IITJEE 2011 paper1 solution

Sri Chaitanya IIT Academy.A.P. 1 :: IIT JEE 2011 Paper - I ::
Sri Chaitanya “IIT” Academy, Plot No 512, Ayyappa Society, Madhapur, Hyderabad - 500081
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Sri Chaitanya IIT Academy
SriChaitanya’s Solutions to
IIT –JEE - 2011
(PAPER – 1)
Time: 3 Hours Maximum Marks: 240
Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose
INSTRUCTIONS
A. General :
1. The question paper CODE is printed on the right hand top corner of this sheet and also on the back
page (page no 36 of this booklet)
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7. Do not break the seals of the question paper booklet before instrucuted to do so by the invigilators.
8. This question paper contains 36 pages having 69 questions.
9. On breaking the seals, please check that all the questions are legible.
B. Filling the bootom half of the ORS :
10. The ORS has CODE printed on its lower and upper Parts.
11. Make sure the CODE on the ORS is the same as that on this booklet. If the Codes do not match, ask for
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12. Write your Registration No., Name and Name of centre and sign with pen in appropriate boxes. Do not
write these anywhere else. Darken the appropriate bubbles below your registration number with HB
pencil.
C. Question paper format and marking scheme :
13. The question paper consists of 3 Parts (Chemistry, Physics and Mathematics). Each part consists of four
sections
14. In Section I (Total Marks : 21), for each question you will be awarded 3 marks if you darken ONLY the
bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other
cases, minus one (–1) mark will be awarded.
15. In Section II (Total Marks : 16), for each question you will be awarded 4 marks if you darken ALL the
bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise. There are no
negative marks in this section.
16. In Section III (Total marks : 15), for each question you will be awarded 3 marks if you darken ONLY the
bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other
cases, minus one (–1) mark will be awarded.
17. In Section IV(Total Marks : 28), for each question you will be awarded 4 marks if you darken ONLY the
bubble corresponding to the correct answer and zero marks otherwise. There are no negative marks
in this section

IIT JEE 2011 (PAPER - I)
PART I : CHEMISTRY
SECTION – I ( Total Marks : 21)
( SINGLE CORRECT CHOICE TYPE )
This section contains 7 multiple choice questions. Each question has four choices A,B,C and D out of
which ONLY ONE is correct.
1. Dissolving 120 g of urea (mol. wt. 60) in 1000g of water gave a solution of density 1.15
g/mL. The molarity of the solution is
a) 1.78 M b) 2.00M c) 2.05 M d) 2.22 M
Ans : C
Sol : Molarity = 2m
 /
1000
1000 . soluteg cc
M
m
d M MW


2.05M M 
2. AgNO3
(aq). was added to an aqueous KCl solution gradually and the conductivity of the
solution was measured. The plot of conductance   versus the volume ofAgNO3
is
a) (P) b) (Q) c) (R) d) (S
Ans : D
Sol : Equation : 3 3K Cl Ag NO K NO AgCl     
    
In the early stages of the titration, addition of silver nitrate, the conductance does not
change very much because the Cl–
ions are replaced by 3NO
ions; both has almost same
ionic conductance.After the end point is passed, the excess of the added salt causes a sharp
increase in conductance.
2 1
73.52K
S cm mole

  76.34Cl  2
S cm mole
61.92Ag  2
S cm mole
3
71.44NO  2
S cm mole

3. Among the following compounds, the most acidic is
a) p-nitrophenol b) p-hydroxybenzoic acid
c) o-hydroxybenzoic acid d) p-toulic acid
Ans : C
Sol : Ortho effect present in ortho-hydroxy benzoic acid
4. The major product of the following reaction is
a) b) c) d)
Ans :A
Sol :
N-H
O
O
N K+
O
O
-
Br- CH2Cl
N
O
O
KOH

CH2 Br
5. Extra pure N2
can be obtained by heating
a) NH3
with CuO b) NH4
NO3
c)  4 2 72
NH Cr O d)  3 2
Ba N
Ans : D
Sol : Pure N2
is prepared by heating Ba(N3
)2
 3 22
3Ba N N Ba  

6. Geometrical shapes of the complexes formed by the reaction of Ni2+
with ,Cl CN 
and
H2
O, respectively, are
a) octahedral, tetrahedral and square planar
b) tetrahedral, square planar and octahedral
c) square planar, tetrahedral and octahedral
d) octahedral, square planr and octahedral
Ans : B
Sol : 2 8 3
4[ ]NiCl d Sp Tetrahedral
 
