JEE Advanced 2014: Paper 2
Text of page 1
JEE(ADVANCED) – 2014 PAPER-2 Code-1
Questions with Answers
PART-1 PHYSICS
SECTION -1 (Only One Option Correct Type)
This section contains 10 multiple choice questions. Each question has four
choices (A), B), (C) and (D) out of which only one option is correct.
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1. A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original
position after hitting the surface. The force on the ball during the collision is proportional to the
length of compression of the ball. Which one of the following sketches describes the variation of
its kinetic energy KK with timett most appropriately? The figures are only illustrative and not to
the scale.
(A) (B)
K K
t
t
(C) (D)
K
K
t
t
Answer (B)
Text of page 2
2. A wire, whichpasses through the hole in a small bead, is bent in the form of quarter of a circle.
The wire is fixed vertically on ground as shown in the figure. The bead is released from near the
top of the wire and it slides along the wire without friction. As the bead moves from A to B, the
force it applies on the wire is
A
90°
B
(A)always radially outwards.
(B)always radially inwards.
(C)radially outwards initially and radially inwards later.
(D)radially inwards initially and radially outwards later.
Answer (D)
3. During an experiment with ametre bridge, the galvanometer shows a null point when the
jockey is pressed at 40.0 ccmm using a standard resistance of 90 ΩΩ,as shown in the figure. The least
count of the scale used in the metre bridge is1 mmmm. The unknown resistance is
R 90 Ω
40.0 cm
(A)60 ± 0.15Ω
(B)135 ± 0.56Ω
(C)60 ± 0.25Ω
(D)135 ± 0.23Ω
Answer (C)
Text of page 3
4. Charges Q, 2Q and 4Q are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of
radii R/2, R and 2R, respectively, as shown in figure. If magnitudes of the electric fields at point
P at a distance R from the center of spheres 1, 2 and 3 are E1, E2 and E3respectively, then
P P P
R R R
Q 2Q 4Q
R/2
2R
Sphere 1 Sphere 2 Sphere 3
(A) E1>E2>E3
(B) E3>E1>E2
(C) E2>E1>E3
(D) E3>E2>E1
Answer (C)
5. A point source S is placed at the bottom of a transparent block of height 10 mm and refractive
index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found
that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54
mm on the top of the block. The refractive index of the liquid is
Liquid
Block
S
(A)1.21
(B)1.30
(C)1.36
(D)1.42
Answer (C)
Text of page 4
6. Parallel rays of light of intensity II = 912 WWmm−2 are incident on a spherical black body kept in
surroundings of temperature 300 K. Take Stefan-Boltzmann constant
−8 −2 −4
σσ = 5.7 × 10 WWmm KK and assume that the energy exchange with the surroundingsis only
through radiation. The final steady state temperature of the black body is close to
(A) 330 K
(B) 660 K
(C) 990 K
(D) 1550 K
Answer (A)
7. A metal surface is illuminated by light of two different wavelengths 248nm and 310 nm. The
maximum speeds of the photoelectrons corresponding to these wavelengths areu1 and
u2,respectively. If the ratio uu1 : uu2 = 2: 1and hc = 1240 eV nm, the work function of the metal is
nearly
(A) 3.7 eV
(B) 3.2 eV
(C) 2.8 eV
(D) 2.5eV
Answer (A)
8. If λλCCCC is the wavelength of KKαα X-ray line of copper (atomic number 29) and λλMMMM is the
wavelength of the KKαα X-ray line of molybdenum (atomic number 42), then the ratio λλCCCC /λλMMMM is
close to
(A) 1.99
(B) 2.14
(C) 0.50
(D) 0.48
Answer (B)
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1
9. A planet of radius RR = × (radius of Earth) has the same mass density as Earth. Scientists
10
RR
dig a well of depth on it and lower a wire of the same length and of linear mass density
5
−3 −1
10 kkkkmm into it. If the wire is not touching anywhere, the force applied at the top of the wire
by a person holding it in place is (take the radius of Earth = 6 × 106 mm and the acceleration due
to gravity on Earth is 10 ms ̶ 2)
(A) 96 N
(B) 108 N
(C) 120 N
(D) 150 N
Answer (B)
10. A glass capillary tube is of the shape of a truncated cone with an apex angle ααso that its two
ends have cross sections of different radii. When dipped in water vertically, water rises in it to a
height h, where the radius of its cross section is bb. If the surface tension of water is S, its density
is ρ, and its contact angle with glass is θθ, the value of h will be (g is the acceleration due to
gravity)
h
2SS
(A) cos(θθ − αα)
bbbbbb
2SS
(B) cos(θθ + αα)
bbbbbb
2SS
(C) cos(θθ − αα⁄2)
bbbbbb
2SS
(D) cos(θθ + αα⁄2)
bbbbbb
Answer (D)
Text of page 6
Section -2 Comprehension Type
This section contains 3 paragraphs each describing theory, experiments, data etc. Six questions
relate to the three paragraphs with two questions on each paragraph. Each question has only one
correct answer among the four given options (A), (B), (C) and (D).
