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Prepare for the IITJEE with past papers solved by the coaching experts at Sri Chaitanya Junior College. Set your sights on the IITJEE 2014 Entrance examinations. To know more - visit www.srichaitanya.net or call 040 66060606. You can also stay in touch with us at www.facebook.com/SriChaitanyaEducationalInstitutes
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INSTRUCTIONSA. General
1. This booklet is your Question Paper. Do not break the seals of this boolket before being instructed to do so by the
invigilators.
2. The question paper CODE is printed on the right hand top corner of this page and on the back page (Page No. 28)
of this booklet.
3. Blank spaces and blank pages are provided in this booklet for your rough work. No additional sheets will be
provided for rough work.
4. Blank papers, clipboards , log tables, slide rules, calculators, cameras, cellular phones, pagers and electronic
gadgets are NOT allowed inside the examination hall.
5. Answer to the questions and personal details are to be filled on a two-part carbon-less paper, which is provided
separately. You should not separate these parts. The invigilator will separate them at the end of examination. The
upper sheet is a machine-gradable Ob jective Response Sheet (ORS) which will be taken back up the invigilator.
You will be allowed to take away the bottom sheet at the end of the examination.
6. Using a black ball point pen, darken the bubbles on the upper original sheet. Apply sufficinet pressure so that the
impression is created on the bottom sheet.
7. DO NOT TAMPER WITH / MUTILATE THE ORS OR THE BOOKLET.
8. On breaking the seals of the booklet check that it contains 28 pages and all the 60 questions and correspond-
ing answer choices are legible. Read carefully the instructions printed at the beginining of each section.B. Filling the Right Part of the ORS
9. The ORS has CODES printed on its left and right parts.
10. Check that the same CODE is printed on the ORS and on this boolet. IF IT NOT THEN ASK FOR A CHANGE OF THEBOOKLET. Sign at the place provided on the ORS affirming that you have verified that all the codes are same.
11. Write your name , Registration Number and the name of examination centre and sign with pen in the boxes providedon the right part of the ORS. Do not write any of this information anywhere else. Darken the appropriate bubbleUNDER each digit of your Registration Number in such a way that the impression is created on the bottom sheet. Also
darken the paper CODE given on the right side of ORS (R4)
C Question Paper FormatThe questions paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part consist of three sections.
12. Section I contains 10 multiple choice questions. Each question has four choices. (A), (B), (C) and (D) out of
which ONLY ONE is correct.
13. Section II contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which
ONE or MORE are correct.
14. Section III contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both
inclusive).
D.Marking Scheme15. For each question in Section I, you will be awarded 3 marks if you darken the bubble corresponding to the correct
answer ONLY and zero marks if no bubbles are darkened. In all ohter cases, minus one (–1) mark will be awarded
in this section.
16. For each question in Section II, you will be awareded 4 marks if you darken ALL the bubble(s) corresponding to the
correct answer(s) ONLY. In all other cases zero (0) marks will be awarded. No negative marks will be awarded for
incorrect answers in this section.
17. For each question in Section III, you will be awarded 4 marks if you darken the bubble corresponding to the correct
answer ONLY. In all other cases zero (0) will be awarded. No negative marks will be awarded for incorrect answers
in this section.
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PHYSICS SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
1. A small block is connected to one end of massless spring of un–stretched length 4.9 m. The
other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless
surface. The block is stretched by 0.2 m and released from rest at t= 0. It then executes
simple harmonic motion with angular frequency /3
rad s . Simultaneously at t = 0, a
small pebble is projected with speed v from point P at an angle of 450 as shown in the figure.
Point P is at a horizontal distance to 10 m from O. If the pebble hits the block at t = 1s, the
value of v is (take g = 10 m/s2)
a) 50 /m s b) 51 /m s c) 52 /m s d) 53 /m s
Ans : A
Equation of SHM cosy A t
At 1t 0.2cos (1) 0.2cos 0.13
y m
Block is at a distance 5m
Range of pebble is R =5m
22sin90 50
10VR V
50 /V m s
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2. In the determination of Young’s modulus 24MLgY
ld
by using Searle’s method, a wire of
length L = 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension
l = 0.25 mm in the length of the wire is observed. Quantities d and l are measured using a
screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The num-
ber of divisions on their circular scale is 100. The contributions to the maximum probable
error of the Y measurement.
a) due to the errors in the measurements of d and l are the same
b) due to the error in the measurement of d is twice that due to the error in the measurement of l.
c) due to the error in the measurement of l is twice that due to the error in the measurement of d.
d) due to the error in the measurement of d is four times that due to the error in the measure-
ment of l.
Ans :A
2
2 2 4d d dd v
40.25
l l ll
3. A small mass m is attached to a massless string whose other end is fixed at P as shown in the
figure. The mass is undergoing circular motion in the x–y plane with centre at O and constant
angular speed . If the angular momentum of the system, calculated about O and P are de-
noted by 0L
and PL
respectively, then
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a) 0L
and PL
do not vary with time
b) 0L
varies with time while PL
remains constant
c) 0L
remains constant while PL
varies with time
d) 0L
and PL
both vary with time
Ans :C
Torque about O is zero angular momentum will be constant about ‘O’. Torque about P is nonzero. Angular momentum about P varies with time.
0L m r v
PL m OP v
direction of PL
changes with OP and PL OP
4. Young’s double slit experiment is carried out by using green, red and blue light, one color at
a time. The fringe widths recorded are ,G R and B , respectively. Then
a) G B R b) B G R c) R B G d) R G B
Ans :D
Dd
R G B
R G B
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5. Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform
positive surface charge density. The variation of the magnitude of the electric field E r
and the electric potential V(r) with the distance r from the centre, is best represented bywhich graph ?
a) b)
c) d)
Ans : D
From centre to surface E = 0, then 1Er
From centre to surface V = k, then 1Vr
6. A bi–convex lens is formed with two thin plano–convex lenses as shown in the figure. Re-fractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curvedsurfaces are of the same radius of curvature R = 14 cm. For this bi–convex lens, for an objectdistance of 40 cm, the image distance will be
a) – 280. 0 cm b) 40.0 cm c) 21. 5 cm d) 13.3 cm
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Ans :B
1 2
1 1 1F f f =
1 21 1R
=
0.5 0.2 0.7 114 14 20
20F cm
40R cm
1 1 1 1 1 1 1 1 1 2 1 120 40 20 40 40 40F v u v v
40v cm
7. Two large vertical and parallel metal plates having as separation of 1 cm are connected to aDC voltage source of potential difference X. A proton is released at rest midway between thetwo plates. It is found to move at 450 to the vertical JUST after release. Then X is nearly
a) 51 10 V b) 71 10 V c) 91 10 V d) 101 10 V
Ans :C
mg Eq
45
mg
EqpVqd
910mgdV V approxq
8. Three very large plates of same area are kept parallel and close to each other. They are con-sidered as ideal black surfaces and have very high thermal conductivity. The first and thirdplates are maintained at temperatures 2T and 3T respectively. The temperature of the middle(i.e. second) plate under steady state condition is
a) 1465
2T
b) 1497
4T
c) 1497
2T
d) 1497 T
Ans : C
4 4 4 41 2 2 3A T T A T T
144 44 4
2 2 2972 32
T T T T T T
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9. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angularspeed , as shown in the figure. At time t = 0, a small insect starts from O and moves withconstant speed v with respect to the rod towards the other end. It reaches the end of the rod att = T and stops. The angular speed of the system remains throughout. The magnitude of the
torque of the torque on the system about O, as a function of time is best represented bywhich plot ?
a) b)
c) d)
Ans : B
dL d dIIdt dt dt
constantrodI
2red
d I mrdt
= 0 2 drm rdt
v
r = vt
22 2mr v m v t
t
when the insect reaches the end it stops v = 0
0
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10. A mixture of 2 moles of helium gas (atomic mass = 4amu) and 1 mole of argon gas (atomic
mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds heliumargon
rms
rms
vv
is
a) 0.32 b) 0.45 c) 2.24 d) 3.16Ans : D
rmsTVM
1 2
2 1
40 104
He
Ar
VV MV V M
SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
11. A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle with the horizontal. A horizontal force of 1 N acts on the block through its center of mass asshown in the figure. The block remains stationary if (take 210 /g m s )
a) 045
b) 045 and a frictional force acts on the block towards P.
c) 045 and a frictional force acts on the block towards Q.
d) 045 and a frictional force acts on the block towards Q.
Ans : A (or) AC
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1cos sinmg
45
when 45
friction force is upwards
If friction force sufficient block will be in state of rest
12. Consider the motion of a positive point charge in a region where there are simulataneousuniform electric and magnetic fields 0
ˆE E j
and 0ˆB B j
. At time t = 0 , this charge has
velcoity v in the x – y plane, making an angle with the x – axis. Which of the followingoption(s) is(are) correct for time t > 0 ?
a) If 00 , the charge moves in a circular path in the x – z plane
b) If 00 , the charge undergoes helical motion with constant pitch along the y – axis.
c) If 010 , the charge undergoes helical motion with its pitch increasing with time, along the y – axis
d) If 090 , the charge undergoes linear but accelerated motion along the y–axis
Ans :C,D
when 90
0mF
when 10
path is helical and with increasing pitch.
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13. For the resistance network shown in the figure, choose the correct option(s).
a) The current through PQ is zero b) 1 3I A
c) The potential at S is less than that at Q. d) 2 2I A
Ans : A,B,C,D
112 34
I
212 318
I 2A
P QV V
S QalsoV V
14. A person blows into open – end of a long pipe. A s a result, a high – pressure pulse of airtravels down the pipe.When this pulse reaches the other end of the pipe,
a) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
b) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
c) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is closedd) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is closed
Ans : B, D
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Reflected pressure pulse
Incident pressure pulse
Open end
At open end pressure should not change pressure wave will get phase change as
Reflected pressure pulse
Incident pressure pulse
Closed end
At enclosed end pressure wave will get reflected without phase change15. A cubical region of side a has its centre at the origin. It encloses three fixed point charges,
q at 0, / 4,0 , 3a q at (0, 0, 0) and q at 0, / 4,0a . Choose the correctoption(s).
a) The net electric flux crossing the plane / 2x a is equal to the net electric flux crossingthe plane / 2x a
b) The net electric flux crossing the plane / 2y a is more than the net electric flux crossingthe plane / 2y a .
c) The net electric flux crossing the entire region is 0
q .
d) The net electric flux crossing the plane / 2z a is equal to the net electric flux crossingthe plane / 2x a
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Ans : ACD
Net flux passing the plane 2ax and 2
ax is same
Net flux through cube is 0
q
Flux through plane 2az and 2
ax is same
SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).16. A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform
mass density and radius 2R, as shown in the figure. The moment of inertia of this laminaabout axes passing through O and P is 0I and PI , respectively. Both these axes are perpen-
dicular to the plane of the lamina. The ratio 0
PII to the nearest integer is
Ans : 3
2 2 2 2 2 20
1 12 22 2
I R R R R R R
4 4382
R R
4 213 132 2
R MR
2 2 2 21 14 2 4 2 52 2PI M R M R MR M R
2 2 2118 162
MR MR MR
2 211 37242 2
MR MR
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17. A circular wire loop of radius R is placed in the x - y plane centered at the origin O. A squareloop of side a a R having two turns is placed with its center at 3z R along the axis ofthe circular wire loop, as shown in figure. The plane of the square loop makes an angle of
45with respect to the z - axis. If the mutual inductance between the loops is given by 2
0/ 22p
aR
,
then the value of p is
Ans : 7
2
203/222
2 14 23
iRB ai R R
(n = 2)
22 20 0
9/21
2 22aM R a
18. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the
figure. Both the cylinder and the cavity are infinetely long. A uniform current density J flows
along the length. If the magnitude of the magnetic field at the point P is given by 012N aJ ,
then the value of N is
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Ans : 5
01 2
jaB
2
200
22
3 1222
ajjaB
a a
1 2 01 12 12
B B ja
0 512ja
19. A proton is fired from very far away towards a nucleus with charge 120Q e , where e is the
electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wave-
length (in units of fm) of the proton at its start is:
(take the proton mass, 275 / 3 10 ;pm kg 15/ 4.2 10 . /H e j S c ;
9
0
1 9 10 / ;4
m F
151 10fm m )
Ans : 7
2 29
151209 10
10 10 2 p
e Pm
9 2 27
215
59 10 120 2 103
10 10
eP
17
153 2 10 3 2
10 10e eP
154.2 106
10
hp
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20. An infinitely long solid cylinder of radius R has a uniform volume charge density . It has a
spherical cavity of radius R/2 with its centre on the axis on the cylinder, as shown in the
figure. The magnitude of the electric field at the point P, which is at the distance 2R from the
axis of the cylinder, is given by the expression 0
2316
Rk . The value of k is
Ans: 6
3
2
2
42 3 8
2 2 p
RKk R ER R
4 113 32pE k R
0
1 114 24
R
0
234 24p
RE
24 4 616
K
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CHEMISRY SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct21. The number of aldol reaction(s) that occurs in the given transformation is
. .3 4 conc aq NaOHCH CHO CHO
OH
OHHO
OH
a) 1 b) 2 c) 3 d) 4Ans : C
3 2OHCH CHO CH CHO
2 2 2H C H CH CHO O CH CH CHO
O
H
2 2OHCH CHO HO CH CH CHO H - C - H
OH
'HCHO Cannizaro sNaOH reactionCH CH
CHO
OH
OH CHO
HO
OH
OH OH OH
OHHO
Number of aldol reaction = 3
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22. A compound M PXP has cubic close packing (ccp) arrangement of X. Its unit cell structure isshown below. The empirical formula of the compound is
a) MX b) MX2 c) M2X d) M5X14
Ans : B
Corner
Face center 16 32
18 18
4
X :
Edge centre
Body center 1 1 1
14 14
2
M :
4 2X M
4 2X M 2MX
23. The carbonyl functional group (–COOH) is present ina) picric acid b) barbituric acid c) ascorbic acid d) aspirin
Ans : DAcetyle salicylic acid = Asprin
COOH
OCOCH3
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24. The colour of light absorbed by an aqueous solution of CuSO4 is :
a) organe-red b) blue-green c) yellow d) violetAns : A
Orange - red is absorbedBased up on Munshell wagon wheel
V
B
G
O
R
Y
25. The number of optically active products obtained from the complete ozonolysis of thegiven compound is :
CH3 - CH = CH - C - CH = CH - C - CH = CH - CH3
HCH3
H CH3
a) 0 b) 1 c) 2 d) 4Ans : A
As the ozonolysis products do not have chiral centres, number of optically active products = 0
26. As per IUPAC nomenclautre, the name of the complex 2 3 34 2Co H O NH Cl is
a) Tetraaqudiaminecobalt (III) chloride b) Tetraaquadiamminecobalt (III) chloridec) Diammineteraaquacobalt (III) chloride d) Diamminetetraaquacobalt (III) chloride
Ans : D
2 3 34 2Co H O NH Cl
Diammineteraaquacobalt (III) chloride
27. Which ordering of compound is according to the decreasing order of the oxidation state ofnitrogen ?
a) 3 4 2, , ,HNO NO NH Cl N b) 3 2 4, , ,HNO NO N NH Cl
c) 3 4 2, , ,HNO NH Cl NO N d) 3 4 2, , ,NO HNO NH Cl N
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Ans : B
3 5HNO
2NO
2 0N
4 3NH Cl
Decreasing order of
O -S = 3 2 4, , ,HNO NO N NH Cl
28. For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. I/V plot is shownbelow. The value of the van der Waals constant a (atm. liter2 mol–2) is
a) 1.0 b) 4.5 c) 1.5 d) 3.0Ans : C
2
2
n aP v nb nRTv
’
n = 1, b = 0
2
qp v b RTv
apv RTv
apv RTv
2 221.6 20.1 1.5 .3 2
y c mx m a slop atm t mole
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29. In allene 3 4C H , the type(s) of hybridisation of the carbon atoms is (are)
a) sp and sp3 b) sp and sp2 c) only sp2 d) sp2 and sp3
Ans : B
2 2H C C CH
2 2sp sp sp
30. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0 is Bohrradius]
a) 2
2 204
hma b)
2
2 2016
hma c)
2
2 2032
hma d)
2
2 2064
hma
Ans : C
mvr= 2nh
v= 2nh
mr
for second Bohr’s orbit
2r =4 0a
V=04
hma
22
2 2 2016
hVm a
K.E2
22 2
0
12 32
hmvma
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SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
31. Which of the following molecules, in prue form, is (are) unstable at room temperature ?
a) b) c)
O
d)
O
Ans : BC
cyclobutadiene and cyclopentadien are anti aromatic.