2 8 2
4[ ]NiCN d dsp Square planar
 
  2 8 3 2
2 6
[ ]Ni H O d sp d Octahedral Shape
 
7. Bombardement of aluminumby particle  leads toits artifical disintegration intwo ways,
(i) and (II) as shown. Products X,Y and Z respectively are,
30
14 Si Z
30
15 P Y27
13 Al
(ii)
30
14 Si X
(i)
a) proton, neutron, positron
b) neutron, positron, proton
c) proton, positron, neutron
d) positron, proton, neutro
Ans : A
Sol : 1
1X= H , 1
0Y= n , 0
+1Z= e

SECTION – II ( Total Marks : 16)
( MULTIPLE CORRECT ANSWERS TYPE)
This section contains 4 multiple choice questions. Each question has four choices A,B,C and D out of
which ONLY or More may be correct.
8. Amongst the given options, the compound(s) in which all the atoms are in one plane in all
the possible conformations (if any), is (are)
a)
C - C
H2C CH2
HH
b) H2C = C = O c) H - C C - C
CH2
H
d) H2C = C = CH2
Ans : BC
Sol : CH2 = C = CH2 allene system the compound is non planar
9. According to kinetic theory of gases
a) collisions are always elastic
b) heavier molecules transfer more momentum to the wall of the container.
c) only a small number of molecules have very high velocity
d) between collisions, the molecules move in straight line with constant velocities.
Ans :A,B,C,D
Sol :Conceptual
10. The correct statement(s) pertaining to the adsorption of a gas on a solid surface is (are)
a)Adsorption is always exothermic
b) Physisorption may transform into chemisorption at high temperature
c) Physisorption increases with increasing temperature but chemisorption decreases with
increasing temperature
d) Chemisorption is more exothermic than physisorption, however it is very slow due to
higher energy of activation.
Ans :ABD
Sol: Physicaladsorptiondecreaseswithincreasingof temperatureaccordingLeachtlier‘s Principle

11. Extraction of metal from the ore cassiterite involves
a) carbon reduction of an oxide re b) self-reduction of a sulphide ore
c) removal of copper impurity d) removal iron impurity
Ans :ACD
Sol : Sn is extracted by the reduction of SnO2
with carbon
2 2SnO C Sn CO  
Sn mainly contains Iron impurities which are removed by blowing air to convert Fe into
FeO. Sn also contains tracer of Cu
SECTION – III ( Total Marks : 15)
( PARAGRAPH TYPE)
This section contains 2 paragraphs. Based upon one of the paragraphs 2 multiple choice questions
and based on the other paragraph 3 multiple choice questions have to be answered. Each of these
questions has four choices A,B,C and D out of which ONLY ONE is correct.
Passage-I :
An acylic hydrocarbon P, having molecular formula 6 10C H , gave acetone as the only or--
ganic product through the following sequence of reactions, in which Q is an intermediate
organic compound.
12. The structure of compound P is
a) CH3CH2CH2CH2 - C C - H b) H3CH2C - C C - CH2CH3
c) d)
Ans : D

13. The structure of the compound Q is
a) b)
c) d)
Ans : B
Sol for Q no 12 to 13 :
CH3 - C - C C - H
CH3
CH3
(P)
CH3 - C
CH3
CH3
C - CH3
O
2 4 4. /dil H SO HgSO
 CH3 - C
CH3
CH3
CH - CH3
OH
(Q)
CH3 - C
CH3
CH3
CH - CH3
 
 
41 /
2 .
NaBH ethanol
dil acid

2 4.con H SO
(catalytic amount)
CH3 - C
CH3
CH - CH3
CH3
CH3 - C
CH3
C - CH3
CH3  31 O
  22 /Zn H O
CH3 - C
CH3
O2

Passage- II :
When a metal rod M is dipped into an aqueous colourless concentrated solution of
compound N, the solution turns light blue. Addition of aqueous NaCl to the blue solution
gives a white precipitate O.Addition of aqueous NH3
dissolves O and given an intense blue
solution
14. The metal rod M is
a) Fe b) Cu c) Ni d) Co
Ans : B
15. The compound N is
a)AgNO3 b) Zn(NO3)2 c)Al(NO3)3 d) Pb(NO3)2
Ans :A
16. The final solution contains
a) [Pb(NH3)4]2+ and [CoCl3)4]2- b) [Al(NH3)4]3+ and [Cu(NH3)4]2+
c) [Ag(NH3)2]+ and [Cu(NH3)4]2+ d) [Ag(NH3)2]+ and [Ni(NH3)6]2+
Ans : C
Sol : for Q. no 14 to 16 :
 
 
 