Paragraph For Questions 11 & 12
In the figure a container is shown to have a movable (without friction) piston on top. The
container and the piston are all made of perfectly insulating material allowing no heat transfer
between outside and inside the container. The container is divided into two compartments by a
rigid partition made of a thermally conducting material that allows slow transfer of heat. The
lower compartment of the container is filled with 2 moles of an ideal monatomic gas at 700 K
and the upper compartment is filled with 2 moles of an ideal diatomic gas at 400 K. The heat
3 5
capacities per mole of an ideal monatomic gas areCCVV = 2 RR, CCPP = 2 RR, and those for an ideal
5 7
diatomic gas are CCVV = 2 RR,CCPP = 2 RR.
11. Consider the partition to be rigidly fixed so that it does not move. When equilibrium is
achieved, the final temperature of the gases will be
(A) 550 K (B) 525 K (C) 513 K (D) 490 K
Answer (D)
12. Now consider the partition to be free to move without friction so that the pressure of gases in
both compartments is the same. Then total work done by the gases till the time they achieve
equilibrium will be
(A) 250 R (B) 200 R (C)100 R (D) −100R
Answer (D)
Text of page 7
Paragraph For Questions 13 & 14
A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of
uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid
container. As the piston pushes air through the nozzle, the liquid from the container rises into
the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are
20 mm and 1mm, respectively. The upper end of the container is open to the atmosphere.
̶1
13. If the piston is pushed at a speed of 5 mms , the air comes out of the nozzle with a speed of
̶1 ̶1 ̶1 ̶1
(A)0.1 ms (B)1 ms (C)2 ms (D)8 ms
Answer (C)
14. If the density of air is ρρaa and that of the liquid ρρℓ , then for a given piston speed the rate
(volume per unit time) at which the liquid is sprayed will be proportional to
ρρ aa
(A)�
ρρ ℓ
(B)�ρρaa ρρℓ
ρρ ℓ
(C)�
ρρ aa
(D)ρρℓ
Answer (A)
Text of page 8
Paragraph For Questions 15 & 16
The figure shows a circular loop of radius a with two long parallel wires (numbered 1 and 2) all
in the plane of the paper. The distance of each wire from the centre of the loop is d. The loop and
the wires are carrying the same currentI. The current in the loop is in the counterclockwise
direction if seen from above.
Q S
d
Wire1 a Wire2
P R
15. When d≈a but wires are not touching the loop, it is found that the net magnetic field on the
axis of the loop is zero at a height h above the loop. In that case
(A)current in wire 1 and wire 2 is the direction PQ and RS, respectively and ℎ ≈ aa
(B) current in wire 1 and wire 2 is the direction PQ and SR, respectively and ℎ ≈ aa
(C)current in wire 1 and wire 2 is the direction PQ and SR, respectively and ℎ ≈ 1.2aa
(D)current in wire 1 and wire 2 is the direction PQ and RS, respectively and ℎ ≈ 1.2aa
Answer (C)
16. Consider dd ≫ aa, and the loop is rotated about its diameter parallel to the wires by 30° from
the position shown in the figure. If the currents in the wires are in the opposite directions, the
torque on the loop at its new position will be (assume that the net field due to the wires is
constant over the loop)
μμ 0 II 2 aa 2 μμ 0 II 2 aa 2 √3μμ 0 II 2 aa 2 √3μμ 0 II 2 aa 2
(A) (B) (C) (D)
dd 2dd dd 2dd
Answer (B)
Text of page 9
Section – 3 (Matching List Type)
This section contains four questions, each having two matching lists. Choices for the correct
combination of elements from List-I and List-II are given as options (A), (B), (C) and (D), out of
which one is correct.
