symmetric )
So option (C) is wrong
(D) adjM adjM adj MN Given
So option D is wrong55 A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is
converted into an open rectangular box by folding after removing squares of equal area from all
four corners If the total area of removed squares is 100, the resulting box has maximum volumeThen the lengths of the sides of the rectangular sheet are
(A) 24 (B) 32 (C) 45 (D) 60
Key (A, C)
Sol
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a a
aa
8x 2a 8x 2a
15x 2a Volume of the box V 8x 2a 15x 2a .a
3 2 2V 4a 46a x 120x .a For max volume
dV0
daand
2
2
d V0
da
2dV
12a 92ax 120ax 0da 2
2
d V24a 92x 0
daat
5xa
3
Now given that sum of Area 24a 100 a 5
So from5x
a3
x 3
SECTION
3 (Integer Value correct Type)
This section contains 5 questions The answer to each question is a single digit integer, ranging from 0
to 9 (both inclusive)
56 Consider the set of eight vectors V ai bj ck : a, b,c 1,1 There non-coplanar vectors
can be chosen from V in 2Pways Then p is
Ans 5Sol Consider a cube of side length 2 unit, whose centre is at origin (assumed sides parallel to
coordinate axis)
Required no of Triplets of non-coplanar vectors 4 53C .2.2.2 32 2
P 5 57 Of the three independent events E1, E2and E3, the probability that only E1occurs is , only E2
occurs is and only E3occurs is Let the probability p that none of events E1, E2or E3occurs
satisfy the equations 2 p and 3 p 2 All the given probabilities are
assumed to lie in the interval (0, 1)
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Then 1
3
Probability of occurrence of E
Probability of occurrence of E
Ans 6Sol Let 1 2 3P E x,P E y,P E z
x 1 y 1 z
y 1 x 1 z
z 1 x 1 y and p 1 x 1 y 1 z
Given equations are
2 p and 3 p 2
1 2 1
p (1)
1 3 2
p (2)
(1) x 2y and (2) y 3z x 6z
x6
z
1
3
P E6
P E
58 The coefficients of three consecutive terms ofn 5
1 x are in the ratio 5 : 10 : 14 Then n =
Ans 6
Sol n 5 n 5 n 5r 1 r r 1
C C C 5:10:14
n 5 ! n 5 ! n 5 !: : 5 :10 :14
r 1 ! n r 6 ! r! n r 5 ! r 1 : n r 4 !
n n r 5:1: 5 :10 :14
n r 6 r 1 r 1
2r n r 6n r 6 2
n 3r 6 0 (I)
n r 5 7
r 1 5 5n 5r 25 7r 7
5n 12r 18 0 (II)4n 12r 24 0 (III)____________
n = 659 A pack contains n cards numbered from 1 to n Two consecutive numbered cards are removed
from the pack and the sum of the numbers on the remaining cards is 1224 If the smaller of the
numbers on the removed cards is k, then k 20 Ans 5
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Sol
2n n2k 1 1224
2
2n n 4k 2450 02n n 2450
k4
But 1 k n 124 n n 2450 4n 4
1
9785n
2
n 49, n N
But for n 49,51, k NSo n = 50,
2n 3n 2459 0
3
9817n
2
n 51
K = 25
k 20 5
60 A vertical line passing through the point h,0 intersects the ellipse2 2x y
14 3
at the points P
and Q Let the tangents to the ellipse at P and Q meet at the point R If h area of the triangle
PQR, max min1 1/2 h 1 2 1/2 h 1h and h , then 1 28
85
Ans 9
Sol
p 2 cos , 3 sin
R 2 sec , 0
Q 2 cos , 3 sin
(h, 0)
Area of 1PQR 2 3 sin 2sec 2cos2
32 3sin
cos
Which is decreasing function of cos
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1 1h 1 2cos
2 2
1 1cos
4 2
3
1
152 3 4
4
2 3 15.
15
16
45 5
8
1
845
5
3
2
32 3 2
2
2
3 3 3 9
4 2
28 36
So
1 2
88 9
5
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ANSWER KEY
PHYSICS CHEMISTRY MATHEMATICS
Q.No. Answer Q.No. Answer Q.No. Answer
1 D 21 B 41 A
2 A 22 A 42 B
3 D 23 A 43 C
4 A 24 C 44 B
5 B 25 D 45 A
6 B 26 B 46 D
7 C 27 B 47 C
8 C 28 D 48 D
9 A 29 B 49 A
10 B 30 D 50 C
11 B,D 31 A 51 B, D
12 A,C 32 A 52 B, C
13 B,C 33 B,D 53 A, D
14 A,D 34 A,B,C,D 54 C, D
15 B,D 35 B,C,D 55 A, C
16 5 36 5 56 5
17 5 378
57 6
18 1 38 2 58 6
19 4 39 4 59 5
20 8 40 6 60 9