3 3 2M
N
Cu +2Ag NO Cu NO 2Ag
 
   NaCl Na aq Cl aq 
 
 White O
Ag Cl AgCl 
  
   
22
3 3 4
Deep blue colour
4Cu NH Cu NH

    

SECTION – IV ( Total Marks : 28)
( INTEGER ANSWER TYPE)
This section contains 7 questions. The answer to each of the questions is a single-digit integer, ranging
from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.
17. The work function   of some metals is listed below. The number of metals which will
show photoelectric effect when light of 300 nm wavelength falls on the metal is
Li KNaMetal Mg Cu Ag Fe Pt
 eV 2.3
W
2.4 2.2 3.7 4.8 6.34.3 4.7 4.75
Ans : 4
Sol : E= +KE
For photo electric effect, condition is E > 
Metals exhibits : Li, Na, K, Mg
18. To an evacuated vessel with movable piston under external pressure of 1 atm., 0.1 mol of He
and 1.0 mol of an unknown compound (vapour pressure 0.68 atm. at 00C) are introduced.
Considering the ideal gas behaviour, the total volume (inlitre) of the gases at 0 0C is close to
Ans : 7
Sol : Unknon Compound may be Soid (or) liquid
Let given volume of vessel : ‘V’
for unknown compound
PV= nRT
0.68 X V = ng (0.0821 X 273)____ 1
For total gaseous mixture pressure = external. pressure
PV= nRT
1 X V =(ng + nHe)(0.0821 X 273)____ 2
from 1 and 2
ng = 0.2125, ntotal = 0.3125
V = 7 lit

19. Reaction of Br2 with Na2CO3 in aqueoussolution givessodiumbromideand sodiumbromate
with evolution of CO2 gas. The number of sodium bromide molecules involved in the
balanced chemical equation is
Ans : 5
Sol : 2 2 3 3 23 3 5 3Br Na CO NaBr NaBrO CO   
20. The difference in the oxidation numbers of the two types of sulphur atoms in Na2S4O6 is
Ans : 5
Sol :
NaO - S - S - S - S - ONa
O
O
(+5)(+5)
(0) (0)
O
O
The difference between the O.S of two different types of sulphur atoms = +5
21. A decapeptide (Mol. Wt. 796) on complete hydrolysis gives glycine (Mol. Wt. 75), alanine
andphenylalanine. Glycinecontributes 47.0%to the total weightof thehydrolysed products.
The number of glycine units present in the decapeptide is
Ans : 6
Sol : 100 ________ 958
47 ________ ?
47
958 450.12
100
 
The weight of glycine present in 958 decapeptide is 450.12.
the molecular weight of one glycine is 75
no. of glycines =
450.12
6.00 6
75
 
22. The total number of alkenes possible by dehydrobromination of 3-bromo
3-cyclopentylhexane using alcoholic KOH is
Ans : 5

Sol :
C - C - C - C - C - C
Br
C - C - C - C - C - C C - C - C - C - C - C C - C - C - C = C - C
cis + trans
C - C - C = C - C - C
+
+
cis + trans
23. The maximum number of electrons that can have principal quantumnumber, n=3, and spin
quantumnumber, sm =-1/2, is
Ans : 9
Sol : In third orbit 3s, 3p, 3d sub levels are present and a total of nine orbitals are present. Each
orbital has one electron with ms = –1/2

PART II : PHYSICS
==============================================================
SECTION – I (Total Marks : 21)
( SINGLE CORRECT CHOICE TYPE )
This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) for
its answer, out of which ONLY ONE is correct
24. The wavelength of the first spectral line in the Balmer series of hydrogen atom is 6561 0
A ,
The wavelength of the second spectral line in the Balmer series of singly-ionized helium
atom is
a)
0
1215A b)
0
1640A c)
0
2430A d)
0
4687 A
Ans :A
Sol :  
2
2 2
1
1 1 1
1
2 3
 
  
 
 
2
2 2
2
1 1 1
2
2 4
 
  
 
2
1
1 5 4 16 5
4 4 9 12 9 3



   
 
0
2
6561 5
1215
9 3
A

 

25. Aball ofmass (m) 0.5 kgis attachedto the end ofa stringhavinglength (L) 0.5 m. The ball is
rotatedon ahorizontal circular path aboutvertical axis. The maximumtension that the string
can bear is 324 N. The maximumpossible value of angular velocity of ball (in radian/s) is
a) 9 b) 18 c) 27 d) 36
Ans : D
Sol :
21 1
324
2 2
w  2 18w   w= 36