17. Four charges Q1, Q2, Q3 and Q4 of same magnitude are fixed along the x axis at xx = −2aa,
−aa, +aa and +2aa, respectively. A positive charge q is placed on the positive y axis at a distance
bb > 0. Four options of the signs of these charges are given in List I. The direction of the forces
on the charge q is given in List II. Match List I with List II and select the correct answer using
the code given below the lists.
q (0, b)
Q1 Q2 Q3 Q4
( ̶2a, 0) ( ̶a, 0) (+a,0) (+2a,0)
List I List II
P.QQ1 , QQ2 , QQ3 , QQ4 all positive 1.+xx
Q.QQ1 , QQ2 positive; QQ3 , QQ4 negative 2.−xx
R.QQ1 , QQ4 positive; QQ2 , QQ3 negative 3.+yy
S.QQ1 , QQ3 positive; QQ2 , QQ4 negative 4.−yy
(A) P-3, Q-1, R-4, S-2
(B) P-4, Q-2, R-3, S-1
(C) P-3, Q-1, R-2, S-4
(D) P-4, Q-2, R-1, S-3
Answer (A)
Text of page 10
18. Four combinations of two thin lenses are given in List I. The radius of curvature of all
curved surfaces is r and the refractive index of all the lenses is 1.5. Match lens combinations in
List I with their focal length in List II and select the correct answer using the code given below
the lists.
List I List II
P.
1.2rr
Q.
2.rr/2
R.
3.−rr
S.
4. rr
Choices:
(A) P-1, Q-2, R-3, S-4
(B) P-2, Q-4, R-3, S-1
(C) P-4, Q-1, R-2, S-3
(D) P-2, Q-1, R-3, S-4
Answer (B)
Text of page 11
19. A block of mass m1 = 1 kg another mass m2 = 2 kg, are placed together (see figure) on an
inclined plane with angle of inclination θ. Various values of θ are given in List I. The coefficient
of friction between the block m1 and the plane is always zero. The coefficient of static and
dynamic friction between the block m2 and the plane are equal to µ = 0.3. In List II expressions
for the friction on block m2 are given. Match the correct expression of the friction in List II with
the angles given in List I, and choose the correct option. The acceleration due to gravity is
denoted by g.
[Useful information :tan(5.5o ) ≈ 0.1; tan(11.5o ) ≈ 0.2; tan(16.5o ) ≈ 0.3]
m1
m2
θ
List I
List II
P. θ = 5o 1.mm2 gg sinθθ
Q.θ = 10
o 2.(mm1 + mm2 )gg sinθθ
R.θ = 15
o 3.μμmm2 gg cosθθ
4.μμ(mm1 + mm2 )gg cosθθ
S.θ = 20o
Choices:
(A) P-1, Q-1, R-1, S-3
(B) P-2, Q-2, R-2, S-3
(C) P-2, Q-2, R-2, S-4
(D) P-2, Q-2, R-3, S-3
Answer (D)
Text of page 12
20. A person in a lift is holding a water jar, which has a small hole at the lower end of its side.
When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance
of 1.2 m from the person. In the following, state of the lift’s motion is given in List I and the
distance where the water jet hits the floor of the lift is given in List II. Match the statements from
List I with those in List II and select the correct answer using the code given below the list.
List I List II
P. Lift is accelerating vertically up. 1. dd = 1.2 m
Q. Lift is accelerating vertically down with an 2. dd > 1.2 m
acceleration less than the gravitational acceleration.
R. Lift is moving vertically up with constant speed. 3. dd < 1.2 m
S. Lift is falling freely. 4.No water leaks out of the jar
(A) P-2 Q-3 R-2 S-4
(B) P-2 Q -3 R-1 S-4
(C)P-1 Q-1 R-1 S-4
(D)P-2 Q-3 R-1 S-1
Answer (C)
Text of page 13
PART-2 CHEMISTRY
SECTION -1 (Only One Option Correct Type)
This section contains 10 multiple choice questions. Each question has four
choices (A), B), (C) and (D) out of which only one option is correct.
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21. The acidic hydrolysis of ether (X) shown below is fastest when
[Figure]
(A) one phenyl group is replaced by a methyl group.
(B) one phenyl group is replaced by a para-methoxyphenyl group.
(C) two phenyl groups are replaced by two para-methoxyphenyl groups.
(D) no structural change is made to X.
Answer (C)
Text of page 14
22. Isomers of hexane, based on their branching, can be divided into three distinct
classes as shown in the figure.
[Figure]
The correct order of their boiling point is
(A) I > II > III
(B) III > II > I
(C) II > III > I
(D) III > I > II
Answer (B)
Text of page 15
23. The major product in the following reaction is
[Figure]
(A)
(B)
(C)
(D)
Answer (D)
24. Hydrogen peroxide in its reaction with KIO4 and NH2OH respectively, is acting as a
(A) reducing agent, oxidising agent
(B) reducing agent, reducing agent
(C) oxidising agent, oxidising agent
(D) oxidising agent, reducing agent
Answer (A)
Text of page 16
25. The product formed in the reaction of SOCl2 with white phosphorous is
(A) PCl3
(B) SO2Cl2
(C) SCl2
(D) POCl3
Answer (A)
26. Under ambient conditions, the total number of gases released as products in the final
step of the reaction scheme shown below is
(A) 0
(B) 1
(C) 2
(D) 3
Answer (C)
Text of page 17
27. For the identification of β-naphthol using dye test, it is necessary to use
(A) dichloromethane solution of β-naphthol.