32. Which of the following hydrogen halides react(s) with 3AgNO aq to give a precipitate that
dissolves in 2 2 3Na S O aq ?
a) HCl b) HF c) HBr d) HI
Ans : ACD
3 3AgNO HX AgX HNO aq aq
2 2 3 3 2 3 22AgX Na S O Na Ag S O NaX aq aqs
AgF is soluble
33. Choose the correct reason(s) for the stability of the lyophobic colloidal particles >
a) Preferential adsorption of ions on their surface from the solution
b) Preferential adsorption of solvent of their surface the solution
c) Attraction between different particles having opposite charges on their surface
d) Potential difference between the fixed layer and the diffused layer of opposite chargesaround the coolidal particles
Ans : AD
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34. For an ideal gas, consider only P-V work in going from an
ini tial state X to the f inal state Z . The can be reached by
either of the two paths shown in the figure. Which of the
following choice(s) is (are) correct ? [Take S as change
in entropy and w as work done]
a) x z x y y zS S S b) x z x y y zW w w
c) x y z x yw w d) x y z x yS S
Ans : AC
Based up on the given graph
x y x y y zS S S
x y z x yW W
35. Identify the binary mixtrue(s) that can be separated into individual compounds, by differen-tial extraction,a s shown in the given scheme.
a) 6 5 6 5C H OH and C H COOH b) 6 5 6 5 2C H COOH and C H CH OH
c) 6 5 2 6 5C H CH OH and C H OH d) 6 5 2 6 5 2C H CH OH and C H CH COOH
Ans : BD
Benzyl alcohol is insoluble in both NaOH and 3NaHCO ,
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SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).
36. When the following aldohexose exists in its B-configuration, the total number C
CH2
CHOH
CHOH
CHO
CH2OH
OHH
of stereoisomrs in its pyranose form is :
Ans : 8
As the configration at C-5 is fixed, (D-configration), then only 4 stero isomers are possiblesolution
37. 29.2% (w/w) HCl stock solution has a densityh of 1.25 g mL–1. The molecular weight of HCl
is 36.5 g mol–1. The volume (mL) of stock solution required to prepare a 200 mL solution of
0.4 M HCl is :
Ans : 8
Molarity of stock solution =10 d x
GMW
=10 1.25 29.2
36.5
=10M
1 1 2 2
1
1
10 200 0.48 ( )
V M V MVV ML approx
38. The periodic table consists of 18 groups. An isotope of copper, on bombardment with pro-
tons, undergoes a nuclear reaction yielding element X as shown below. To which
group,element X belongs int he periodic table ?
Ans : 863 1 1 4 1 5229 1 0 2 1 266 2CU H n He H X
26 Fe 8 th group
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39. An organic compound undergoes first-order decomposition. The time taken for its decom-
position to 1/8 and 1/10 of its initial concentration are t1/8 and t1/10 respectively. What is the
Ans : 9
18
110
2.303 log
82.303 log
10
at ak
at ak
18
110
log8log10
t
t
18
110
3log 2t
t
18
110
10 9t
t
40. The substituents R1 and R2 for nine peptides are listed in the table given below. How many of
these peptides are positively charged at pH = 7.0 ?
Ans : 4
At pH below pI(Isoelectric point) Aminoacid exist as Cation. As BasicAminoacids of pHaround 9-8, these are expected to exists as cation of pH=7IV,VI,VIII,IX.
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MATHEMATICSSECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
41. Let z be a complex number such that the imaginary part of z is nonzero and and 2 1a z z is real. Then a connot take the value
a) –1 b) 13 c)
12 d)
34
Ans. D
Let Z x iy 0y
2 1a z z becomes
2 1a x iy x iy
2 2 1 2 0x y x i xy y a i
Comparing real & imaginary parts
2 2 1x y x a ––––––(1)
2 0xy y –––––––(2)
From (2) 1
2x as 0y
Put value of x in (1)
221 1 1
2 2y a
21 14 2
y a
2 3 04
y a
34
a
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42. If 2 1lim 4
1x
x x ax bx
, then
a) a = 1, b = 4 b) a = 1, b = –4 c) a = 2, b = –3 d) a = 2, b = 3
Ans. B
2 1lim 41x
x x ax bx
2 1 1 14
1x
x a x a b blim
x
For existense of limit coefficient of 2 0x
1 – a = 0
a = 1
1 1lim 4
1x
x a b bx
1 4a b
b = – 4
43. Let ijP a be a 3 3 matrix and let ijQ b , where 2i jij ijb a for 1 , 3i j . If the deter-
minant of P is 2, then the determinant of the matrix Q is
a) 210 b) 211 c) 212 d) 213
Ans. D
as 2i jij ijb a
11 11 12 12 13 134 , 8 , 16b a b a b a
21 2 22 22 23 238 , 16 , 32b a b a b a
31 31 32 32 33 3316 , 32 , 64b a b a b a
11 12 13
21 22 23
31 32 33
2a a a
A a a aa a a
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11 12 13
21 22 23
31 32 33
4 8 168 16 32
16 32 64
a a aB a a a
a a a
11 12 13
21 22 23
31 32 33
4 8 16 2 2 24 4 4
a a aa a aa a a
11 12 13
21 22 23
31 32 33
4 8 16 2 4a a aa a aa a a
4 8 16 2 4 2
132B
44. The ellipse 2 2
1 : 19 4x yE is inscribed in a rectangle R whose sides are parallel to the coor-
dinates axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangleR. The eccentricity of the ellipse E2 is
a) 22
b) 32
c) 12 d)
34
Ans. C
Given Ellipse 2 2
19 4x y
As ellipse is bounded by &x a y b
Sides of rectangle are
3 & 2x y
3,2 , 3,2C D
D(0,2)
(3,0)
(0,-2)
(-3,0)
A B
C
3, 2 , 3, 2A B
As new ellipse
Circumscribe the rectangle & passes through (0 ,4).
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Let new ellipse is 2 2
2 2 1x ya b
Passes through 22
160,4 1 16bb
Passes through 3,2 2
9 4 116a
2 12a
So, Vertical ellipse 2 2
112 16x y
2 2 21a b e
212 16 1 e
2 12 1116 4
e
12
e
45. The function : 0,3 1, 29f , defined by 3 22 15 36 1f x x x x is
a) one-one and onto b) onto but not one-one
c) one-one but not onto d) neither one-one nor onto
Ans. B
: 0,3 1, 29f
3 22 15 36 1f x x x x
' 26 30 36f x x x
6 2 3x x
'' 6 2 5f x x
2x is pt of maxima.
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3x is pt of minima.
& 2 29 0f
3 0f
so curve is
nf is not one one
2 3Min value 0 1f
Max value 2 29f
so nf is Onto.
46. The locus of the mid-point of the chord of contact of tangents drawn from points lying onthe straight line 4 5 20x y to the circle 2 2 9x y is
a) 2 220 36 45 0x y x y b) 2 220 36 45 0x y x y
c) 2 236 20 45 0x y x y d) 2 236 20 45 0x y x y
Ans. A
Let 1 1, & ,P h k Q x y
As P lies of 4x-5y=20
4 20,5
hP h
P
A
B
Q
Equation of chord of contact
1 0AB S
4 20 95
hxh y
5 4 20 45xh y h ------------------(1)
Equation of chord AB whose mid point is 1 1,Q x y
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2 21 1 1 1xx yy x y ---------------(2)
Compare (1) & (2)
2 21 1 1 1
5 4 20 45h hx y x y
1 12 2 2 21 1 1 1
9 45& 4 20x yh hx y x y
On eliminating h
1 12 2 2 21 1 1 1
49 4520x yx y x y
2 21 1 1 120 36 45x y x y
Locus is
2 220 36 45 0x y x y
47. The point P is the intersection of the straight line joining the points Q (2, 3, 5) and
R (1, –1, 4) with the plane 5 4 1x y z . If S is the foot of the perpendicular drawn from thepoint T (2, 1, 4) to QR, then the length of the line segement PS is
a) 12 b) 2 c) 2 d) 2 2
Ans. A
Equation of 1 1 4
1 4 1x y zQR
Dr’s of 1, 4,1QR
Co-ordinates of 1 , 1 4 , 4P
As P is point of Intersection of QR & given plane, so
5 1 4 1 4 4 1
13
T(2,1,4)
Q S R
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4 1 13, ,3 3 3
P
Let 1 , 1 4 ,4S
Dr’s of 1, 4 2,TS
TS QR So,
1 1 4 4 2 0
12
3 9,1,2 2
S
2 2 24 3 1 13 913 2 3 3 2
PS
12
48. Let 2 cos , 0
0, 0
x xf x x
x
, x IR , then f is
a) differentiable both at x = 0 and at x = 2
b) differentiable at x = 0 but not differentiable at x = 2
c) not differentiable at x = 0 but differentiable at x = 2
d) differentiable neither at x = 0 nor at x = 2
Ans. B
2 cos ; 0
0 ; 0
x xf x x
x
At x=0
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2
0 0
cos 00 0lim limh h
hf h f hRHDh h
0lim cos 0h
hh
2
0 0
cos 00 0
lim limh h
hf h f hLHD
h h
0lim cos 0h
hh
Differentiable at x=0
At x=2
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hRHDh h
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hLHDh h
Not Differentiable at x=2.
49. The total number of ways in which 5 balls of different colours can be distributed among 3persons so that each person gets at least one ball is
a) 75 b) 150 c) 210 d) 243
Ans. B
5 Balls to 3 persons, so that all persons get atleast one ball.
Case-I
Total ways 2 2 1
2 1 2
1 2 2
5 3 12 2 1 3C C C
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10 3 1 3 90
Case-II
Total ways
5 2 13 1 1 3C C C
11
1
13
3
1
3110 2 1 3 60
Total ways = 90 + 60 = 150.
50. The integral
2
9/ 2sec
sec tanx dx
x x equals (for some arbitrary constant K)
a) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
b) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
c) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
d) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
Ans. C
9
2
sec
sec tan
x dxx x
Let sec tanx x t -------------(1)
So, 2sec tan secx x x dx dt
sec dtx dxt
------------(2)
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1sec tanx xt
----------(3)
From (1) & (3)
1 1sec2
x tt
So, Integral becomes
9
2
sec .sec
sec tan
x x dxx x
92
1 12
dtt ttt
2
132
1 12
t dtt
9 132 21
2t dt t dt
7 112 21
7 112 22
t t K
7 112 2
7 11t t K
7 11
22
1 1
11 sec tan7 sec tan x xx x
2
112
1 1 1 sec tan11 7sec tan
x x Kx x
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SECTION – II( MULTIPLE CORRECT CHOICE TYPE )
This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONE OR MORE is/ are correct
51. A ship is fitted with three engines E1 , E2 and E3. The engines function independently of each
other with respective probabilities 1 1,2 4 and
14 . For the ship to be operational at least two of
its engines must function. Let X denote the event that the ship is operational and let X1, X2and X3
denote respectively the events that the engines E1, E2 and E3 are functioning. Which ofthe following is (are) true ?
a) 13|
16cP x X b) P [Exactly two engines of the ship are functioning | X]
78
c) 25|
16P X X d) 1
7|16
P X X
Ans. B,D
A)
1
1 1 112 4 4
1 84
cP X XP X
1 1 3 1 1 1 12 22 4 4 2 4 4 4
P X
B) 1 1 3 1 1 12
72 4 4 2 4 4/ 1 84
P Exactly twoengines of the ship are functioning X
C) 2
22
1 1 3 1 1 1 252 4 4 2 4 4/ 1 8
4
P X XP X X
P X
D)
11
1
1 1 3 1 1 1272 4 4 2 4 4/ 1 16
2
P X XP X X
P X
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52. Tangents are drawn to the hyperbola 2 2
19 4x y
, parallel to the straight line 2 1x y . The
points of contact of the tangents on the hyperbola are
a) 9 1,
2 2 2
b) 9 1,
2 2 2
c) 3 3, 2 2 d) 3 3,2 2
Ans. A, B
2 2
19 4x y
Tangent to the hyperbola is of the form2y x
2 9 4 4 36 4 32
4 2
Tangent is : 2 4 2 0x y
Point of contact is 2 2 9 1 9 1, , , ,
2 2 2 2 2 2a l b mn n
53. Let S be the area of the region enclosed by 2xy e , y = 0, x = 0, and x = 1. Then
a) 1Se
b) 11Se
c) 1 114
Se
d)
1 1 112 2
Se
Ans. A, B, D
21 1
0 0
11x xI e dx e dxe
11Ie
21 1xe e
21 1 1
1
0 0 0
xe dx e dx dx
21
0
1xe dxe
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54. If y (x) satisfies the differential equation ' tan 2 secy y x x x and y (0) = 0, then
a) 2
4 8 2y
b) 2
'4 18
y
c) 2
3 9y
d) 24 2'
3 3 3 3y
Ans. A,D
tan 2 secdy y x x xdx
tan. cos
dxI F e x
General solution is cos 2 sec cosy x x x x dx 2x c
2 sec secy x x c x
0,0
0 0 0c c .
2 secy x x
2
24 16
y
2
8 2
2
23 9
y
2 sec tan 2 secdy x x x x xdx
.
2
1 2 3 2 23 9 3
y
22 433 3
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55. Let , 0, 2 be such that
22cos 1 sin sin tan cot cos 12 2
,
tan 2 0 and 31 sin2
.
Then cannot satisfy
a) 0 2 b)
42 3 c)
4 33 2 d)
3 22
Ans. A, C, D
3tan 2 0, 1 sin2
3 5 1, 0 cos2 3 2
______ (1)
22cos 1 sin sin tan / 2 cot / 2 cos 1
1sin cos2
1 sin 12
(from (1))
13 176 6
2 73 6 (from (1))
cannot satisfied by A, C, D
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SECTION –III( INTEGER ANSWER TYPE)
This section contains 5 questions . The answer to each of the questions is a single digit integer, ranging from 0 to9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened.
56. Let p (x) be a real polynomial of least degree which has a local maximum at x = 1 and a localminimum at x =3. If p (1) = 6 and p (3) = 2, then ' 0p is
Ans. 9
3 2P x ax bx cx d
1 6 6P a b c d
3 2 27 9 3 2P a b c d
1 1 0 3 2 0P a b c
1 3 0 27 6 0P a b c
1, 6, 9a b c
1 23 2P x ax bx c
1 0 9P c
57. If ,a b and c are unit vectors satisfying
2 2 2 9a b b c c a ,then 2 5 5a b c
is\
Ans. 3
sol. 1a b c
2 2 2 9a b b c c a
3.
2a b b c c a
3 2 . . 0a b b c c a
22 2 2 . . . 0a b c a b b c c a
2
0a b c
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0a b c ––––––(1)
Taking dot product with (1) by , &a b c
then 1 1 1. , . , .
2 2 2a b b c c a
2 5 5 3a b c
58. The value of 32
1 1 1 16 log 4 4 4 ...3 2 3 2 3 2 3 2
is
Ans. 4
sol.1 4
3 2x x
3 2 4x x
squaring on both sides218 4x x 218 4 0x x 218 9 8 4 0x x x
9 2 1 4 2 1 0x x x .
2 1 9 4 0x x
1 4,2 9
x
49
x 0x
Required answer is 2
3/ 236 log2
= 6 – 2 = 4
59. Let S be the focus of the parabola 2 8y x and let PQ be the common chord of the circle2 2 2 4 0x y x y and the given parabola. The area of the triangle PQS is
Ans. 4
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sol.P
Q
S(0, 0)
(2,4)
(2, 0)2
(0,4)
2 8 2 4 2 0x x x x
0,2 0,0 , 2,4x P Q
area of SPQ
= 1 2 42
= 4
60. Let :f IR IR be defined as 2 1f x x x . The total number of points at which f attainseither a local maximum or a local minimum is
Ans. 5sol. 2 1, 1f x x x x
21 , 1 0x x x
21 , 0 1x x x
2 1, 1x x x
' 2 1, 1f x x x
2 1, 1 0x x
0-1 1x
y
1 2 , 0 1x x
1 2 , 1x x
differentiable at 1 1,2 2
x ,
not differentiable at –1, 0, 1
The number of points = 5
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IIT-JEE 2012Paper - 1 along with Key
& Solutions
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INSTRUCTIONSA. General
1. This booklet is your Question Paper. Do not break the seals of this boolket before being instructed to do so by the
invigilators.