26. A meter bridge isset-up as shown, to determine an unknown resistance ‘X’ usinga standard
10 ohm resistor. The galvanometer shows null point when tapping - key is at 52 cm mark.
The end-corrections are 1 cm and 2cm respectively for the endsAand B. The determined
value of ‘X’ is
a) 10.2 ohm b) 10.6 ohm c) 10.8 ohm d) 11.1 ohm
Ans :B
Sol :
 10
2
2
10
52 1 48 2
Y

 
53
10 x
50
X 
X =10.6
27. A 2 F capacitor is charged as shown in figure. The percentage of its stored energy dissi-
pated after the switch S is turned to position 2 is
a) 0% b) 20% c) 75% d) 80%
Ans : D
Sol : 0 2q V
1 2
2 8
q q


1 24q q
1 2 2q q V 
15 2q V
1
2
5
q V
 
  
 
2
8
5
q V
 
  
 
2 2
2
4 64
1 4 1 125 25
2 2 2 2 2 8
V V
V
 
2 2 24 16
4V = V + V
25 25
2 1 4
4 1
2 25
V   
2 25 1 4
4
25
V
 
 
20
x100 = 80
25
28. A police car with a siren of frequency 8kHz is moving with uniform velocity 36km/hr to-
wards atall buildingwhich reflectsthesound waves. The speedof sound inair is320 m/s.The
frequency of the siren heard by the car driver is
a) 8.50 kHz b) 8.25 kHz c) 7.75 kHz d) 7.50 kHz
Ans:A
Aol : frequency reflucted by wall ‘f’
3 320
' 8 10
320 10
f
 
    
frequency heared by car driver
320 10
" '
320
f f
 
  
 
3 32 33
" 8 10
31 32
f   
3 33
8 10
31
  
8.50kHz

29. 5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial
temperature to be 1T , the work done in the process is
a) 1
9
8
RT b) 1
3
2
RT c) 1
15
8
RT d) 1
9
2
RT
Ans: A
Sol:  1 2
R
W T T
l
 
1 1
l
TV k


1
1
1 1 2
8
l V
T V T



 
  
 
2 14T T
W=
19
8
RT
30. Consider an electric field 0
ˆE E x

, where 0E is a constant. The flux through the shaded
area (as shown in the figure) due to this field is
a) 02E a
b) 02E a
c) 0E a
d)
0
2
E a
Ans : C
Sol :
2 a
z
x
0
45
area 2
2a 2 0
. 2 cos45E A E a
 2
Ea

SECTION – II (Total Marks : 16)
( MULTIPLE CORRECT CHOICE TYPE )
This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D)
for its answer, out of which ONE or MORE may be correct
31. Ametal rod of length ‘L’and mass ‘m’is pivoted at one end. Athin disk of mass ‘M’and
radius ‘R’(<L) is attached at its center to the free end of the rod. Consider two ways the
disc is attached; (caseA) the disc is not free to rotate about its center and (case B) the disc
is free to rotate about its center. The rod - disc system performs SHM in vertical plane
after being released from the same displaced position. Which of the following statement(s)
is (are) true ?
a) Restoring torque in case A= Restoring torque in case B
b) Restoring torque in case A< Restoring torque in case B
c)Angular frequency for caseA>Angular frequency for case B
d)Angular frequency for caseA<Angular frequency for case B
Ans: AD
Sol : Torque about point of suspenstion is same in both case
case A:
2 2
2
2 3
MR ml
ML k  
 
    
 
case B ; In case B ; Disc is not going to get any torque about its centre.
all of it’s point have acceleration that is equal to acceleration of end point of rod.
2
2
3
l
ML m k  
 
   
 
2 2
2
3
k
W
ml
mL
 
 
 
 
1 2 2
2
2 3
k
W
MR ml
ML

 
  
 
2 1W W

32. A composite block is made of slabs A, B, C, D and E of different thermal conductivities
(given in terms of a constant K) and sizes (given in terms of length, L) as shown in the
figure. All slabs are of same width. Heat ‘Q’ flows only from left to right through the
blocks. Then in steady state
a) heat flow throughAand E slabs are same
b) heat flow through slab E is maximum
c) temperature difference across slab E is smallest
d) heat flow through C = heat flow through B + heat flow through D
Ans :ABCD
33. An electron and a proton are moving on straight parallel paths with same velocity. They
enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which
of the following statement(s) is/are true ?
a) They will never come out of the magnetic field region
b) They will come out travelling alongparallel paths
c) They will come out at the same time
d) They will come out a different times
Ans : BD
Sol :
mv
r
qB