(B) acidic solution of β-naphthol.
(C) neutral solution of β-naphthol.
(D) alkaline solution of β-naphthol.
Answer (D)
28. For the elementary reaction M → N, the rate of disappearance of M increases by a
factor of 8 upon doubling the concentration of M. The order of the reaction with respect
to M is
(A) 4
(B) 3
(C) 2
(D) 1
Answer (B)
29. For the process
H2O (l) → H2O (g)
at T = 100°C and 1 atmosphere pressure, the correct choice is
(A) ∆Ssystem > 0 and ∆Ssurroundings > 0
(B) ∆Ssystem > 0 and ∆Ssurroundings < 0
(C) ∆Ssystem < 0 and ∆Ssurroundings > 0
(D) ∆Ssystem < 0 and ∆Ssurroundings < 0
Answer (B)
Text of page 18
30. Assuming 2s-2p mixing is NOT operative, the paramagnetic species among the
following is
(A) Be2
(B) B2
(C) C2
(D) N2
Answer (C)
Text of page 19
Section -2 Comprehension Type
This section contains 3 paragraphs each describing theory, experiments, data etc. Six questions
relate to the three paragraphs with two questions on each paragraph. Each question has only one
correct answer among the four given options (A), (B), (C) and (D).
Paragraph For Question 31 and 32
Schemes 1 and 2 describe sequential transformation of alkynes M and N. Consider only
the major products formed in each step for both the schemes.
31. The product X is
(A)
(B)
Text of page 20
(C)
(D)
Answer (A)
32. The correct statement with respect to product Y is
(A) It gives a positive Tollens test and is a functional isomer of X.
(B) It gives a positive Tollens test and is a geometrical isomer of X.
(C) It gives a positive iodoform test and is a functional isomer of X.
(D) It gives a positive iodoform test and is a geometrical isomer of X.
Answer (C)
Text of page 21
Paragraph For Question 33 and 34
An aqueous solution of metal ion M1 reacts separately with reagents Q and R in excess
to give tetrahedral and square planar complexes, respectively. An aqueous solution of
another metal ion M2 always forms tetrahedral complexes with these reagents.
Aqueous solution of M2 on reaction with reagent S gives white precipitate which
dissolves in excess of S. The reactions are summarized in the scheme given below:
33. M1, Q and R, respectively are
2+
(A) Zn , KCN and HCl
2+
(B) Ni , HCl and KCN
2+
(C) Cd , KCN and HCl
2+
(D) Co , HCl, and KCN
Answer (B)
Text of page 22
34. Reagent S is
(A) K4[Fe(CN)6]
(B) Na2HPO4
(C) K2CrO4
(D) KOH
Answer (D)
Text of page 23
Paragraph For Question 35 and 36
X and Y are two volatile liquids with molar weights of 10 g mol-1 and 40 g mol-1
respectively. Two cotton plugs, one soaked in X and the other soaked in Y, are
simultaneously placed at the ends of a tube of length L = 24 cm, as shown in the figure.
The tube is filled with an inert gas at 1 atmosphere pressure and a temperature of 300
K. Vapours of X and Y react to form a product which is first observed at a distance d cm
from the plug soaked in X. Take X and Y to have equal molecular diameters and
assume ideal behaviour for the inert gas and the two vapours.
35. The value of d in cm (shown in the figure), as estimated from Graham’s law, is
(A) 8
(B) 12
(C) 16
(D) 20
Answer (C)
Text of page 24
36. The experimental value of d is found to be smaller than the estimate obtained using
Graham’s law. This is due to
(A) larger mean free path for X as compared to that of Y.
(B) larger mean free path for Y as compared to that of X.
(C) increased collision frequency of Y with the inert gas as compared to that of X with
the inert gas.
(D) increased collision frequency of X with the inert gas as compared to that of Y with
the inert gas.
Answer (D)
Text of page 25
Section – 3 (Matching List Type)
This section contains four questions, each having two matching lists. Choices for the correct
combination of elements from List-I and List-II are given as options (A), (B), (C) and (D), out of
which one is correct.
37. Different possible thermal decomposition pathways for peroxyesters are shown
below. Match each pathway from List I with an appropriate structure from List II and
select the correct answer using the code given below the lists.