2. The question paper CODE is printed on the right hand top corner of this page and on the back page (Page No. 28)
of this booklet.
3. Blank spaces and blank pages are provided in this booklet for your rough work. No additional sheets will be
provided for rough work.
4. Blank papers, clipboards , log tables, slide rules, calculators, cameras, cellular phones, pagers and electronic
gadgets are NOT allowed inside the examination hall.
5. Answer to the questions and personal details are to be filled on a two-part carbon-less paper, which is provided
separately. You should not separate these parts. The invigilator will separate them at the end of examination. The
upper sheet is a machine-gradable Ob jective Response Sheet (ORS) which will be taken back up the invigilator.
You will be allowed to take away the bottom sheet at the end of the examination.
6. Using a black ball point pen, darken the bubbles on the upper original sheet. Apply sufficinet pressure so that the
impression is created on the bottom sheet.
7. DO NOT TAMPER WITH / MUTILATE THE ORS OR THE BOOKLET.
8. On breaking the seals of the booklet check that it contains 28 pages and all the 60 questions and correspond-
ing answer choices are legible. Read carefully the instructions printed at the beginining of each section.B. Filling the Right Part of the ORS
9. The ORS has CODES printed on its left and right parts.
10. Check that the same CODE is printed on the ORS and on this boolet. IF IT NOT THEN ASK FOR A CHANGE OF THEBOOKLET. Sign at the place provided on the ORS affirming that you have verified that all the codes are same.
11. Write your name , Registration Number and the name of examination centre and sign with pen in the boxes providedon the right part of the ORS. Do not write any of this information anywhere else. Darken the appropriate bubbleUNDER each digit of your Registration Number in such a way that the impression is created on the bottom sheet. Also
darken the paper CODE given on the right side of ORS (R4)
C Question Paper FormatThe questions paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part consist of three sections.
12. Section I contains 10 multiple choice questions. Each question has four choices. (A), (B), (C) and (D) out of
which ONLY ONE is correct.
13. Section II contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which
ONE or MORE are correct.
14. Section III contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both
inclusive).
D.Marking Scheme15. For each question in Section I, you will be awarded 3 marks if you darken the bubble corresponding to the correct
answer ONLY and zero marks if no bubbles are darkened. In all ohter cases, minus one (–1) mark will be awarded
in this section.
16. For each question in Section II, you will be awareded 4 marks if you darken ALL the bubble(s) corresponding to the
correct answer(s) ONLY. In all other cases zero (0) marks will be awarded. No negative marks will be awarded for
incorrect answers in this section.
17. For each question in Section III, you will be awarded 4 marks if you darken the bubble corresponding to the correct
answer ONLY. In all other cases zero (0) will be awarded. No negative marks will be awarded for incorrect answers
in this section.
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PHYSICS SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
1. A small block is connected to one end of massless spring of un–stretched length 4.9 m. The
other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless
surface. The block is stretched by 0.2 m and released from rest at t= 0. It then executes
simple harmonic motion with angular frequency /3
rad s . Simultaneously at t = 0, a
small pebble is projected with speed v from point P at an angle of 450 as shown in the figure.
Point P is at a horizontal distance to 10 m from O. If the pebble hits the block at t = 1s, the
value of v is (take g = 10 m/s2)
a) 50 /m s b) 51 /m s c) 52 /m s d) 53 /m s
Ans : A
Equation of SHM cosy A t
At 1t 0.2cos (1) 0.2cos 0.13
y m
Block is at a distance 5m
Range of pebble is R =5m
22sin90 50
10VR V
50 /V m s
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2. In the determination of Young’s modulus 24MLgY
ld
by using Searle’s method, a wire of
length L = 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension
l = 0.25 mm in the length of the wire is observed. Quantities d and l are measured using a
screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The num-
ber of divisions on their circular scale is 100. The contributions to the maximum probable
error of the Y measurement.
a) due to the errors in the measurements of d and l are the same
b) due to the error in the measurement of d is twice that due to the error in the measurement of l.
c) due to the error in the measurement of l is twice that due to the error in the measurement of d.
d) due to the error in the measurement of d is four times that due to the error in the measure-
ment of l.
Ans :A
2
2 2 4d d dd v
40.25
l l ll
3. A small mass m is attached to a massless string whose other end is fixed at P as shown in the
figure. The mass is undergoing circular motion in the x–y plane with centre at O and constant
angular speed . If the angular momentum of the system, calculated about O and P are de-
noted by 0L
and PL
respectively, then
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a) 0L
and PL
do not vary with time
b) 0L
varies with time while PL
remains constant
c) 0L
remains constant while PL
varies with time
d) 0L
and PL
both vary with time
Ans :C
Torque about O is zero angular momentum will be constant about ‘O’. Torque about P is nonzero. Angular momentum about P varies with time.
0L m r v
PL m OP v
direction of PL
changes with OP and PL OP
4. Young’s double slit experiment is carried out by using green, red and blue light, one color at
a time. The fringe widths recorded are ,G R and B , respectively. Then
a) G B R b) B G R c) R B G d) R G B
Ans :D
Dd
R G B
R G B
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5. Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform
positive surface charge density. The variation of the magnitude of the electric field E r
and the electric potential V(r) with the distance r from the centre, is best represented bywhich graph ?
a) b)
c) d)
Ans : D
From centre to surface E = 0, then 1Er
From centre to surface V = k, then 1Vr
6. A bi–convex lens is formed with two thin plano–convex lenses as shown in the figure. Re-fractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curvedsurfaces are of the same radius of curvature R = 14 cm. For this bi–convex lens, for an objectdistance of 40 cm, the image distance will be
a) – 280. 0 cm b) 40.0 cm c) 21. 5 cm d) 13.3 cm
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Ans :B
1 2
1 1 1F f f =
1 21 1R
=
0.5 0.2 0.7 114 14 20
20F cm
40R cm
1 1 1 1 1 1 1 1 1 2 1 120 40 20 40 40 40F v u v v
40v cm
7. Two large vertical and parallel metal plates having as separation of 1 cm are connected to aDC voltage source of potential difference X. A proton is released at rest midway between thetwo plates. It is found to move at 450 to the vertical JUST after release. Then X is nearly
a) 51 10 V b) 71 10 V c) 91 10 V d) 101 10 V
Ans :C
mg Eq
45
mg
EqpVqd
910mgdV V approxq
8. Three very large plates of same area are kept parallel and close to each other. They are con-sidered as ideal black surfaces and have very high thermal conductivity. The first and thirdplates are maintained at temperatures 2T and 3T respectively. The temperature of the middle(i.e. second) plate under steady state condition is
a) 1465
2T
b) 1497
4T
c) 1497
2T
d) 1497 T
Ans : C
4 4 4 41 2 2 3A T T A T T
144 44 4
2 2 2972 32
T T T T T T
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9. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angularspeed , as shown in the figure. At time t = 0, a small insect starts from O and moves withconstant speed v with respect to the rod towards the other end. It reaches the end of the rod att = T and stops. The angular speed of the system remains throughout. The magnitude of the
torque of the torque on the system about O, as a function of time is best represented bywhich plot ?
a) b)
c) d)
Ans : B
dL d dIIdt dt dt
constantrodI
2red
d I mrdt
= 0 2 drm rdt
v
r = vt
22 2mr v m v t
t
when the insect reaches the end it stops v = 0
0
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10. A mixture of 2 moles of helium gas (atomic mass = 4amu) and 1 mole of argon gas (atomic
mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds heliumargon
rms
rms
vv
is
a) 0.32 b) 0.45 c) 2.24 d) 3.16Ans : D
rmsTVM
1 2
2 1
40 104
He
Ar
VV MV V M
SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
11. A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle with the horizontal. A horizontal force of 1 N acts on the block through its center of mass asshown in the figure. The block remains stationary if (take 210 /g m s )
a) 045
b) 045 and a frictional force acts on the block towards P.
c) 045 and a frictional force acts on the block towards Q.
d) 045 and a frictional force acts on the block towards Q.
Ans : A (or) AC
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1cos sinmg
45
when 45
friction force is upwards
If friction force sufficient block will be in state of rest
12. Consider the motion of a positive point charge in a region where there are simulataneousuniform electric and magnetic fields 0
ˆE E j
and 0ˆB B j
. At time t = 0 , this charge has
velcoity v in the x – y plane, making an angle with the x – axis. Which of the followingoption(s) is(are) correct for time t > 0 ?
a) If 00 , the charge moves in a circular path in the x – z plane
b) If 00 , the charge undergoes helical motion with constant pitch along the y – axis.
c) If 010 , the charge undergoes helical motion with its pitch increasing with time, along the y – axis
d) If 090 , the charge undergoes linear but accelerated motion along the y–axis
Ans :C,D
when 90
0mF
when 10
path is helical and with increasing pitch.
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13. For the resistance network shown in the figure, choose the correct option(s).
a) The current through PQ is zero b) 1 3I A
c) The potential at S is less than that at Q. d) 2 2I A
Ans : A,B,C,D
112 34
I
212 318
I 2A
P QV V
S QalsoV V
14. A person blows into open – end of a long pipe. A s a result, a high – pressure pulse of airtravels down the pipe.When this pulse reaches the other end of the pipe,
a) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
b) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
c) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is closedd) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is closed
Ans : B, D
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Reflected pressure pulse
Incident pressure pulse
Open end
At open end pressure should not change pressure wave will get phase change as
Reflected pressure pulse
Incident pressure pulse
Closed end
At enclosed end pressure wave will get reflected without phase change15. A cubical region of side a has its centre at the origin. It encloses three fixed point charges,
q at 0, / 4,0 , 3a q at (0, 0, 0) and q at 0, / 4,0a . Choose the correctoption(s).
a) The net electric flux crossing the plane / 2x a is equal to the net electric flux crossingthe plane / 2x a
b) The net electric flux crossing the plane / 2y a is more than the net electric flux crossingthe plane / 2y a .
c) The net electric flux crossing the entire region is 0
q .
d) The net electric flux crossing the plane / 2z a is equal to the net electric flux crossingthe plane / 2x a
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Ans : ACD
Net flux passing the plane 2ax and 2
ax is same
Net flux through cube is 0
q
Flux through plane 2az and 2
ax is same
SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).16. A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform
mass density and radius 2R, as shown in the figure. The moment of inertia of this laminaabout axes passing through O and P is 0I and PI , respectively. Both these axes are perpen-
dicular to the plane of the lamina. The ratio 0
PII to the nearest integer is
Ans : 3
2 2 2 2 2 20
1 12 22 2
I R R R R R R
4 4382
R R
4 213 132 2
R MR
2 2 2 21 14 2 4 2 52 2PI M R M R MR M R
2 2 2118 162
MR MR MR
2 211 37242 2
MR MR
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17. A circular wire loop of radius R is placed in the x - y plane centered at the origin O. A squareloop of side a a R having two turns is placed with its center at 3z R along the axis ofthe circular wire loop, as shown in figure. The plane of the square loop makes an angle of
45with respect to the z - axis. If the mutual inductance between the loops is given by 2
0/ 22p
aR
,
then the value of p is
Ans : 7
2
203/222
2 14 23
iRB ai R R
(n = 2)
22 20 0
9/21
2 22aM R a
18. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the
figure. Both the cylinder and the cavity are infinetely long. A uniform current density J flows
along the length. If the magnitude of the magnetic field at the point P is given by 012N aJ ,
then the value of N is
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Ans : 5
01 2
jaB
2
200
22
3 1222
ajjaB
a a
1 2 01 12 12
B B ja
0 512ja
19. A proton is fired from very far away towards a nucleus with charge 120Q e , where e is the
electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wave-
length (in units of fm) of the proton at its start is:
(take the proton mass, 275 / 3 10 ;pm kg 15/ 4.2 10 . /H e j S c ;
9
0
1 9 10 / ;4
m F
151 10fm m )
Ans : 7
2 29
151209 10
10 10 2 p
e Pm
9 2 27
215
59 10 120 2 103
10 10
eP
17
153 2 10 3 2
10 10e eP
154.2 106
10
hp
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20. An infinitely long solid cylinder of radius R has a uniform volume charge density . It has a
spherical cavity of radius R/2 with its centre on the axis on the cylinder, as shown in the
figure. The magnitude of the electric field at the point P, which is at the distance 2R from the
axis of the cylinder, is given by the expression 0
2316
Rk . The value of k is
Ans: 6
3
2
2
42 3 8
2 2 p
RKk R ER R
4 113 32pE k R
0
1 114 24
R
0
234 24p
RE
24 4 616
K
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CHEMISRY SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct21. The number of aldol reaction(s) that occurs in the given transformation is
. .3 4 conc aq NaOHCH CHO CHO
OH
OHHO
OH
a) 1 b) 2 c) 3 d) 4Ans : C
3 2OHCH CHO CH CHO
2 2 2H C H CH CHO O CH CH CHO
O
H
2 2OHCH CHO HO CH CH CHO H - C - H
OH
'HCHO Cannizaro sNaOH reactionCH CH
CHO
OH
OH CHO
HO
OH
OH OH OH
OHHO
Number of aldol reaction = 3
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22. A compound M PXP has cubic close packing (ccp) arrangement of X. Its unit cell structure isshown below. The empirical formula of the compound is
a) MX b) MX2 c) M2X d) M5X14
Ans : B
Corner
Face center 16 32
18 18
4
X :
Edge centre
Body center 1 1 1
14 14
2
M :
4 2X M
4 2X M 2MX
23. The carbonyl functional group (–COOH) is present ina) picric acid b) barbituric acid c) ascorbic acid d) aspirin
Ans : DAcetyle salicylic acid = Asprin
COOH
OCOCH3
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24. The colour of light absorbed by an aqueous solution of CuSO4 is :
a) organe-red b) blue-green c) yellow d) violetAns : A
Orange - red is absorbedBased up on Munshell wagon wheel
V
B
G
O
R
Y
25. The number of optically active products obtained from the complete ozonolysis of thegiven compound is :
CH3 - CH = CH - C - CH = CH - C - CH = CH - CH3
HCH3
H CH3
a) 0 b) 1 c) 2 d) 4Ans : A
As the ozonolysis products do not have chiral centres, number of optically active products = 0
26. As per IUPAC nomenclautre, the name of the complex 2 3 34 2Co H O NH Cl is
a) Tetraaqudiaminecobalt (III) chloride b) Tetraaquadiamminecobalt (III) chloridec) Diammineteraaquacobalt (III) chloride d) Diamminetetraaquacobalt (III) chloride
Ans : D
2 3 34 2Co H O NH Cl
Diammineteraaquacobalt (III) chloride
27. Which ordering of compound is according to the decreasing order of the oxidation state ofnitrogen ?
a) 3 4 2, , ,HNO NO NH Cl N b) 3 2 4, , ,HNO NO N NH Cl
c) 3 4 2, , ,HNO NH Cl NO N d) 3 4 2, , ,NO HNO NH Cl N
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Ans : B
3 5HNO
2NO
2 0N
4 3NH Cl
Decreasing order of
O -S = 3 2 4, , ,HNO NO N NH Cl
28. For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. I/V plot is shownbelow. The value of the van der Waals constant a (atm. liter2 mol–2) is
a) 1.0 b) 4.5 c) 1.5 d) 3.0Ans : C
2
2
n aP v nb nRTv
’
n = 1, b = 0
2
qp v b RTv
apv RTv
apv RTv
2 221.6 20.1 1.5 .3 2
y c mx m a slop atm t mole
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29. In allene 3 4C H , the type(s) of hybridisation of the carbon atoms is (are)
a) sp and sp3 b) sp and sp2 c) only sp2 d) sp2 and sp3
Ans : B
2 2H C C CH
2 2sp sp sp
30. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0 is Bohrradius]
a) 2
2 204
hma b)
2
2 2016
hma c)
2
2 2032
hma d)
2
2 2064
hma
Ans : C
mvr= 2nh
v= 2nh
mr
for second Bohr’s orbit
2r =4 0a
V=04
hma
22
2 2 2016
hVm a
K.E2
22 2
0
12 32
hmvma
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SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
31. Which of the following molecules, in prue form, is (are) unstable at room temperature ?
a) b) c)
O
d)
O
Ans : BC
cyclobutadiene and cyclopentadien are anti aromatic.