They go in different direction but come out along parallel paths.
2 m
T
qB


as ‘m’ is different ‘T’ also different

34. Aspherical metal shell Aof radius AR and a solid metal sphere B of radius  B AR R are
kept far apart and each is given charge ‘+Q’. Now they are connected by a thin metal wire.
Then
a) 0inside
AE  b) A BQ Q c)
A B
B A
R
R


 d) onsurface onsurface
A BE E
Ans :ABCD
Sol : a) EA
= 0
b)
A B
A B
Q Q
R R

A BR R A BQ Q
c) R V 
d)
02
E


 so E 
SECTION – III (Total Marks : 15)
(COMPREHENSION TYPE)
and based on the paragraph 3 multiple choice questions have to be answered. Each of these questions
has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
Paragraph for Question Nos. 35 to 36
A dense collection of equal number of electrons and positive ions is called neutral plasma.
Certain solids containing fixed positive ions surrounded by free electrons can be treated as
neutral plasma. Let ‘N’ be the number density of free electrons, each of mass ‘m’. When
the electrons are subjected to an to electric field, they are displaced relatively away from
the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate
about the positive ions with a natural angular frequency ' 'p which is called the palsma
frequency. To sustain the oscillations, a time varying electric field needs to be applied that
has an angular frequency  , where a part of the energy is absorbed and a part of it is re-
flected. As  approaches p , all the free electrons are set to resonance together and all
the energy is reflected. This is the explanation of high reflectivity metals.

35. Taking the electric charge as ' 'e and the permittivity as 0' ' , use dimesional analysis to
determine the correct expression for p .
a)
0
Ne
m b) 0m
Ne

c)
2
0
Ne
m
d) 0
2
m
Ne

Ans : C
Sol :
1
T
T

 
 
2 2 1 3 2
2 3
1 2 2
0
1
. .
I T M L T
Ne LD f of
m M I T

 


D.f. = 1
T 
36. Estimate the wavelength at which plasma reflection will occur for a metal havingthe den-
sity of electrons 27 3
4 10N m
  . Take 11
0 10 
 and 30
10m 
 , where these quantities are
in proper SI units.
a) 800 nm b) 600nm c) 300nm d) 200nm
Ans : B
Sol : 2
2
n n

 

  
2c c
c n
n

 


   
 
8
2 227 19
30 110
2 3 10 2
4 10 1.6 10
10 10
c
Ne
m
 



 
  
 
  

9
585 10 600nm 
  

Paragraph for Question Nos. 37 to 39
Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They
are especially useful in studying the changes in motion as initial position and momentum
are changed. Here we consider some simple dynamical systems in one-dimension. For
such systems, phase space is a plane in which position is plotted along horizontal axis and
momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in
this plane. The arrow on the curve indicates the time flow. For example, the phase space
diagram for a particle moving with constant velocity is a straight line as shown in the fig-
ure. We use the sign convention in which position or momentum upwards (or to right) is
positive and downwards (or to left) is negative
37. The phase spacemdiagram for a ball thrown vertically up fromground is
a) b)
c) d)
Ans : D
Sol : Vertically thrown particle comes back to original position till the particle reaches the high-
est position of its path, its momentum is +ve. After that, it terns to zero and them negative

38. The phase space diagram fro simple harmonic motion is a circle centrred at the origin. In
the figure, the two circles represent the same oscillator but for different initial conditions,
and E1 and E2 are the total mechanical energies respectively. Then
a) 1 2E = 2 E b) 1 2E = 2 E c) 1 2E = 4 E d) 1 2E =16 E
Ans : C
Sol :
2 21
2
Energy ka a
2
1 1
2
2 2
4
E a
E a
 
1 24E E 
39. Consider the spring-mass system, with the mass submerged in water,
as shown in the figure. The phase space diagram for one cycle of this
system is
a) b)
c) d)
Ans:A
Sol : As the block oscillates, due to viscous effects, its total energy decreases continuously and
so its amplitude decreases continuously. Assuming that the block is initially pulled down
and released, its momentum will increase upwards till it reaches the mean position

SECTION –IV( Total Marks : 28)
( INTEGER ANSWER TYPE)
This section contains 7 questions . The answer to each of the questions is a single -digit integer, ranging
40. Steel wire of length ‘L’at 40°C is suspended from the ceilling and then a mass ‘m’is hung
from its free end. The wire is cooled down from 40°C to 30°C to regain its original length
‘L’. The coefficient of liner thermal expansion of the steel is 10–5/°C, Young’s modulus of
steell is 1011
N/m2
and radius of the wire is 1mm. Assume that L>> diameter of the wire.
Then the value of ‘m’ in kgis nearly
Ans : 3
Sol : Y=
mgL
Ae
, e= L t 
mgL
Y
A L L


11
6 5
10
10
22
1 10 10 10
7
M
 


   
3.14 3m  
41. The activity of a freshly prepared radioactive sample is 1010
disintegrations per second,
whose mean life is 109
s. The mass of an atom of this radioisotope is 10–23
kg. The mass (in
mg) of the radioactive sample is
Ans : 1
Sol :
10
9
1
10
10
N 
19
10N 
 total mass
25 19
10 10 kg
 
6
10 1kg mg
 

42. Ablock moving on an inclined plane making an angle 45° with the horizontal and the coef-
ficient of friction is . The force required to just push it up the inclined plane is 3 times
the force required to just prevent from sliging down. If we define N =10 then N is
Ans : 5
Sol : 3
2 2 2 2
mg mg mg mg

   
     
    
   3 1 1   
3 3 1   
4 2 
1/ 2 
10 5 
43. Aboy is pushing a ring of mass 2 kg and radius 0.5 m with a sticka shown in the figure. The
stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of
0.3m/s2
. The coefficient of friction between the ground and the ring is large enough that
olling always occurs and the coefficient of friction between the stick and the ring is (P/
10). The value of P is
Ans : 4
Sol : 2
F = N + (i) 
  
 2
2 iip
a
NR NR mR
R
 
 
    
 

N
solving (i)and (ii),
and 0.36  
but    is not a possible solution
   
3.6 4p  

44. Four solid spheres each of diameter 5 cm and mass 0.5 kg are placed with their centers at
the corners of a square of side 4cm. The moment of inertia of the system about the diago-
nal of the square is 4 2
N x10 kg m ,-
- then N is
Ans : 9
Sol :
2
22
4 2
5 2
a
MR M I
 
    
 
2 22
22 5 4 10
4 5 10 2 5
5 2 2


   
          
  
11 4
10N 

N = 9
45. A long circular tube of length 10 m and radius 0.3 m carries a current I along its curved
surface as shown.Awire-loop of resistance 0.005 ohmand of radius 0.1 m is placed inside
the tube with its axis coinciding with the axis of the tube. The current varies as
0I = I cos(300 )t where 0I is constant. If the magnetic moment of the loop is
 0 0 sin 300 ,N I t then ‘N’ is
Ans : 6
Sol :
I
ni
l

0 0 cos
300
I
B t
l


 0 0 2300
sin300
Id
V t R
dt l

 
 2
0 0 300
sin300
I R
V t
l
 

V
i
R
 2
m i R

46. Four point chargs, each of +q, are rigidly fixed at the four corners of a square planar soap
film of side ‘’. The surface tension of the soap film is  . The system of charges and
planar film are in equilibrium, the
1/2
q
a = ,
N

 
 
 
where ‘k’ is a constant. Then N is
Ans : 3
Sol :
q q
qq
a
 
2 2
22
2 2 2
.
2
Aq q
y a
a a
 
1
2 3q
a
y
 
   
 
3N 

PART III : MATHEMATICS
SECTION – I (Total Marks : 21)
(Single Correct Answer Type)
This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out
of which ONLY ONE is correct
47. Let  :sin cos 2 cosP       and  :sin cos 2 sinQ       be two sets. Then
a) p Q and Q P   b) Q P
c) P Q d) P = Q
Ans: D
Sol: sin cos 2 cosP     
sin 2 cos cos   
 sin 2 1 cos  
    sin 2 1 2 1 2 1 cos    
2 sin sin cos   
2 sin cos sin   
P Q 
48. Let the straight line x b divide the area enclosed by  2
1 , 0y x y   , and 0x  into two
parts  1 0R x b  and  2 1R b x  such that 1 2
1
4
R R  . Then b equals
a)
3
4
b)
1
2
c)
1
3
d)
1
4
Ans : B
Sol :    
1
2 2
0
1
1 1
4
b
b
x dx x dx    
1
2
b 

49. Let  and  be the roots of 2
6 2 0x x   , with   . If n n
na    for 1n  , then the
value of
10
9
2
2
sa a
a

is
a) 1 b) 2 c) 3 d) 4
Ans : C
Sol :     10 10 9 9 8 8
10a               
10 9 86 2a a a  
50. A straight line L through the point (3,–2) is inclined at an angle 600
to the line 3 1x y  .
also intersects the x–axis, then the equation of L is
a) 3 2 3 3 0y x    b) 3 2 3 3 0y x   
c) 3 3 2 3 0y x    d) 3 3 2 3 0y x   
Ans : B
Sol : Let ‘m’ be the slope of the required line
3
3
1 3
m
m

 