32. Which of the following hydrogen halides react(s) with 3AgNO aq to give a precipitate that
dissolves in 2 2 3Na S O aq ?
a) HCl b) HF c) HBr d) HI
Ans : ACD
3 3AgNO HX AgX HNO aq aq
2 2 3 3 2 3 22AgX Na S O Na Ag S O NaX aq aqs
AgF is soluble
33. Choose the correct reason(s) for the stability of the lyophobic colloidal particles >
a) Preferential adsorption of ions on their surface from the solution
b) Preferential adsorption of solvent of their surface the solution
c) Attraction between different particles having opposite charges on their surface
d) Potential difference between the fixed layer and the diffused layer of opposite chargesaround the coolidal particles
Ans : AD
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34. For an ideal gas, consider only P-V work in going from an
ini tial state X to the f inal state Z . The can be reached by
either of the two paths shown in the figure. Which of the
following choice(s) is (are) correct ? [Take S as change
in entropy and w as work done]
a) x z x y y zS S S b) x z x y y zW w w
c) x y z x yw w d) x y z x yS S
Ans : AC
Based up on the given graph
x y x y y zS S S
x y z x yW W
35. Identify the binary mixtrue(s) that can be separated into individual compounds, by differen-tial extraction,a s shown in the given scheme.
a) 6 5 6 5C H OH and C H COOH b) 6 5 6 5 2C H COOH and C H CH OH
c) 6 5 2 6 5C H CH OH and C H OH d) 6 5 2 6 5 2C H CH OH and C H CH COOH
Ans : BD
Benzyl alcohol is insoluble in both NaOH and 3NaHCO ,
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SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).
36. When the following aldohexose exists in its B-configuration, the total number C
CH2
CHOH
CHOH
CHO
CH2OH
OHH
of stereoisomrs in its pyranose form is :
Ans : 8
As the configration at C-5 is fixed, (D-configration), then only 4 stero isomers are possiblesolution
37. 29.2% (w/w) HCl stock solution has a densityh of 1.25 g mL–1. The molecular weight of HCl
is 36.5 g mol–1. The volume (mL) of stock solution required to prepare a 200 mL solution of
0.4 M HCl is :
Ans : 8
Molarity of stock solution =10 d x
GMW
=10 1.25 29.2
36.5
=10M
1 1 2 2
1
1
10 200 0.48 ( )
V M V MVV ML approx
38. The periodic table consists of 18 groups. An isotope of copper, on bombardment with pro-
tons, undergoes a nuclear reaction yielding element X as shown below. To which
group,element X belongs int he periodic table ?
Ans : 863 1 1 4 1 5229 1 0 2 1 266 2CU H n He H X
26 Fe 8 th group
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39. An organic compound undergoes first-order decomposition. The time taken for its decom-
position to 1/8 and 1/10 of its initial concentration are t1/8 and t1/10 respectively. What is the
Ans : 9
18
110
2.303 log
82.303 log
10
at ak
at ak
18
110
log8log10
t
t
18
110
3log 2t
t
18
110
10 9t
t
40. The substituents R1 and R2 for nine peptides are listed in the table given below. How many of
these peptides are positively charged at pH = 7.0 ?
Ans : 4
At pH below pI(Isoelectric point) Aminoacid exist as Cation. As BasicAminoacids of pHaround 9-8, these are expected to exists as cation of pH=7IV,VI,VIII,IX.
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MATHEMATICSSECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
41. Let z be a complex number such that the imaginary part of z is nonzero and and 2 1a z z is real. Then a connot take the value
a) –1 b) 13 c)
12 d)
34
Ans. D
Let Z x iy 0y
2 1a z z becomes
2 1a x iy x iy
2 2 1 2 0x y x i xy y a i
Comparing real & imaginary parts
2 2 1x y x a ––––––(1)
2 0xy y –––––––(2)
From (2) 1
2x as 0y
Put value of x in (1)
221 1 1
2 2y a
21 14 2
y a
2 3 04
y a
34
a
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42. If 2 1lim 4
1x
x x ax bx
, then
a) a = 1, b = 4 b) a = 1, b = –4 c) a = 2, b = –3 d) a = 2, b = 3
Ans. B
2 1lim 41x
x x ax bx
2 1 1 14
1x
x a x a b blim
x
For existense of limit coefficient of 2 0x
1 – a = 0
a = 1
1 1lim 4
1x
x a b bx
1 4a b
b = – 4
43. Let ijP a be a 3 3 matrix and let ijQ b , where 2i jij ijb a for 1 , 3i j . If the deter-
minant of P is 2, then the determinant of the matrix Q is
a) 210 b) 211 c) 212 d) 213
Ans. D
as 2i jij ijb a
11 11 12 12 13 134 , 8 , 16b a b a b a
21 2 22 22 23 238 , 16 , 32b a b a b a
31 31 32 32 33 3316 , 32 , 64b a b a b a
11 12 13
21 22 23
31 32 33
2a a a
A a a aa a a
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11 12 13
21 22 23
31 32 33
4 8 168 16 32
16 32 64
a a aB a a a
a a a
11 12 13
21 22 23
31 32 33
4 8 16 2 2 24 4 4
a a aa a aa a a
11 12 13
21 22 23
31 32 33
4 8 16 2 4a a aa a aa a a
4 8 16 2 4 2
132B
44. The ellipse 2 2
1 : 19 4x yE is inscribed in a rectangle R whose sides are parallel to the coor-
dinates axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangleR. The eccentricity of the ellipse E2 is
a) 22
b) 32
c) 12 d)
34
Ans. C
Given Ellipse 2 2
19 4x y
As ellipse is bounded by &x a y b
Sides of rectangle are
3 & 2x y
3,2 , 3,2C D
D(0,2)
(3,0)
(0,-2)
(-3,0)
A B
C
3, 2 , 3, 2A B
As new ellipse
Circumscribe the rectangle & passes through (0 ,4).
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Let new ellipse is 2 2
2 2 1x ya b
Passes through 22
160,4 1 16bb
Passes through 3,2 2
9 4 116a
2 12a
So, Vertical ellipse 2 2
112 16x y
2 2 21a b e
212 16 1 e
2 12 1116 4
e
12
e
45. The function : 0,3 1, 29f , defined by 3 22 15 36 1f x x x x is
a) one-one and onto b) onto but not one-one
c) one-one but not onto d) neither one-one nor onto
Ans. B
: 0,3 1, 29f
3 22 15 36 1f x x x x
' 26 30 36f x x x
6 2 3x x
'' 6 2 5f x x
2x is pt of maxima.
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3x is pt of minima.
& 2 29 0f
3 0f
so curve is
nf is not one one
2 3Min value 0 1f
Max value 2 29f
so nf is Onto.
46. The locus of the mid-point of the chord of contact of tangents drawn from points lying onthe straight line 4 5 20x y to the circle 2 2 9x y is
a) 2 220 36 45 0x y x y b) 2 220 36 45 0x y x y
c) 2 236 20 45 0x y x y d) 2 236 20 45 0x y x y
Ans. A
Let 1 1, & ,P h k Q x y
As P lies of 4x-5y=20
4 20,5
hP h
P
A
B
Q
Equation of chord of contact
1 0AB S
4 20 95
hxh y
5 4 20 45xh y h ------------------(1)
Equation of chord AB whose mid point is 1 1,Q x y
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2 21 1 1 1xx yy x y ---------------(2)
Compare (1) & (2)
2 21 1 1 1
5 4 20 45h hx y x y
1 12 2 2 21 1 1 1
9 45& 4 20x yh hx y x y
On eliminating h
1 12 2 2 21 1 1 1
49 4520x yx y x y
2 21 1 1 120 36 45x y x y
Locus is
2 220 36 45 0x y x y
47. The point P is the intersection of the straight line joining the points Q (2, 3, 5) and
R (1, –1, 4) with the plane 5 4 1x y z . If S is the foot of the perpendicular drawn from thepoint T (2, 1, 4) to QR, then the length of the line segement PS is
a) 12 b) 2 c) 2 d) 2 2
Ans. A
Equation of 1 1 4
1 4 1x y zQR
Dr’s of 1, 4,1QR
Co-ordinates of 1 , 1 4 , 4P
As P is point of Intersection of QR & given plane, so
5 1 4 1 4 4 1
13
T(2,1,4)
Q S R
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4 1 13, ,3 3 3
P
Let 1 , 1 4 ,4S
Dr’s of 1, 4 2,TS
TS QR So,
1 1 4 4 2 0
12
3 9,1,2 2
S
2 2 24 3 1 13 913 2 3 3 2
PS
12
48. Let 2 cos , 0
0, 0
x xf x x
x
, x IR , then f is
a) differentiable both at x = 0 and at x = 2
b) differentiable at x = 0 but not differentiable at x = 2
c) not differentiable at x = 0 but differentiable at x = 2
d) differentiable neither at x = 0 nor at x = 2
Ans. B
2 cos ; 0
0 ; 0
x xf x x
x
At x=0
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2
0 0
cos 00 0lim limh h
hf h f hRHDh h
0lim cos 0h
hh
2
0 0
cos 00 0
lim limh h
hf h f hLHD
h h
0lim cos 0h
hh
Differentiable at x=0
At x=2
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hRHDh h
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hLHDh h
Not Differentiable at x=2.
49. The total number of ways in which 5 balls of different colours can be distributed among 3persons so that each person gets at least one ball is
a) 75 b) 150 c) 210 d) 243
Ans. B
5 Balls to 3 persons, so that all persons get atleast one ball.
Case-I
Total ways 2 2 1
2 1 2
1 2 2
5 3 12 2 1 3C C C
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10 3 1 3 90
Case-II
Total ways
5 2 13 1 1 3C C C
11
1
13
3
1
3110 2 1 3 60
Total ways = 90 + 60 = 150.
50. The integral
2
9/ 2sec
sec tanx dx
x x equals (for some arbitrary constant K)
a) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
b) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
c) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
d) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
Ans. C
9
2
sec
sec tan
x dxx x
Let sec tanx x t -------------(1)
So, 2sec tan secx x x dx dt
sec dtx dxt
------------(2)
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1sec tanx xt
----------(3)
From (1) & (3)
1 1sec2
x tt
So, Integral becomes
9
2
sec .sec
sec tan
x x dxx x
92
1 12
dtt ttt
2
132
1 12
t dtt
9 132 21
2t dt t dt
7 112 21
7 112 22
t t K
7 112 2
7 11t t K
7 11
22
1 1
11 sec tan7 sec tan x xx x
2
112
1 1 1 sec tan11 7sec tan
x x Kx x
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SECTION – II( MULTIPLE CORRECT CHOICE TYPE )
This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONE OR MORE is/ are correct
51. A ship is fitted with three engines E1 , E2 and E3. The engines function independently of each
other with respective probabilities 1 1,2 4 and
14 . For the ship to be operational at least two of
its engines must function. Let X denote the event that the ship is operational and let X1, X2and X3
denote respectively the events that the engines E1, E2 and E3 are functioning. Which ofthe following is (are) true ?
a) 13|
16cP x X b) P [Exactly two engines of the ship are functioning | X]
78
c) 25|
16P X X d) 1
7|16
P X X
Ans. B,D
A)
1
1 1 112 4 4
1 84
cP X XP X
1 1 3 1 1 1 12 22 4 4 2 4 4 4
P X
B) 1 1 3 1 1 12
72 4 4 2 4 4/ 1 84
P Exactly twoengines of the ship are functioning X
C) 2
22
1 1 3 1 1 1 252 4 4 2 4 4/ 1 8
4
P X XP X X
P X
D)
11
1
1 1 3 1 1 1272 4 4 2 4 4/ 1 16
2
P X XP X X
P X
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52. Tangents are drawn to the hyperbola 2 2
19 4x y
, parallel to the straight line 2 1x y . The
points of contact of the tangents on the hyperbola are
a) 9 1,
2 2 2
b) 9 1,
2 2 2
c) 3 3, 2 2 d) 3 3,2 2
Ans. A, B
2 2
19 4x y
Tangent to the hyperbola is of the form2y x
2 9 4 4 36 4 32
4 2
Tangent is : 2 4 2 0x y
Point of contact is 2 2 9 1 9 1, , , ,
2 2 2 2 2 2a l b mn n
53. Let S be the area of the region enclosed by 2xy e , y = 0, x = 0, and x = 1. Then
a) 1Se
b) 11Se
c) 1 114
Se
d)
1 1 112 2
Se
Ans. A, B, D
21 1
0 0
11x xI e dx e dxe
11Ie
21 1xe e
21 1 1
1
0 0 0
xe dx e dx dx
21
0
1xe dxe
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54. If y (x) satisfies the differential equation ' tan 2 secy y x x x and y (0) = 0, then
a) 2
4 8 2y
b) 2
'4 18
y
c) 2
3 9y
d) 24 2'
3 3 3 3y
Ans. A,D
tan 2 secdy y x x xdx
tan. cos
dxI F e x
General solution is cos 2 sec cosy x x x x dx 2x c
2 sec secy x x c x
0,0
0 0 0c c .
2 secy x x
2
24 16
y
2
8 2
2
23 9
y
2 sec tan 2 secdy x x x x xdx
.
2
1 2 3 2 23 9 3
y
22 433 3
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55. Let , 0, 2 be such that
22cos 1 sin sin tan cot cos 12 2
,
tan 2 0 and 31 sin2
.
Then cannot satisfy
a) 0 2 b)
42 3 c)
4 33 2 d)
3 22
Ans. A, C, D
3tan 2 0, 1 sin2
3 5 1, 0 cos2 3 2
______ (1)
22cos 1 sin sin tan / 2 cot / 2 cos 1
1sin cos2
1 sin 12
(from (1))
13 176 6
2 73 6 (from (1))
cannot satisfied by A, C, D
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SECTION –III( INTEGER ANSWER TYPE)
This section contains 5 questions . The answer to each of the questions is a single digit integer, ranging from 0 to9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened.
56. Let p (x) be a real polynomial of least degree which has a local maximum at x = 1 and a localminimum at x =3. If p (1) = 6 and p (3) = 2, then ' 0p is
Ans. 9
3 2P x ax bx cx d
1 6 6P a b c d
3 2 27 9 3 2P a b c d
1 1 0 3 2 0P a b c
1 3 0 27 6 0P a b c
1, 6, 9a b c
1 23 2P x ax bx c
1 0 9P c
57. If ,a b and c are unit vectors satisfying
2 2 2 9a b b c c a ,then 2 5 5a b c
is\
Ans. 3
sol. 1a b c
2 2 2 9a b b c c a
3.
2a b b c c a
3 2 . . 0a b b c c a
22 2 2 . . . 0a b c a b b c c a
2
0a b c
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0a b c ––––––(1)
Taking dot product with (1) by , &a b c
then 1 1 1. , . , .
2 2 2a b b c c a
2 5 5 3a b c
58. The value of 32
1 1 1 16 log 4 4 4 ...3 2 3 2 3 2 3 2
is
Ans. 4
sol.1 4
3 2x x
3 2 4x x
squaring on both sides218 4x x 218 4 0x x 218 9 8 4 0x x x
9 2 1 4 2 1 0x x x .
2 1 9 4 0x x
1 4,2 9
x
49
x 0x
Required answer is 2
3/ 236 log2
= 6 – 2 = 4
59. Let S be the focus of the parabola 2 8y x and let PQ be the common chord of the circle2 2 2 4 0x y x y and the given parabola. The area of the triangle PQS is
Ans. 4
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sol.P
Q
S(0, 0)
(2,4)
(2, 0)2
(0,4)
2 8 2 4 2 0x x x x
0,2 0,0 , 2,4x P Q
area of SPQ
= 1 2 42
= 4
60. Let :f IR IR be defined as 2 1f x x x . The total number of points at which f attainseither a local maximum or a local minimum is
Ans. 5sol. 2 1, 1f x x x x
21 , 1 0x x x
21 , 0 1x x x
2 1, 1x x x
' 2 1, 1f x x x
2 1, 1 0x x
0-1 1x
y
1 2 , 0 1x x
1 2 , 1x x
differentiable at 1 1,2 2
x ,
not differentiable at –1, 0, 1
The number of points = 5
Sri Chaitanya IIT Academy., A.P.
IIT-JEE 2012Paper - 1 along with Key
& Solutions
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INSTRUCTIONSA. General
1. This booklet is your Question Paper. Do not break the seals of this boolket before being instructed to do so by the
invigilators.
2. The question paper CODE is printed on the right hand top corner of this page and on the back page (Page No. 28)
of this booklet.