0m  or 3m 
51. Let  0 0,x y be the solution of the following equations
   
ln 2 ln3
2 3x y
ln ln
3 2x y

Then 0x is
a)
1
6
b)
1
3
c)
1
2
d) 6
Ans : C
Sol :    log 2 log2 log log3 log3 logx y    --------1
   log3 log log2 logx y -------------2
From 1 & 2
log log 2x  
1
2
x 

52. The value of
 
ln3 2
2 2
ln 2
sin
sin sin ln 6
x x
dx
x x  is
a)
1 3
ln
4 2
b)
1 3
ln
2 2
c)
3
ln
2
d)
1 3
ln
6 2
Ans : A
Sol : Put 2
x t
 
log3
log2
1 sin
2 sin sin log
t
dt
t b t

 
1 log3 log 2 1 3
log
2 2 4 2
   
    
   
53. Let  a i j k  

 ,  b i j k  

 and ˆˆ ˆc i j k  

be three vectors. Avector v

in the plane of a

and b

, whose projection on c

is
1
3
, is given by
a)  3 3i j k  b)  3 3i j k  c)  3 3i j k  d)  3 3i j k 
Ans : C
Sol : Let v a b 
  
Projection v

on c

 . 1
3
a b c
c


  

2 

SECTION - II (Total Marks : 16)
(Multiple Correct Answer Type)
This Section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out
of which ONE or MORE may be correct.
54. L et f : :f   be a function such that f(x + y) = f(x) + f(y), ,x y  .
If f(x) is differentiable at x = 0, then
a) f(x) is differentiable only in a finite interval containing zero
b) f(x) is continuous x 
c)  'f x is constant x 
d) f(x) is differentiable except at finitely many points
Ans : B,C
Sol : Fromthe given data  f x be the linear function.
55. Let the eccentiricty of the hyperbola
2 2
2 2
1
x y
a b
  be reciprocal to that of the ellipse x2
+
4y2
= 4. If the hyperbola passes thorugh a focus of the ellipse, then
a) the equation of the hyperbola is
2 2
1
3 2
x y
 
b) a focus of the hyperbola is (2, 0)
c) the eccentricity of the hyperbola is
5
3
d) the equation of the hyperbola is x2
– 3y2
= 3
Ans : B,D
Sol : Ecentricity of ellipse
3
2
Ecentricity of Hyper bola =
2
3
Focus of ellipse =    3,0 , 3,0
Focus lies on the hyper bola then 3a  , b = 1

56. Let M and N be two 3 3 non - singular skew - symmetric matrices such that MN = NM.
If PT
denotes the transpose of P, then M2
N2
(MT
N)–1
(MN–1
)T
is equal to
a) M2
b) –N2
c) –M2
d) MN
Ans : wrong
Reason : 3 3 non singular skew -symmetrices does’t exist.
Sol : Among the given options option C may be correct.
57. The vector(s) which is/are coplanar with vectors ˆˆ ˆ 2i j k  and ˆˆ ˆ2i j k  , and perpen-
dicular to the vector ˆˆ î j k  is /are
a) ˆˆj k b) ˆ î j  c) ˆ î j d) ˆˆj k 
Ans :A,D
Sol: Required vector =   a b c  
  
SECTION - III (Total Marks : 15)
(Paragraph Type)
and based on the other paragraph 3 multiple choice questions have to be answered. Each of these
questions has four choices (A), (B), (C) and (D) out of with ONLY ONE is correct.
Paragraph for Question Nos. 58 and 59
Let U1
and U2
be two urns suchthat U1
contains3 whiteand 2 red balls, and U2
contains only
1 white ball.Afair coin is tossed. If head appearsthen 1 ballis drawnat randomfrom U1
and
put into U2
. However, if fail appears then 2 balls are drawn at random from U1
and put into
U2
. Now 1 ball is drawn at random from U2
.
58. The probability of the drawn ball from U2
being white is
a)
13
30
b)
23
30
c)
19
30
d)
11
30
Ans : B
Probability =
2 2 1 1
2 2 2
1 3 2 2 1 1 3 3 2 1 3 .2 2
x x x x x
2 5 2 5 2 2 5 3 5 3 5 3
C C C C
C C C
  
        
23
30


59. Given that the drawn ball from U2
is white, the probability that head appeared on the coin is
a)
17
23
b)
11
23
c)
15
23
d)
12
23
Ans : D
Sol : Required conditional Probability =
1 3 2 2 1
x x
2 5 2 5 2
23
30
 
  
Paragraph for Question Nos. 60 and 62
Let a, b and c be three real numbers satisfying
   