3. Blank spaces and blank pages are provided in this booklet for your rough work. No additional sheets will be
provided for rough work.
4. Blank papers, clipboards , log tables, slide rules, calculators, cameras, cellular phones, pagers and electronic
gadgets are NOT allowed inside the examination hall.
5. Answer to the questions and personal details are to be filled on a two-part carbon-less paper, which is provided
separately. You should not separate these parts. The invigilator will separate them at the end of examination. The
upper sheet is a machine-gradable Ob jective Response Sheet (ORS) which will be taken back up the invigilator.
You will be allowed to take away the bottom sheet at the end of the examination.
6. Using a black ball point pen, darken the bubbles on the upper original sheet. Apply sufficinet pressure so that the
impression is created on the bottom sheet.
7. DO NOT TAMPER WITH / MUTILATE THE ORS OR THE BOOKLET.
8. On breaking the seals of the booklet check that it contains 28 pages and all the 60 questions and correspond-
ing answer choices are legible. Read carefully the instructions printed at the beginining of each section.B. Filling the Right Part of the ORS
9. The ORS has CODES printed on its left and right parts.
10. Check that the same CODE is printed on the ORS and on this boolet. IF IT NOT THEN ASK FOR A CHANGE OF THEBOOKLET. Sign at the place provided on the ORS affirming that you have verified that all the codes are same.
11. Write your name , Registration Number and the name of examination centre and sign with pen in the boxes providedon the right part of the ORS. Do not write any of this information anywhere else. Darken the appropriate bubbleUNDER each digit of your Registration Number in such a way that the impression is created on the bottom sheet. Also
darken the paper CODE given on the right side of ORS (R4)
C Question Paper FormatThe questions paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part consist of three sections.
12. Section I contains 10 multiple choice questions. Each question has four choices. (A), (B), (C) and (D) out of
which ONLY ONE is correct.
13. Section II contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which
ONE or MORE are correct.
14. Section III contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both
inclusive).
D.Marking Scheme15. For each question in Section I, you will be awarded 3 marks if you darken the bubble corresponding to the correct
answer ONLY and zero marks if no bubbles are darkened. In all ohter cases, minus one (–1) mark will be awarded
in this section.
16. For each question in Section II, you will be awareded 4 marks if you darken ALL the bubble(s) corresponding to the
correct answer(s) ONLY. In all other cases zero (0) marks will be awarded. No negative marks will be awarded for
incorrect answers in this section.
17. For each question in Section III, you will be awarded 4 marks if you darken the bubble corresponding to the correct
answer ONLY. In all other cases zero (0) will be awarded. No negative marks will be awarded for incorrect answers
in this section.
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PHYSICS SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
1. A small block is connected to one end of massless spring of un–stretched length 4.9 m. The
other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless
surface. The block is stretched by 0.2 m and released from rest at t= 0. It then executes
simple harmonic motion with angular frequency /3
rad s . Simultaneously at t = 0, a
small pebble is projected with speed v from point P at an angle of 450 as shown in the figure.
Point P is at a horizontal distance to 10 m from O. If the pebble hits the block at t = 1s, the
value of v is (take g = 10 m/s2)
a) 50 /m s b) 51 /m s c) 52 /m s d) 53 /m s
Ans : A
Equation of SHM cosy A t
At 1t 0.2cos (1) 0.2cos 0.13
y m
Block is at a distance 5m
Range of pebble is R =5m
22sin90 50
10VR V
50 /V m s
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2. In the determination of Young’s modulus 24MLgY
ld
by using Searle’s method, a wire of
length L = 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension
l = 0.25 mm in the length of the wire is observed. Quantities d and l are measured using a
screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The num-
ber of divisions on their circular scale is 100. The contributions to the maximum probable
error of the Y measurement.
a) due to the errors in the measurements of d and l are the same
b) due to the error in the measurement of d is twice that due to the error in the measurement of l.
c) due to the error in the measurement of l is twice that due to the error in the measurement of d.
d) due to the error in the measurement of d is four times that due to the error in the measure-
ment of l.
Ans :A
2
2 2 4d d dd v
40.25
l l ll
3. A small mass m is attached to a massless string whose other end is fixed at P as shown in the
figure. The mass is undergoing circular motion in the x–y plane with centre at O and constant
angular speed . If the angular momentum of the system, calculated about O and P are de-
noted by 0L
and PL
respectively, then
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a) 0L
and PL
do not vary with time
b) 0L
varies with time while PL
remains constant
c) 0L
remains constant while PL
varies with time
d) 0L
and PL
both vary with time
Ans :C
Torque about O is zero angular momentum will be constant about ‘O’. Torque about P is nonzero. Angular momentum about P varies with time.
0L m r v
PL m OP v
direction of PL
changes with OP and PL OP
4. Young’s double slit experiment is carried out by using green, red and blue light, one color at
a time. The fringe widths recorded are ,G R and B , respectively. Then
a) G B R b) B G R c) R B G d) R G B
Ans :D
Dd
R G B
R G B
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5. Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform
positive surface charge density. The variation of the magnitude of the electric field E r
and the electric potential V(r) with the distance r from the centre, is best represented bywhich graph ?
a) b)
c) d)
Ans : D
From centre to surface E = 0, then 1Er
From centre to surface V = k, then 1Vr
6. A bi–convex lens is formed with two thin plano–convex lenses as shown in the figure. Re-fractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curvedsurfaces are of the same radius of curvature R = 14 cm. For this bi–convex lens, for an objectdistance of 40 cm, the image distance will be
a) – 280. 0 cm b) 40.0 cm c) 21. 5 cm d) 13.3 cm
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Ans :B
1 2
1 1 1F f f =
1 21 1R
=
0.5 0.2 0.7 114 14 20
20F cm
40R cm
1 1 1 1 1 1 1 1 1 2 1 120 40 20 40 40 40F v u v v
40v cm
7. Two large vertical and parallel metal plates having as separation of 1 cm are connected to aDC voltage source of potential difference X. A proton is released at rest midway between thetwo plates. It is found to move at 450 to the vertical JUST after release. Then X is nearly
a) 51 10 V b) 71 10 V c) 91 10 V d) 101 10 V
Ans :C
mg Eq
45
mg
EqpVqd
910mgdV V approxq
8. Three very large plates of same area are kept parallel and close to each other. They are con-sidered as ideal black surfaces and have very high thermal conductivity. The first and thirdplates are maintained at temperatures 2T and 3T respectively. The temperature of the middle(i.e. second) plate under steady state condition is
a) 1465
2T
b) 1497
4T
c) 1497
2T
d) 1497 T
Ans : C
4 4 4 41 2 2 3A T T A T T
144 44 4
2 2 2972 32
T T T T T T
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9. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angularspeed , as shown in the figure. At time t = 0, a small insect starts from O and moves withconstant speed v with respect to the rod towards the other end. It reaches the end of the rod att = T and stops. The angular speed of the system remains throughout. The magnitude of the
torque of the torque on the system about O, as a function of time is best represented bywhich plot ?
a) b)
c) d)
Ans : B
dL d dIIdt dt dt
constantrodI
2red
d I mrdt
= 0 2 drm rdt
v
r = vt
22 2mr v m v t
t
when the insect reaches the end it stops v = 0
0
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10. A mixture of 2 moles of helium gas (atomic mass = 4amu) and 1 mole of argon gas (atomic
mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds heliumargon
rms
rms
vv
is
a) 0.32 b) 0.45 c) 2.24 d) 3.16Ans : D
rmsTVM
1 2
2 1
40 104
He
Ar
VV MV V M
SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
11. A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle with the horizontal. A horizontal force of 1 N acts on the block through its center of mass asshown in the figure. The block remains stationary if (take 210 /g m s )
a) 045
b) 045 and a frictional force acts on the block towards P.
c) 045 and a frictional force acts on the block towards Q.
d) 045 and a frictional force acts on the block towards Q.
Ans : A (or) AC
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1cos sinmg
45
when 45
friction force is upwards
If friction force sufficient block will be in state of rest
12. Consider the motion of a positive point charge in a region where there are simulataneousuniform electric and magnetic fields 0
ˆE E j
and 0ˆB B j
. At time t = 0 , this charge has
velcoity v in the x – y plane, making an angle with the x – axis. Which of the followingoption(s) is(are) correct for time t > 0 ?
a) If 00 , the charge moves in a circular path in the x – z plane
b) If 00 , the charge undergoes helical motion with constant pitch along the y – axis.
c) If 010 , the charge undergoes helical motion with its pitch increasing with time, along the y – axis
d) If 090 , the charge undergoes linear but accelerated motion along the y–axis
Ans :C,D
when 90
0mF
when 10
path is helical and with increasing pitch.
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13. For the resistance network shown in the figure, choose the correct option(s).
a) The current through PQ is zero b) 1 3I A
c) The potential at S is less than that at Q. d) 2 2I A
Ans : A,B,C,D
112 34
I
212 318
I 2A
P QV V
S QalsoV V
14. A person blows into open – end of a long pipe. A s a result, a high – pressure pulse of airtravels down the pipe.When this pulse reaches the other end of the pipe,
a) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
b) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
c) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is closedd) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is closed
Ans : B, D
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Reflected pressure pulse
Incident pressure pulse
Open end
At open end pressure should not change pressure wave will get phase change as
Reflected pressure pulse
Incident pressure pulse
Closed end
At enclosed end pressure wave will get reflected without phase change15. A cubical region of side a has its centre at the origin. It encloses three fixed point charges,
q at 0, / 4,0 , 3a q at (0, 0, 0) and q at 0, / 4,0a . Choose the correctoption(s).
a) The net electric flux crossing the plane / 2x a is equal to the net electric flux crossingthe plane / 2x a
b) The net electric flux crossing the plane / 2y a is more than the net electric flux crossingthe plane / 2y a .
c) The net electric flux crossing the entire region is 0
q .
d) The net electric flux crossing the plane / 2z a is equal to the net electric flux crossingthe plane / 2x a
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Ans : ACD
Net flux passing the plane 2ax and 2
ax is same
Net flux through cube is 0
q
Flux through plane 2az and 2
ax is same
SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).16. A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform
mass density and radius 2R, as shown in the figure. The moment of inertia of this laminaabout axes passing through O and P is 0I and PI , respectively. Both these axes are perpen-
dicular to the plane of the lamina. The ratio 0
PII to the nearest integer is
Ans : 3
2 2 2 2 2 20
1 12 22 2
I R R R R R R
4 4382
R R
4 213 132 2
R MR
2 2 2 21 14 2 4 2 52 2PI M R M R MR M R
2 2 2118 162
MR MR MR
2 211 37242 2
MR MR
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17. A circular wire loop of radius R is placed in the x - y plane centered at the origin O. A squareloop of side a a R having two turns is placed with its center at 3z R along the axis ofthe circular wire loop, as shown in figure. The plane of the square loop makes an angle of
45with respect to the z - axis. If the mutual inductance between the loops is given by 2
0/ 22p
aR
,
then the value of p is
Ans : 7
2
203/222
2 14 23
iRB ai R R
(n = 2)
22 20 0
9/21
2 22aM R a
18. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the
figure. Both the cylinder and the cavity are infinetely long. A uniform current density J flows
along the length. If the magnitude of the magnetic field at the point P is given by 012N aJ ,
then the value of N is
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Ans : 5
01 2
jaB
2
200
22
3 1222
ajjaB
a a
1 2 01 12 12
B B ja
0 512ja
19. A proton is fired from very far away towards a nucleus with charge 120Q e , where e is the
electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wave-
length (in units of fm) of the proton at its start is:
(take the proton mass, 275 / 3 10 ;pm kg 15/ 4.2 10 . /H e j S c ;
9
0
1 9 10 / ;4
m F
151 10fm m )
Ans : 7
2 29
151209 10
10 10 2 p
e Pm
9 2 27
215
59 10 120 2 103
10 10
eP
17
153 2 10 3 2
10 10e eP
154.2 106
10
hp
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20. An infinitely long solid cylinder of radius R has a uniform volume charge density . It has a
spherical cavity of radius R/2 with its centre on the axis on the cylinder, as shown in the
figure. The magnitude of the electric field at the point P, which is at the distance 2R from the
axis of the cylinder, is given by the expression 0
2316
Rk . The value of k is
Ans: 6
3
2
2
42 3 8
2 2 p
RKk R ER R
4 113 32pE k R
0
1 114 24
R
0
234 24p
RE
24 4 616
K
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CHEMISRY SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct21. The number of aldol reaction(s) that occurs in the given transformation is
. .3 4 conc aq NaOHCH CHO CHO
OH
OHHO
OH
a) 1 b) 2 c) 3 d) 4Ans : C
3 2OHCH CHO CH CHO
2 2 2H C H CH CHO O CH CH CHO
O
H
2 2OHCH CHO HO CH CH CHO H - C - H
OH
'HCHO Cannizaro sNaOH reactionCH CH
CHO
OH
OH CHO
HO
OH
OH OH OH
OHHO
Number of aldol reaction = 3
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22. A compound M PXP has cubic close packing (ccp) arrangement of X. Its unit cell structure isshown below. The empirical formula of the compound is
a) MX b) MX2 c) M2X d) M5X14
Ans : B
Corner
Face center 16 32
18 18
4
X :
Edge centre
Body center 1 1 1
14 14
2
M :
4 2X M
4 2X M 2MX
23. The carbonyl functional group (–COOH) is present ina) picric acid b) barbituric acid c) ascorbic acid d) aspirin
Ans : DAcetyle salicylic acid = Asprin
COOH
OCOCH3
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24. The colour of light absorbed by an aqueous solution of CuSO4 is :
a) organe-red b) blue-green c) yellow d) violetAns : A
Orange - red is absorbedBased up on Munshell wagon wheel
V
B
G
O
R
Y
25. The number of optically active products obtained from the complete ozonolysis of thegiven compound is :
CH3 - CH = CH - C - CH = CH - C - CH = CH - CH3
HCH3
H CH3
a) 0 b) 1 c) 2 d) 4Ans : A
As the ozonolysis products do not have chiral centres, number of optically active products = 0
26. As per IUPAC nomenclautre, the name of the complex 2 3 34 2Co H O NH Cl is
a) Tetraaqudiaminecobalt (III) chloride b) Tetraaquadiamminecobalt (III) chloridec) Diammineteraaquacobalt (III) chloride d) Diamminetetraaquacobalt (III) chloride
Ans : D
2 3 34 2Co H O NH Cl
Diammineteraaquacobalt (III) chloride
27. Which ordering of compound is according to the decreasing order of the oxidation state ofnitrogen ?
a) 3 4 2, , ,HNO NO NH Cl N b) 3 2 4, , ,HNO NO N NH Cl
c) 3 4 2, , ,HNO NH Cl NO N d) 3 4 2, , ,NO HNO NH Cl N
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Ans : B
3 5HNO
2NO
2 0N
4 3NH Cl
Decreasing order of
O -S = 3 2 4, , ,HNO NO N NH Cl
28. For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. I/V plot is shownbelow. The value of the van der Waals constant a (atm. liter2 mol–2) is
a) 1.0 b) 4.5 c) 1.5 d) 3.0Ans : C
2
2
n aP v nb nRTv
’
n = 1, b = 0
2
qp v b RTv
apv RTv
apv RTv
2 221.6 20.1 1.5 .3 2
y c mx m a slop atm t mole
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29. In allene 3 4C H , the type(s) of hybridisation of the carbon atoms is (are)
a) sp and sp3 b) sp and sp2 c) only sp2 d) sp2 and sp3
Ans : B
2 2H C C CH
2 2sp sp sp
30. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0 is Bohrradius]
a) 2
2 204
hma b)
2
2 2016
hma c)
2
2 2032
hma d)
2
2 2064
hma
Ans : C
mvr= 2nh
v= 2nh
mr
for second Bohr’s orbit
2r =4 0a
V=04
hma
22
2 2 2016
hVm a
K.E2
22 2
0
12 32
hmvma
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SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
31. Which of the following molecules, in prue form, is (are) unstable at room temperature ?
a) b) c)
O
d)
O
Ans : BC
cyclobutadiene and cyclopentadien are anti aromatic.
32. Which of the following hydrogen halides react(s) with 3AgNO aq to give a precipitate that
dissolves in 2 2 3Na S O aq ?
a) HCl b) HF c) HBr d) HI
Ans : ACD
3 3AgNO HX AgX HNO aq aq
2 2 3 3 2 3 22AgX Na S O Na Ag S O NaX aq aqs
AgF is soluble
33. Choose the correct reason(s) for the stability of the lyophobic colloidal particles >
a) Preferential adsorption of ions on their surface from the solution
b) Preferential adsorption of solvent of their surface the solution
c) Attraction between different particles having opposite charges on their surface
d) Potential difference between the fixed layer and the diffused layer of opposite chargesaround the coolidal particles
Ans : AD
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34. For an ideal gas, consider only P-V work in going from an
ini tial state X to the f inal state Z . The can be reached by
either of the two paths shown in the figure. Which of the
following choice(s) is (are) correct ? [Take S as change
in entropy and w as work done]
a) x z x y y zS S S b) x z x y y zW w w
c) x y z x yw w d) x y z x yS S
Ans : AC
Based up on the given graph
x y x y y zS S S
x y z x yW W
35. Identify the binary mixtrue(s) that can be separated into individual compounds, by differen-tial extraction,a s shown in the given scheme.
a) 6 5 6 5C H OH and C H COOH b) 6 5 6 5 2C H COOH and C H CH OH
c) 6 5 2 6 5C H CH OH and C H OH d) 6 5 2 6 5 2C H CH OH and C H CH COOH
Ans : BD
Benzyl alcohol is insoluble in both NaOH and 3NaHCO ,
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SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).
36. When the following aldohexose exists in its B-configuration, the total number C
CH2
CHOH
CHOH
CHO
CH2OH
OHH
of stereoisomrs in its pyranose form is :
Ans : 8
As the configration at C-5 is fixed, (D-configration), then only 4 stero isomers are possiblesolution
37. 29.2% (w/w) HCl stock solution has a densityh of 1.25 g mL–1. The molecular weight of HCl
is 36.5 g mol–1. The volume (mL) of stock solution required to prepare a 200 mL solution of
0.4 M HCl is :
Ans : 8
Molarity of stock solution =10 d x
GMW
=10 1.25 29.2
36.5
=10M
1 1 2 2
1
1
10 200 0.48 ( )
V M V MVV ML approx
38. The periodic table consists of 18 groups. An isotope of copper, on bombardment with pro-
tons, undergoes a nuclear reaction yielding element X as shown below. To which
group,element X belongs int he periodic table ?
Ans : 863 1 1 4 1 5229 1 0 2 1 266 2CU H n He H X
26 Fe 8 th group
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39. An organic compound undergoes first-order decomposition. The time taken for its decom-
position to 1/8 and 1/10 of its initial concentration are t1/8 and t1/10 respectively. What is the
Ans : 9
18
110
2.303 log
82.303 log
10
at ak
at ak
18
110
log8log10
t
t
18
110
3log 2t
t
18
110
10 9t
t
40. The substituents R1 and R2 for nine peptides are listed in the table given below. How many of
these peptides are positively charged at pH = 7.0 ?
Ans : 4
At pH below pI(Isoelectric point) Aminoacid exist as Cation. As BasicAminoacids of pHaround 9-8, these are expected to exists as cation of pH=7IV,VI,VIII,IX.
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MATHEMATICSSECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
41. Let z be a complex number such that the imaginary part of z is nonzero and and 2 1a z z is real. Then a connot take the value
a) –1 b) 13 c)
12 d)
34
Ans. D
Let Z x iy 0y
2 1a z z becomes
2 1a x iy x iy
2 2 1 2 0x y x i xy y a i
Comparing real & imaginary parts
2 2 1x y x a ––––––(1)
2 0xy y –––––––(2)
From (2) 1
2x as 0y
Put value of x in (1)
221 1 1
2 2y a
21 14 2
y a
2 3 04
y a
34
a
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42. If 2 1lim 4
1x
x x ax bx
, then
a) a = 1, b = 4 b) a = 1, b = –4 c) a = 2, b = –3 d) a = 2, b = 3
Ans. B
2 1lim 41x
x x ax bx
2 1 1 14
1x
x a x a b blim
x
For existense of limit coefficient of 2 0x
1 – a = 0
a = 1
1 1lim 4
1x
x a b bx
1 4a b
b = – 4
43. Let ijP a be a 3 3 matrix and let ijQ b , where 2i jij ijb a for 1 , 3i j . If the deter-
minant of P is 2, then the determinant of the matrix Q is
a) 210 b) 211 c) 212 d) 213
Ans. D
as 2i jij ijb a
11 11 12 12 13 134 , 8 , 16b a b a b a
21 2 22 22 23 238 , 16 , 32b a b a b a
31 31 32 32 33 3316 , 32 , 64b a b a b a
11 12 13
21 22 23
31 32 33
2a a a
A a a aa a a
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11 12 13
21 22 23
31 32 33
4 8 168 16 32
16 32 64
a a aB a a a
a a a
11 12 13
21 22 23
31 32 33
4 8 16 2 2 24 4 4
a a aa a aa a a
11 12 13
21 22 23
31 32 33
4 8 16 2 4a a aa a aa a a
4 8 16 2 4 2
132B
44. The ellipse 2 2
1 : 19 4x yE is inscribed in a rectangle R whose sides are parallel to the coor-
dinates axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangleR. The eccentricity of the ellipse E2 is
a) 22
b) 32
c) 12 d)
34
Ans. C
Given Ellipse 2 2
19 4x y
As ellipse is bounded by &x a y b
Sides of rectangle are
3 & 2x y
3,2 , 3,2C D
D(0,2)
(3,0)
(0,-2)
(-3,0)
A B
C
3, 2 , 3, 2A B
As new ellipse
Circumscribe the rectangle & passes through (0 ,4).
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Let new ellipse is 2 2
2 2 1x ya b
Passes through 22
160,4 1 16bb
Passes through 3,2 2
9 4 116a
2 12a
So, Vertical ellipse 2 2
112 16x y
2 2 21a b e
212 16 1 e
2 12 1116 4
e
12
e
45. The function : 0,3 1, 29f , defined by 3 22 15 36 1f x x x x is
a) one-one and onto b) onto but not one-one
c) one-one but not onto d) neither one-one nor onto
Ans. B
: 0,3 1, 29f
3 22 15 36 1f x x x x
' 26 30 36f x x x
6 2 3x x
'' 6 2 5f x x
2x is pt of maxima.
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3x is pt of minima.
& 2 29 0f
3 0f
so curve is
nf is not one one
2 3Min value 0 1f
Max value 2 29f
so nf is Onto.
46. The locus of the mid-point of the chord of contact of tangents drawn from points lying onthe straight line 4 5 20x y to the circle 2 2 9x y is
a) 2 220 36 45 0x y x y b) 2 220 36 45 0x y x y
c) 2 236 20 45 0x y x y d) 2 236 20 45 0x y x y
Ans. A
Let 1 1, & ,P h k Q x y
As P lies of 4x-5y=20
4 20,5
hP h
P
A
B
Q
Equation of chord of contact
1 0AB S
4 20 95
hxh y
5 4 20 45xh y h ------------------(1)
Equation of chord AB whose mid point is 1 1,Q x y
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2 21 1 1 1xx yy x y ---------------(2)
Compare (1) & (2)
2 21 1 1 1
5 4 20 45h hx y x y
1 12 2 2 21 1 1 1
9 45& 4 20x yh hx y x y
On eliminating h
1 12 2 2 21 1 1 1
49 4520x yx y x y
2 21 1 1 120 36 45x y x y
Locus is
2 220 36 45 0x y x y
47. The point P is the intersection of the straight line joining the points Q (2, 3, 5) and
R (1, –1, 4) with the plane 5 4 1x y z . If S is the foot of the perpendicular drawn from thepoint T (2, 1, 4) to QR, then the length of the line segement PS is
a) 12 b) 2 c) 2 d) 2 2
Ans. A
Equation of 1 1 4
1 4 1x y zQR
Dr’s of 1, 4,1QR
Co-ordinates of 1 , 1 4 , 4P
As P is point of Intersection of QR & given plane, so
5 1 4 1 4 4 1
13
T(2,1,4)
Q S R
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4 1 13, ,3 3 3
P
Let 1 , 1 4 ,4S
Dr’s of 1, 4 2,TS
TS QR So,
1 1 4 4 2 0
12
3 9,1,2 2
S
2 2 24 3 1 13 913 2 3 3 2
PS
12
48. Let 2 cos , 0
0, 0
x xf x x
x
, x IR , then f is
a) differentiable both at x = 0 and at x = 2
b) differentiable at x = 0 but not differentiable at x = 2
c) not differentiable at x = 0 but differentiable at x = 2
d) differentiable neither at x = 0 nor at x = 2
Ans. B
2 cos ; 0
0 ; 0
x xf x x
x
At x=0
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2
0 0
cos 00 0lim limh h
hf h f hRHDh h
0lim cos 0h
hh
2
0 0
cos 00 0
lim limh h
hf h f hLHD
h h
0lim cos 0h
hh
Differentiable at x=0
At x=2
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hRHDh h
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hLHDh h
Not Differentiable at x=2.
49. The total number of ways in which 5 balls of different colours can be distributed among 3persons so that each person gets at least one ball is
a) 75 b) 150 c) 210 d) 243
Ans. B
5 Balls to 3 persons, so that all persons get atleast one ball.
Case-I
Total ways 2 2 1
2 1 2
1 2 2
5 3 12 2 1 3C C C
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10 3 1 3 90
Case-II
Total ways
5 2 13 1 1 3C C C
11
1
13
3
1
3110 2 1 3 60
Total ways = 90 + 60 = 150.
50. The integral
2
9/ 2sec
sec tanx dx
x x equals (for some arbitrary constant K)
a) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
b) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
c) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
d) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
Ans. C
9
2
sec
sec tan
x dxx x
Let sec tanx x t -------------(1)
So, 2sec tan secx x x dx dt
sec dtx dxt
------------(2)
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1sec tanx xt
----------(3)
From (1) & (3)
1 1sec2
x tt
So, Integral becomes
9
2
sec .sec
sec tan
x x dxx x
92
1 12
dtt ttt
2
132
1 12
t dtt
9 132 21
2t dt t dt
7 112 21
7 112 22
t t K
7 112 2
7 11t t K
7 11
22
1 1
11 sec tan7 sec tan x xx x
2
112
1 1 1 sec tan11 7sec tan
x x Kx x
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SECTION – II( MULTIPLE CORRECT CHOICE TYPE )
This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONE OR MORE is/ are correct
51. A ship is fitted with three engines E1 , E2 and E3. The engines function independently of each
other with respective probabilities 1 1,2 4 and
14 . For the ship to be operational at least two of
its engines must function. Let X denote the event that the ship is operational and let X1, X2and X3
denote respectively the events that the engines E1, E2 and E3 are functioning. Which ofthe following is (are) true ?
a) 13|
16cP x X b) P [Exactly two engines of the ship are functioning | X]
78
c) 25|
16P X X d) 1
7|16
P X X
Ans. B,D
A)
1
1 1 112 4 4
1 84
cP X XP X
1 1 3 1 1 1 12 22 4 4 2 4 4 4
P X
B) 1 1 3 1 1 12
72 4 4 2 4 4/ 1 84
P Exactly twoengines of the ship are functioning X
C) 2
22
1 1 3 1 1 1 252 4 4 2 4 4/ 1 8
4
P X XP X X
P X
D)
11
1
1 1 3 1 1 1272 4 4 2 4 4/ 1 16
2
P X XP X X
P X
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52. Tangents are drawn to the hyperbola 2 2
19 4x y
, parallel to the straight line 2 1x y . The
points of contact of the tangents on the hyperbola are
a) 9 1,
2 2 2
b) 9 1,
2 2 2
c) 3 3, 2 2 d) 3 3,2 2
Ans. A, B
2 2
19 4x y
Tangent to the hyperbola is of the form2y x
2 9 4 4 36 4 32
4 2
Tangent is : 2 4 2 0x y
Point of contact is 2 2 9 1 9 1, , , ,
2 2 2 2 2 2a l b mn n
53. Let S be the area of the region enclosed by 2xy e , y = 0, x = 0, and x = 1. Then
a) 1Se
b) 11Se
c) 1 114
Se
d)
1 1 112 2
Se
Ans. A, B, D
21 1
0 0
11x xI e dx e dxe
11Ie
21 1xe e
21 1 1
1
0 0 0
xe dx e dx dx
21
0
1xe dxe
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54. If y (x) satisfies the differential equation ' tan 2 secy y x x x and y (0) = 0, then
a) 2
4 8 2y
b) 2
'4 18
y
c) 2
3 9y
d) 24 2'
3 3 3 3y
Ans. A,D
tan 2 secdy y x x xdx
tan. cos
dxI F e x
General solution is cos 2 sec cosy x x x x dx 2x c
2 sec secy x x c x
0,0
0 0 0c c .
2 secy x x
2
24 16
y
2
8 2
2
23 9
y
2 sec tan 2 secdy x x x x xdx
.
2
1 2 3 2 23 9 3
y
22 433 3
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55. Let , 0, 2 be such that
22cos 1 sin sin tan cot cos 12 2
,
tan 2 0 and 31 sin2
.
Then cannot satisfy
a) 0 2 b)
42 3 c)
4 33 2 d)
3 22
Ans. A, C, D
3tan 2 0, 1 sin2
3 5 1, 0 cos2 3 2
______ (1)
22cos 1 sin sin tan / 2 cot / 2 cos 1
1sin cos2
1 sin 12
(from (1))
13 176 6
2 73 6 (from (1))
cannot satisfied by A, C, D
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SECTION –III( INTEGER ANSWER TYPE)
This section contains 5 questions . The answer to each of the questions is a single digit integer, ranging from 0 to9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened.
56. Let p (x) be a real polynomial of least degree which has a local maximum at x = 1 and a localminimum at x =3. If p (1) = 6 and p (3) = 2, then ' 0p is
Ans. 9
3 2P x ax bx cx d
1 6 6P a b c d
3 2 27 9 3 2P a b c d
1 1 0 3 2 0P a b c
1 3 0 27 6 0P a b c
1, 6, 9a b c
1 23 2P x ax bx c
1 0 9P c
57. If ,a b and c are unit vectors satisfying
2 2 2 9a b b c c a ,then 2 5 5a b c
is\
Ans. 3
sol. 1a b c
2 2 2 9a b b c c a
3.
2a b b c c a
3 2 . . 0a b b c c a
22 2 2 . . . 0a b c a b b c c a
2
0a b c
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0a b c ––––––(1)
Taking dot product with (1) by , &a b c
then 1 1 1. , . , .
2 2 2a b b c c a
2 5 5 3a b c
58. The value of 32
1 1 1 16 log 4 4 4 ...3 2 3 2 3 2 3 2
is
Ans. 4
sol.1 4
3 2x x
3 2 4x x
squaring on both sides218 4x x 218 4 0x x 218 9 8 4 0x x x
9 2 1 4 2 1 0x x x .
2 1 9 4 0x x
1 4,2 9
x
49
x 0x
Required answer is 2
3/ 236 log2
= 6 – 2 = 4
59. Let S be the focus of the parabola 2 8y x and let PQ be the common chord of the circle2 2 2 4 0x y x y and the given parabola. The area of the triangle PQS is
Ans. 4
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sol.P
Q
S(0, 0)
(2,4)
(2, 0)2
(0,4)
2 8 2 4 2 0x x x x
0,2 0,0 , 2,4x P Q
area of SPQ
= 1 2 42
= 4
60. Let :f IR IR be defined as 2 1f x x x . The total number of points at which f attainseither a local maximum or a local minimum is
Ans. 5sol. 2 1, 1f x x x x
21 , 1 0x x x
21 , 0 1x x x
2 1, 1x x x
' 2 1, 1f x x x
2 1, 1 0x x
0-1 1x
y
1 2 , 0 1x x
1 2 , 1x x
differentiable at 1 1,2 2
x ,
not differentiable at –1, 0, 1
The number of points = 5
Sri Chaitanya IIT Academy., A.P.
IIT-JEE 2012Paper - 1 along with Key
& Solutions
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INSTRUCTIONSA. General
1. This booklet is your Question Paper. Do not break the seals of this boolket before being instructed to do so by the
invigilators.
2. The question paper CODE is printed on the right hand top corner of this page and on the back page (Page No. 28)
of this booklet.
3. Blank spaces and blank pages are provided in this booklet for your rough work. No additional sheets will be
provided for rough work.
4. Blank papers, clipboards , log tables, slide rules, calculators, cameras, cellular phones, pagers and electronic
gadgets are NOT allowed inside the examination hall.