1 9 7
8 2 7 0 0 0
7 3 7
a b c
 
   
  
..................(E)
60. If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the
value of 7a + b + c is
a) 0 b) 12 c) 7 d) 6
Ans : D
61. Let  be a solution of x3
– 1 = 0 with Im( )>0. If a = 2 with b and c satisfying (E), then
the value of
3 1 3
a b c
  
  is equal to
a) –2 b) 2 c) 3 d) –3
Ans :A
62. Let b = 6, with a and c satisfying (E). If  and  are the roots of the quadratic equation
ax2
+ bx + c = 0, then
0
1 1
n
n  


 
 
 

a) 6 b) 7 c)
6
7
d) 
Ans : B

Sol : 60, 61, 62.
8 7 0a b c  
9 2 3 0a b c  
0a b c  
, 6 , 7a x b c     
60. Now 2 1a b c  
2 6 7 1     
1 
Hence a = 1, b = 6, c = –7
7 6a b c   
61. 2 2a   
12, 14b C   
3
3 1 3
2a b
W W W
    
62. 6 1b   
1, 7a c   
2 2
0 6 7 0ax bx c x x       
1, =-7  
1 1 6
7 
  
0
1 1
7
n
n  


 
   
 


SECTION - IV (Total Marks : 28)
(Integer Answer Type)
This section contains 7 questions. The answer to each of the questions is a single - digit integer, ranging
63. Let f : [1, ) [2, )   be a differentiable function such that f(1) = 2.
If     3
1
6 3
x
f t dt x f x x  for all 1x  , then the value of f(2) is
Ans : 6
Sol :      1 2
2.6 3. 3 3f x xf x f x x   
   1 2
xf x f x x 
   1 1
f x f x x
x
 
1
If
x

 
1
. 1f x dx
x
 
 f x
x c
n
 
  2
f x x c 
 1 2f 
2 1 c 
1c 
  2
f x x x 
  4 2f x c  
64. If z is any complex number satisfying 3 2 2z i   , then the minimum value of 2 6 5z i 
is
Ans : 5
Sol :
5 5
2 2 3 2 3 2 2
2 2
i
i z i i      
9
2 1 3 2
2
i
z i   
9
2
2
  5

65. Let a1
, a2
, a3
, ..........,a100
be an arithmetic progression with a1
= 3 and SP
=
1
, 1 100
p
i
i
a p

  .
For any integer n with 1 20n  , let m = 5n. If
m
n
S
S does not depend on n, then a2
is
Ans : 9
Sol :
 
 
6 1
2
6 1
2
m
n
m
m d
S
nS n d
   

   
 
 
5 6 5 1
6 1
n d
n d
   
 
s
Which is independed of n if 16 6d a 
2 1 9a a d   
66. Consider the parabola y2
= 8x. Let 1 be the area of the triangle formed by the end points of
its latus rectum and the point
1
, 2
2
P
 
 
 
on the parabola, and 2 be the area of the triangle
formed by drawing tangents at P and at the end points of the latus rectum. Then
1
2

 is
Ans : 2
Sol :    1 1 2 2 3 3 1
1
8
y y y y y y
a
    
   1 1 2 2 3 3 1
1
16
y y y y y y
a
    
1
2
2

 

67. Let   1 sin
sin tan
cos2
f



  
   
  
, where
4 4
 
   . Then the value of  
  tan
d
f
d

 is
Ans : 1
Sol : Put 1 sin
tan
cos 2



  
 
 
sin
tan
cos2



 
sin tan  
 
 
 tan 1
tan
d
f f
d
  

   

68. The minimum value of the sum of real numbers a–5
, a–4
, 3a–3
, 1, a8
and a10
with a > 0 is
Ans : 8
Sol : AM GM
8 10
5 4 3 3 3
1 1 1 1 1
1 a a
a a a a a
       
8
8 1 8 
69. The positive integer value of n > 3 satisfying the equation
1 1 1
2 3
sin sin sin
n n n
  
 
     
     
     
is
Ans : 7
Sol :
2 3 2 3
.Sin Sin Sin Sin Sin
n n n n n
     
  
 
5 3 2 4
cos cos cos cos cos cos
n n n n n n
     
     
3 5 2 4
cos cos cos cos
n n n n
   
   
4 3
2sin sin 2sin sin
n n n n
   
 
3
sin 4 sin 0
n n
 
  
7
2cos sin 0
2 2n n
    
    
   
 
7
cos 0 n>3
2n

  
 
7
2 1
2 2
k
n
 
  
7
integerodd
n
 
7n 

IITJEE 2011 paper1 solution

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IITJEE 2011 paper1 solution