5. Answer to the questions and personal details are to be filled on a two-part carbon-less paper, which is provided
separately. You should not separate these parts. The invigilator will separate them at the end of examination. The
upper sheet is a machine-gradable Ob jective Response Sheet (ORS) which will be taken back up the invigilator.
You will be allowed to take away the bottom sheet at the end of the examination.
6. Using a black ball point pen, darken the bubbles on the upper original sheet. Apply sufficinet pressure so that the
impression is created on the bottom sheet.
7. DO NOT TAMPER WITH / MUTILATE THE ORS OR THE BOOKLET.
8. On breaking the seals of the booklet check that it contains 28 pages and all the 60 questions and correspond-
ing answer choices are legible. Read carefully the instructions printed at the beginining of each section.B. Filling the Right Part of the ORS
9. The ORS has CODES printed on its left and right parts.
10. Check that the same CODE is printed on the ORS and on this boolet. IF IT NOT THEN ASK FOR A CHANGE OF THEBOOKLET. Sign at the place provided on the ORS affirming that you have verified that all the codes are same.
11. Write your name , Registration Number and the name of examination centre and sign with pen in the boxes providedon the right part of the ORS. Do not write any of this information anywhere else. Darken the appropriate bubbleUNDER each digit of your Registration Number in such a way that the impression is created on the bottom sheet. Also
darken the paper CODE given on the right side of ORS (R4)
C Question Paper FormatThe questions paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part consist of three sections.
12. Section I contains 10 multiple choice questions. Each question has four choices. (A), (B), (C) and (D) out of
which ONLY ONE is correct.
13. Section II contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which
ONE or MORE are correct.
14. Section III contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both
inclusive).
D.Marking Scheme15. For each question in Section I, you will be awarded 3 marks if you darken the bubble corresponding to the correct
answer ONLY and zero marks if no bubbles are darkened. In all ohter cases, minus one (–1) mark will be awarded
in this section.
16. For each question in Section II, you will be awareded 4 marks if you darken ALL the bubble(s) corresponding to the
correct answer(s) ONLY. In all other cases zero (0) marks will be awarded. No negative marks will be awarded for
incorrect answers in this section.
17. For each question in Section III, you will be awarded 4 marks if you darken the bubble corresponding to the correct
answer ONLY. In all other cases zero (0) will be awarded. No negative marks will be awarded for incorrect answers
in this section.
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PHYSICS SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
1. A small block is connected to one end of massless spring of un–stretched length 4.9 m. The
other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless
surface. The block is stretched by 0.2 m and released from rest at t= 0. It then executes
simple harmonic motion with angular frequency /3
rad s . Simultaneously at t = 0, a
small pebble is projected with speed v from point P at an angle of 450 as shown in the figure.
Point P is at a horizontal distance to 10 m from O. If the pebble hits the block at t = 1s, the
value of v is (take g = 10 m/s2)
a) 50 /m s b) 51 /m s c) 52 /m s d) 53 /m s
Ans : A
Equation of SHM cosy A t
At 1t 0.2cos (1) 0.2cos 0.13
y m
Block is at a distance 5m
Range of pebble is R =5m
22sin90 50
10VR V
50 /V m s
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2. In the determination of Young’s modulus 24MLgY
ld
by using Searle’s method, a wire of
length L = 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension
l = 0.25 mm in the length of the wire is observed. Quantities d and l are measured using a
screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The num-
ber of divisions on their circular scale is 100. The contributions to the maximum probable
error of the Y measurement.
a) due to the errors in the measurements of d and l are the same
b) due to the error in the measurement of d is twice that due to the error in the measurement of l.
c) due to the error in the measurement of l is twice that due to the error in the measurement of d.
d) due to the error in the measurement of d is four times that due to the error in the measure-
ment of l.
Ans :A
2
2 2 4d d dd v
40.25
l l ll
3. A small mass m is attached to a massless string whose other end is fixed at P as shown in the
figure. The mass is undergoing circular motion in the x–y plane with centre at O and constant
angular speed . If the angular momentum of the system, calculated about O and P are de-
noted by 0L
and PL
respectively, then
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a) 0L
and PL
do not vary with time
b) 0L
varies with time while PL
remains constant
c) 0L
remains constant while PL
varies with time
d) 0L
and PL
both vary with time
Ans :C
Torque about O is zero angular momentum will be constant about ‘O’. Torque about P is nonzero. Angular momentum about P varies with time.
0L m r v
PL m OP v
direction of PL
changes with OP and PL OP
4. Young’s double slit experiment is carried out by using green, red and blue light, one color at
a time. The fringe widths recorded are ,G R and B , respectively. Then
a) G B R b) B G R c) R B G d) R G B
Ans :D
Dd
R G B
R G B
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5. Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform
positive surface charge density. The variation of the magnitude of the electric field E r
and the electric potential V(r) with the distance r from the centre, is best represented bywhich graph ?
a) b)
c) d)
Ans : D
From centre to surface E = 0, then 1Er
From centre to surface V = k, then 1Vr
6. A bi–convex lens is formed with two thin plano–convex lenses as shown in the figure. Re-fractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curvedsurfaces are of the same radius of curvature R = 14 cm. For this bi–convex lens, for an objectdistance of 40 cm, the image distance will be
a) – 280. 0 cm b) 40.0 cm c) 21. 5 cm d) 13.3 cm
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Ans :B
1 2
1 1 1F f f =
1 21 1R
=
0.5 0.2 0.7 114 14 20
20F cm
40R cm
1 1 1 1 1 1 1 1 1 2 1 120 40 20 40 40 40F v u v v
40v cm
7. Two large vertical and parallel metal plates having as separation of 1 cm are connected to aDC voltage source of potential difference X. A proton is released at rest midway between thetwo plates. It is found to move at 450 to the vertical JUST after release. Then X is nearly
a) 51 10 V b) 71 10 V c) 91 10 V d) 101 10 V
Ans :C
mg Eq
45
mg
EqpVqd
910mgdV V approxq
8. Three very large plates of same area are kept parallel and close to each other. They are con-sidered as ideal black surfaces and have very high thermal conductivity. The first and thirdplates are maintained at temperatures 2T and 3T respectively. The temperature of the middle(i.e. second) plate under steady state condition is
a) 1465
2T
b) 1497
4T
c) 1497
2T
d) 1497 T
Ans : C
4 4 4 41 2 2 3A T T A T T
144 44 4
2 2 2972 32
T T T T T T
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9. A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angularspeed , as shown in the figure. At time t = 0, a small insect starts from O and moves withconstant speed v with respect to the rod towards the other end. It reaches the end of the rod att = T and stops. The angular speed of the system remains throughout. The magnitude of the
torque of the torque on the system about O, as a function of time is best represented bywhich plot ?
a) b)
c) d)
Ans : B
dL d dIIdt dt dt
constantrodI
2red
d I mrdt
= 0 2 drm rdt
v
r = vt
22 2mr v m v t
t
when the insect reaches the end it stops v = 0
0
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10. A mixture of 2 moles of helium gas (atomic mass = 4amu) and 1 mole of argon gas (atomic
mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds heliumargon
rms
rms
vv
is
a) 0.32 b) 0.45 c) 2.24 d) 3.16Ans : D
rmsTVM
1 2
2 1
40 104
He
Ar
VV MV V M
SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
11. A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle with the horizontal. A horizontal force of 1 N acts on the block through its center of mass asshown in the figure. The block remains stationary if (take 210 /g m s )
a) 045
b) 045 and a frictional force acts on the block towards P.
c) 045 and a frictional force acts on the block towards Q.
d) 045 and a frictional force acts on the block towards Q.
Ans : A (or) AC
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1cos sinmg
45
when 45
friction force is upwards
If friction force sufficient block will be in state of rest
12. Consider the motion of a positive point charge in a region where there are simulataneousuniform electric and magnetic fields 0
ˆE E j
and 0ˆB B j
. At time t = 0 , this charge has
velcoity v in the x – y plane, making an angle with the x – axis. Which of the followingoption(s) is(are) correct for time t > 0 ?
a) If 00 , the charge moves in a circular path in the x – z plane
b) If 00 , the charge undergoes helical motion with constant pitch along the y – axis.
c) If 010 , the charge undergoes helical motion with its pitch increasing with time, along the y – axis
d) If 090 , the charge undergoes linear but accelerated motion along the y–axis
Ans :C,D
when 90
0mF
when 10
path is helical and with increasing pitch.
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13. For the resistance network shown in the figure, choose the correct option(s).
a) The current through PQ is zero b) 1 3I A
c) The potential at S is less than that at Q. d) 2 2I A
Ans : A,B,C,D
112 34
I
212 318
I 2A
P QV V
S QalsoV V
14. A person blows into open – end of a long pipe. A s a result, a high – pressure pulse of airtravels down the pipe.When this pulse reaches the other end of the pipe,
a) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
b) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is open
c) a low – pressure pulse starts travelling up the pipe, if the other end of the pipe is closedd) a high – pressure pulse starts travelling up the pipe, if the other end of the pipe is closed
Ans : B, D
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Reflected pressure pulse
Incident pressure pulse
Open end
At open end pressure should not change pressure wave will get phase change as
Reflected pressure pulse
Incident pressure pulse
Closed end
At enclosed end pressure wave will get reflected without phase change15. A cubical region of side a has its centre at the origin. It encloses three fixed point charges,
q at 0, / 4,0 , 3a q at (0, 0, 0) and q at 0, / 4,0a . Choose the correctoption(s).
a) The net electric flux crossing the plane / 2x a is equal to the net electric flux crossingthe plane / 2x a
b) The net electric flux crossing the plane / 2y a is more than the net electric flux crossingthe plane / 2y a .
c) The net electric flux crossing the entire region is 0
q .
d) The net electric flux crossing the plane / 2z a is equal to the net electric flux crossingthe plane / 2x a
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Ans : ACD
Net flux passing the plane 2ax and 2
ax is same
Net flux through cube is 0
q
Flux through plane 2az and 2
ax is same
SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).16. A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform
mass density and radius 2R, as shown in the figure. The moment of inertia of this laminaabout axes passing through O and P is 0I and PI , respectively. Both these axes are perpen-
dicular to the plane of the lamina. The ratio 0
PII to the nearest integer is
Ans : 3
2 2 2 2 2 20
1 12 22 2
I R R R R R R
4 4382
R R
4 213 132 2
R MR
2 2 2 21 14 2 4 2 52 2PI M R M R MR M R
2 2 2118 162
MR MR MR
2 211 37242 2
MR MR
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17. A circular wire loop of radius R is placed in the x - y plane centered at the origin O. A squareloop of side a a R having two turns is placed with its center at 3z R along the axis ofthe circular wire loop, as shown in figure. The plane of the square loop makes an angle of
45with respect to the z - axis. If the mutual inductance between the loops is given by 2
0/ 22p
aR
,
then the value of p is
Ans : 7
2
203/222
2 14 23
iRB ai R R
(n = 2)
22 20 0
9/21
2 22aM R a
18. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the
figure. Both the cylinder and the cavity are infinetely long. A uniform current density J flows
along the length. If the magnitude of the magnetic field at the point P is given by 012N aJ ,
then the value of N is
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Ans : 5
01 2
jaB
2
200
22
3 1222
ajjaB
a a
1 2 01 12 12
B B ja
0 512ja
19. A proton is fired from very far away towards a nucleus with charge 120Q e , where e is the
electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wave-
length (in units of fm) of the proton at its start is:
(take the proton mass, 275 / 3 10 ;pm kg 15/ 4.2 10 . /H e j S c ;
9
0
1 9 10 / ;4
m F
151 10fm m )
Ans : 7
2 29
151209 10
10 10 2 p
e Pm
9 2 27
215
59 10 120 2 103
10 10
eP
17
153 2 10 3 2
10 10e eP
154.2 106
10
hp
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20. An infinitely long solid cylinder of radius R has a uniform volume charge density . It has a
spherical cavity of radius R/2 with its centre on the axis on the cylinder, as shown in the
figure. The magnitude of the electric field at the point P, which is at the distance 2R from the
axis of the cylinder, is given by the expression 0
2316
Rk . The value of k is
Ans: 6
3
2
2
42 3 8
2 2 p
RKk R ER R
4 113 32pE k R
0
1 114 24
R
0
234 24p
RE
24 4 616
K
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CHEMISRY SECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct21. The number of aldol reaction(s) that occurs in the given transformation is
. .3 4 conc aq NaOHCH CHO CHO
OH
OHHO
OH
a) 1 b) 2 c) 3 d) 4Ans : C
3 2OHCH CHO CH CHO
2 2 2H C H CH CHO O CH CH CHO
O
H
2 2OHCH CHO HO CH CH CHO H - C - H
OH
'HCHO Cannizaro sNaOH reactionCH CH
CHO
OH
OH CHO
HO
OH
OH OH OH
OHHO
Number of aldol reaction = 3
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22. A compound M PXP has cubic close packing (ccp) arrangement of X. Its unit cell structure isshown below. The empirical formula of the compound is
a) MX b) MX2 c) M2X d) M5X14
Ans : B
Corner
Face center 16 32
18 18
4
X :
Edge centre
Body center 1 1 1
14 14
2
M :
4 2X M
4 2X M 2MX
23. The carbonyl functional group (–COOH) is present ina) picric acid b) barbituric acid c) ascorbic acid d) aspirin
Ans : DAcetyle salicylic acid = Asprin
COOH
OCOCH3
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24. The colour of light absorbed by an aqueous solution of CuSO4 is :
a) organe-red b) blue-green c) yellow d) violetAns : A
Orange - red is absorbedBased up on Munshell wagon wheel
V
B
G
O
R
Y
25. The number of optically active products obtained from the complete ozonolysis of thegiven compound is :
CH3 - CH = CH - C - CH = CH - C - CH = CH - CH3
HCH3
H CH3
a) 0 b) 1 c) 2 d) 4Ans : A
As the ozonolysis products do not have chiral centres, number of optically active products = 0
26. As per IUPAC nomenclautre, the name of the complex 2 3 34 2Co H O NH Cl is
a) Tetraaqudiaminecobalt (III) chloride b) Tetraaquadiamminecobalt (III) chloridec) Diammineteraaquacobalt (III) chloride d) Diamminetetraaquacobalt (III) chloride
Ans : D
2 3 34 2Co H O NH Cl
Diammineteraaquacobalt (III) chloride
27. Which ordering of compound is according to the decreasing order of the oxidation state ofnitrogen ?
a) 3 4 2, , ,HNO NO NH Cl N b) 3 2 4, , ,HNO NO N NH Cl
c) 3 4 2, , ,HNO NH Cl NO N d) 3 4 2, , ,NO HNO NH Cl N
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Ans : B
3 5HNO
2NO
2 0N
4 3NH Cl
Decreasing order of
O -S = 3 2 4, , ,HNO NO N NH Cl
28. For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. I/V plot is shownbelow. The value of the van der Waals constant a (atm. liter2 mol–2) is
a) 1.0 b) 4.5 c) 1.5 d) 3.0Ans : C
2
2
n aP v nb nRTv
’
n = 1, b = 0
2
qp v b RTv
apv RTv
apv RTv
2 221.6 20.1 1.5 .3 2
y c mx m a slop atm t mole
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29. In allene 3 4C H , the type(s) of hybridisation of the carbon atoms is (are)
a) sp and sp3 b) sp and sp2 c) only sp2 d) sp2 and sp3
Ans : B
2 2H C C CH
2 2sp sp sp
30. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0 is Bohrradius]
a) 2
2 204
hma b)
2
2 2016
hma c)
2
2 2032
hma d)
2
2 2064
hma
Ans : C
mvr= 2nh
v= 2nh
mr
for second Bohr’s orbit
2r =4 0a
V=04
hma
22
2 2 2016
hVm a
K.E2
22 2
0
12 32
hmvma
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SECTION – II( MULTIPLE CORRECT ANSWER TYPE)
This section contains 5 multiple choicse questions. Each question has 4 choices (A), (B), (C) and (D) for its answer,out of which ONE or More are correct.
31. Which of the following molecules, in prue form, is (are) unstable at room temperature ?
a) b) c)
O
d)
O
Ans : BC
cyclobutadiene and cyclopentadien are anti aromatic.
32. Which of the following hydrogen halides react(s) with 3AgNO aq to give a precipitate that
dissolves in 2 2 3Na S O aq ?
a) HCl b) HF c) HBr d) HI
Ans : ACD
3 3AgNO HX AgX HNO aq aq
2 2 3 3 2 3 22AgX Na S O Na Ag S O NaX aq aqs
AgF is soluble
33. Choose the correct reason(s) for the stability of the lyophobic colloidal particles >
a) Preferential adsorption of ions on their surface from the solution
b) Preferential adsorption of solvent of their surface the solution
c) Attraction between different particles having opposite charges on their surface
d) Potential difference between the fixed layer and the diffused layer of opposite chargesaround the coolidal particles
Ans : AD
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34. For an ideal gas, consider only P-V work in going from an
ini tial state X to the f inal state Z . The can be reached by
either of the two paths shown in the figure. Which of the
following choice(s) is (are) correct ? [Take S as change
in entropy and w as work done]
a) x z x y y zS S S b) x z x y y zW w w
c) x y z x yw w d) x y z x yS S
Ans : AC
Based up on the given graph
x y x y y zS S S
x y z x yW W
35. Identify the binary mixtrue(s) that can be separated into individual compounds, by differen-tial extraction,a s shown in the given scheme.
a) 6 5 6 5C H OH and C H COOH b) 6 5 6 5 2C H COOH and C H CH OH
c) 6 5 2 6 5C H CH OH and C H OH d) 6 5 2 6 5 2C H CH OH and C H CH COOH
Ans : BD
Benzyl alcohol is insoluble in both NaOH and 3NaHCO ,
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SECTION – III( INTEGER ANSWER TYPE)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9(both inclusive).
36. When the following aldohexose exists in its B-configuration, the total number C
CH2
CHOH
CHOH
CHO
CH2OH
OHH
of stereoisomrs in its pyranose form is :
Ans : 8
As the configration at C-5 is fixed, (D-configration), then only 4 stero isomers are possiblesolution
37. 29.2% (w/w) HCl stock solution has a densityh of 1.25 g mL–1. The molecular weight of HCl
is 36.5 g mol–1. The volume (mL) of stock solution required to prepare a 200 mL solution of
0.4 M HCl is :
Ans : 8
Molarity of stock solution =10 d x
GMW
=10 1.25 29.2
36.5
=10M
1 1 2 2
1
1
10 200 0.48 ( )
V M V MVV ML approx
38. The periodic table consists of 18 groups. An isotope of copper, on bombardment with pro-
tons, undergoes a nuclear reaction yielding element X as shown below. To which
group,element X belongs int he periodic table ?
Ans : 863 1 1 4 1 5229 1 0 2 1 266 2CU H n He H X
26 Fe 8 th group
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39. An organic compound undergoes first-order decomposition. The time taken for its decom-
position to 1/8 and 1/10 of its initial concentration are t1/8 and t1/10 respectively. What is the
Ans : 9
18
110
2.303 log
82.303 log
10
at ak
at ak
18
110
log8log10
t
t
18
110
3log 2t
t
18
110
10 9t
t
40. The substituents R1 and R2 for nine peptides are listed in the table given below. How many of
these peptides are positively charged at pH = 7.0 ?
Ans : 4
At pH below pI(Isoelectric point) Aminoacid exist as Cation. As BasicAminoacids of pHaround 9-8, these are expected to exists as cation of pH=7IV,VI,VIII,IX.
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MATHEMATICSSECTION – I
( SINGLE CORRECT CHOICE TYPE )This section contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONLY ONE is correct
41. Let z be a complex number such that the imaginary part of z is nonzero and and 2 1a z z is real. Then a connot take the value
a) –1 b) 13 c)
12 d)
34
Ans. D
Let Z x iy 0y
2 1a z z becomes
2 1a x iy x iy
2 2 1 2 0x y x i xy y a i
Comparing real & imaginary parts
2 2 1x y x a ––––––(1)
2 0xy y –––––––(2)
From (2) 1
2x as 0y
Put value of x in (1)
221 1 1
2 2y a
21 14 2
y a
2 3 04
y a
34
a
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42. If 2 1lim 4
1x
x x ax bx
, then
a) a = 1, b = 4 b) a = 1, b = –4 c) a = 2, b = –3 d) a = 2, b = 3
Ans. B
2 1lim 41x
x x ax bx
2 1 1 14
1x
x a x a b blim
x
For existense of limit coefficient of 2 0x
1 – a = 0
a = 1
1 1lim 4
1x
x a b bx
1 4a b
b = – 4
43. Let ijP a be a 3 3 matrix and let ijQ b , where 2i jij ijb a for 1 , 3i j . If the deter-
minant of P is 2, then the determinant of the matrix Q is
a) 210 b) 211 c) 212 d) 213
Ans. D
as 2i jij ijb a
11 11 12 12 13 134 , 8 , 16b a b a b a
21 2 22 22 23 238 , 16 , 32b a b a b a
31 31 32 32 33 3316 , 32 , 64b a b a b a
11 12 13
21 22 23
31 32 33
2a a a
A a a aa a a
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11 12 13
21 22 23
31 32 33
4 8 168 16 32
16 32 64
a a aB a a a
a a a
11 12 13
21 22 23
31 32 33
4 8 16 2 2 24 4 4
a a aa a aa a a
11 12 13
21 22 23
31 32 33
4 8 16 2 4a a aa a aa a a
4 8 16 2 4 2
132B
44. The ellipse 2 2
1 : 19 4x yE is inscribed in a rectangle R whose sides are parallel to the coor-
dinates axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangleR. The eccentricity of the ellipse E2 is
a) 22
b) 32
c) 12 d)
34
Ans. C
Given Ellipse 2 2
19 4x y
As ellipse is bounded by &x a y b
Sides of rectangle are
3 & 2x y
3,2 , 3,2C D
D(0,2)
(3,0)
(0,-2)
(-3,0)
A B
C
3, 2 , 3, 2A B
As new ellipse
Circumscribe the rectangle & passes through (0 ,4).
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Let new ellipse is 2 2
2 2 1x ya b
Passes through 22
160,4 1 16bb
Passes through 3,2 2
9 4 116a
2 12a
So, Vertical ellipse 2 2
112 16x y
2 2 21a b e
212 16 1 e
2 12 1116 4
e
12
e
45. The function : 0,3 1, 29f , defined by 3 22 15 36 1f x x x x is
a) one-one and onto b) onto but not one-one
c) one-one but not onto d) neither one-one nor onto
Ans. B
: 0,3 1, 29f
3 22 15 36 1f x x x x
' 26 30 36f x x x
6 2 3x x
'' 6 2 5f x x
2x is pt of maxima.
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3x is pt of minima.
& 2 29 0f
3 0f
so curve is
nf is not one one
2 3Min value 0 1f
Max value 2 29f
so nf is Onto.
46. The locus of the mid-point of the chord of contact of tangents drawn from points lying onthe straight line 4 5 20x y to the circle 2 2 9x y is
a) 2 220 36 45 0x y x y b) 2 220 36 45 0x y x y
c) 2 236 20 45 0x y x y d) 2 236 20 45 0x y x y
Ans. A
Let 1 1, & ,P h k Q x y
As P lies of 4x-5y=20
4 20,5
hP h
P
A
B
Q
Equation of chord of contact
1 0AB S
4 20 95
hxh y
5 4 20 45xh y h ------------------(1)
Equation of chord AB whose mid point is 1 1,Q x y
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2 21 1 1 1xx yy x y ---------------(2)
Compare (1) & (2)
2 21 1 1 1
5 4 20 45h hx y x y
1 12 2 2 21 1 1 1
9 45& 4 20x yh hx y x y
On eliminating h
1 12 2 2 21 1 1 1
49 4520x yx y x y
2 21 1 1 120 36 45x y x y
Locus is
2 220 36 45 0x y x y
47. The point P is the intersection of the straight line joining the points Q (2, 3, 5) and
R (1, –1, 4) with the plane 5 4 1x y z . If S is the foot of the perpendicular drawn from thepoint T (2, 1, 4) to QR, then the length of the line segement PS is
a) 12 b) 2 c) 2 d) 2 2
Ans. A
Equation of 1 1 4
1 4 1x y zQR
Dr’s of 1, 4,1QR
Co-ordinates of 1 , 1 4 , 4P
As P is point of Intersection of QR & given plane, so
5 1 4 1 4 4 1
13
T(2,1,4)
Q S R
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4 1 13, ,3 3 3
P
Let 1 , 1 4 ,4S
Dr’s of 1, 4 2,TS
TS QR So,
1 1 4 4 2 0
12
3 9,1,2 2
S
2 2 24 3 1 13 913 2 3 3 2
PS
12
48. Let 2 cos , 0
0, 0
x xf x x
x
, x IR , then f is
a) differentiable both at x = 0 and at x = 2
b) differentiable at x = 0 but not differentiable at x = 2
c) not differentiable at x = 0 but differentiable at x = 2
d) differentiable neither at x = 0 nor at x = 2
Ans. B
2 cos ; 0
0 ; 0
x xf x x
x
At x=0
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2
0 0
cos 00 0lim limh h
hf h f hRHDh h
0lim cos 0h
hh
2
0 0
cos 00 0
lim limh h
hf h f hLHD
h h
0lim cos 0h
hh
Differentiable at x=0
At x=2
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hRHDh h
2
0 0
2 cos 4 cos2 2 2 2lim limh h
hf h f hLHDh h
Not Differentiable at x=2.
49. The total number of ways in which 5 balls of different colours can be distributed among 3persons so that each person gets at least one ball is
a) 75 b) 150 c) 210 d) 243
Ans. B
5 Balls to 3 persons, so that all persons get atleast one ball.
Case-I
Total ways 2 2 1
2 1 2
1 2 2
5 3 12 2 1 3C C C
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10 3 1 3 90
Case-II
Total ways
5 2 13 1 1 3C C C
11
1
13
3
1
3110 2 1 3 60
Total ways = 90 + 60 = 150.
50. The integral
2
9/ 2sec
sec tanx dx
x x equals (for some arbitrary constant K)
a) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
b) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
c) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
d) 2
11/ 2
1 1 1 sec tan11 7sec tan
x x Kx x
Ans. C
9
2
sec
sec tan
x dxx x
Let sec tanx x t -------------(1)
So, 2sec tan secx x x dx dt
sec dtx dxt
------------(2)
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1sec tanx xt
----------(3)
From (1) & (3)
1 1sec2
x tt
So, Integral becomes
9
2
sec .sec
sec tan
x x dxx x
92
1 12
dtt ttt
2
132
1 12
t dtt
9 132 21
2t dt t dt
7 112 21
7 112 22
t t K
7 112 2
7 11t t K
7 11
22
1 1
11 sec tan7 sec tan x xx x
2
112
1 1 1 sec tan11 7sec tan
x x Kx x
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SECTION – II( MULTIPLE CORRECT CHOICE TYPE )
This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for itsanswer, out of which ONE OR MORE is/ are correct
51. A ship is fitted with three engines E1 , E2 and E3. The engines function independently of each
other with respective probabilities 1 1,2 4 and
14 . For the ship to be operational at least two of
its engines must function. Let X denote the event that the ship is operational and let X1, X2and X3
denote respectively the events that the engines E1, E2 and E3 are functioning. Which ofthe following is (are) true ?
a) 13|
16cP x X b) P [Exactly two engines of the ship are functioning | X]
78
c) 25|
16P X X d) 1
7|16
P X X
Ans. B,D
A)
1
1 1 112 4 4
1 84
cP X XP X
1 1 3 1 1 1 12 22 4 4 2 4 4 4
P X
B) 1 1 3 1 1 12
72 4 4 2 4 4/ 1 84
P Exactly twoengines of the ship are functioning X
C) 2
22
1 1 3 1 1 1 252 4 4 2 4 4/ 1 8
4
P X XP X X
P X
D)
11
1
1 1 3 1 1 1272 4 4 2 4 4/ 1 16
2
P X XP X X
P X
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52. Tangents are drawn to the hyperbola 2 2
19 4x y
, parallel to the straight line 2 1x y . The
points of contact of the tangents on the hyperbola are
a) 9 1,
2 2 2
b) 9 1,
2 2 2
c) 3 3, 2 2 d) 3 3,2 2
Ans. A, B
2 2
19 4x y
Tangent to the hyperbola is of the form2y x
2 9 4 4 36 4 32
4 2
Tangent is : 2 4 2 0x y
Point of contact is 2 2 9 1 9 1, , , ,
2 2 2 2 2 2a l b mn n
53. Let S be the area of the region enclosed by 2xy e , y = 0, x = 0, and x = 1. Then
a) 1Se
b) 11Se
c) 1 114
Se
d)
1 1 112 2
Se
Ans. A, B, D
21 1
0 0
11x xI e dx e dxe
11Ie
21 1xe e
21 1 1
1
0 0 0
xe dx e dx dx
21
0
1xe dxe
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54. If y (x) satisfies the differential equation ' tan 2 secy y x x x and y (0) = 0, then
a) 2
4 8 2y
b) 2
'4 18
y
c) 2
3 9y
d) 24 2'
3 3 3 3y
Ans. A,D
tan 2 secdy y x x xdx
tan. cos
dxI F e x
General solution is cos 2 sec cosy x x x x dx 2x c
2 sec secy x x c x
0,0
0 0 0c c .
2 secy x x
2
24 16
y
2
8 2
2
23 9
y
2 sec tan 2 secdy x x x x xdx
.
2
1 2 3 2 23 9 3
y
22 433 3
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55. Let , 0, 2 be such that
22cos 1 sin sin tan cot cos 12 2
,
tan 2 0 and 31 sin2
.
Then cannot satisfy
a) 0 2 b)
42 3 c)
4 33 2 d)
3 22
Ans. A, C, D
3tan 2 0, 1 sin2
3 5 1, 0 cos2 3 2
______ (1)
22cos 1 sin sin tan / 2 cot / 2 cos 1
1sin cos2
1 sin 12
(from (1))
13 176 6
2 73 6 (from (1))
cannot satisfied by A, C, D
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SECTION –III( INTEGER ANSWER TYPE)
This section contains 5 questions . The answer to each of the questions is a single digit integer, ranging from 0 to9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened.
56. Let p (x) be a real polynomial of least degree which has a local maximum at x = 1 and a localminimum at x =3. If p (1) = 6 and p (3) = 2, then ' 0p is
Ans. 9
3 2P x ax bx cx d
1 6 6P a b c d
3 2 27 9 3 2P a b c d
1 1 0 3 2 0P a b c
1 3 0 27 6 0P a b c
1, 6, 9a b c
1 23 2P x ax bx c
1 0 9P c
57. If ,a b and c are unit vectors satisfying
2 2 2 9a b b c c a ,then 2 5 5a b c
is\
Ans. 3
sol. 1a b c
2 2 2 9a b b c c a
3.
2a b b c c a
3 2 . . 0a b b c c a
22 2 2 . . . 0a b c a b b c c a
2
0a b c
Sri Chaitanya IIT Academy PAPER-1 KEY & SOLUTIONS
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0a b c ––––––(1)
Taking dot product with (1) by , &a b c
then 1 1 1. , . , .
2 2 2a b b c c a
2 5 5 3a b c
58. The value of 32
1 1 1 16 log 4 4 4 ...3 2 3 2 3 2 3 2
is
Ans. 4
sol.1 4
3 2x x
3 2 4x x
squaring on both sides218 4x x 218 4 0x x 218 9 8 4 0x x x
9 2 1 4 2 1 0x x x .
2 1 9 4 0x x
1 4,2 9
x
49
x 0x
Required answer is 2
3/ 236 log2
= 6 – 2 = 4
59. Let S be the focus of the parabola 2 8y x and let PQ be the common chord of the circle2 2 2 4 0x y x y and the given parabola. The area of the triangle PQS is
Ans. 4
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# 42
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sol.P
Q
S(0, 0)
(2,4)
(2, 0)2
(0,4)
2 8 2 4 2 0x x x x
0,2 0,0 , 2,4x P Q
area of SPQ
= 1 2 42
= 4
60. Let :f IR IR be defined as 2 1f x x x . The total number of points at which f attainseither a local maximum or a local minimum is
Ans. 5sol. 2 1, 1f x x x x
21 , 1 0x x x
21 , 0 1x x x
2 1, 1x x x
' 2 1, 1f x x x
2 1, 1 0x x
0-1 1x
y
1 2 , 0 1x x
1 2 , 1x x
differentiable at 1 1,2 2
x ,
not differentiable at –1, 0, 1
The number of points = 5