List-I List-II
P. Pathway P
Q. Pathway Q
R. Pathway R
S. Pathway S
Codes:
P Q R S
(A) 1 3 4 2
(B) 2 4 3 1
(C) 4 1 2 3
(D) 3 2 1 4
Answer (A)
Text of page 26
38. Match the four starting materials (P, Q, R, S) given in List I with the corresponding
reaction schemes (I, II, III, IV) provided in List II and select the correct answer using the
code given below the lists.
List-I List-II
Codes:
P Q R S
(A) 1 4 2 3
(B) 3 1 4 2
(C) 3 4 2 1
(D) 4 1 3 2
Answer (C)
Text of page 27
39. Match each coordination compound in List-I with an appropriate pair of
characteristics from List-II and select the correct answer using the code given below the
lists.
{en = H2NCH2CH2NH2; atomic numbers: Ti = 22; Cr = 24; Co = 27; Pt = 78}
List-I List-II
P. [Cr(NH3)4Cl2]Cl 1. Paramagnetic and exhibits ionisation isomerism
Q. [Ti(H2O)5Cl](NO3)2 2. Diamagnetic and exhibits cis-trans isomerism
R. [Pt(en)(NH3)Cl]NO3 3. Paramagnetic and exhibits cis-trans isomerism
S. [Co(NH3)4(NO3)2]NO3 4. Diamagnetic and exhibits ionisation isomerism
Codes:
P Q R S
(A) 4 2 3 1
(B) 3 1 4 2
(C) 2 1 3 4
(D) 1 3 4 2
Answer (B)
Text of page 28
40. Match the orbital overlap figures shown in List-I with the description given in List-II
and select the correct answer using the code given below the lists.
List-I List-II
1. p─d π antibonding
2. d─d σ bonding
3. p─d π bonding
4. d─d σ antibonding
Codes:
P Q R S
(A) 2 1 3 4
(B) 4 3 1 2
(C) 2 3 1 4
(D) 4 1 3 2
Answer (C)
Text of page 29
PART-3 MATHEMATICS
SECTION -1 (Only One Option Correct Type)
This section contains 10 multiple choice questions. Each question has four
choices (A), B), (C) and (D) out of which only one option is correct.
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41. The function yy = ff(xx) is the solution of the differential equation
dddd xxxx xx 4 + 2xx
+ 2 =
dddd xx − 1 √1 − xx 2
in (−1, 1) satisfying ff(0) = 0. Then
√3
2
� ff(xx) dddd
√3
−
2
is
ππ √3 ππ √3 ππ √3 ππ √3
(A) − (B) − (C) − (D) −
3 2 3 4 6 4 6 2
Answer (B)
42. The following integral
ππ
2
�(2 cosec xx)17 dddd
ππ
4
is equal to
log (1+√2 )
(A) ∫0 2(ee uu + ee −uu )16 dddd
log (1+√2 )
(B) ∫0 (ee uu + ee −uu )17 dddd
lo g(1+√2 )
(C) ∫0 (ee uu − ee −uu )17 dddd
log (1+√2 )
(D) ∫0 2(ee uu − ee −uu )16 dddd
Answer (A)
Text of page 30
43. Coefficient of xx 11 in the expansion of (1 + xx 2 )4 (1 + xx 3 )7 (1 + xx 4 )12 is
(A) 1051
(B) 1106
(C) 1113
(D) 1120
Answer (C)
44. Let ff: [0, 2] → ℝ be a function which is continuous on [0, 2] and is differentiable on (0, 2) with ff(0) =
1. Let
xx 2
FF(xx) = � ff(√tt ) dddd
0
for xx ∈ [0, 2]. If FF ′ (xx) = ff ′ (xx) for all xx ∈ (0, 2), then FF(2) equals
(A) ee 2 − 1
(B) ee 4 − 1
(C) ee − 1
(D) ee 4
Answer (B)
45. The common tangents to the circle xx 2 + yy 2 = 2 and the parabola yy 2 = 8xx touch the circle at the points
PP, QQ and the parabola at the points RR, SS. Then the area of the quadrilateral PPPPPPPP is
(A) 3
(B) 6
(C) 9
(D) 15
Answer (D)
Text of page 31
46. For xx ∈ (0, ππ), the equation sin xx + 2 sin 2xx − sin 3xx = 3 has
(A) infinitely many solutions
(B) three solutions
(C) one solution
(D) no solution
Answer (D)
47. In a triangle the sum of two sides is xx and the product of the same two sides is yy. If xx 2 − cc 2 = yy,
where cc is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is
3333
(A)
2222(xx+cc)
3333
(B)
2222(xx+cc)
3333
(C)
4444(xx+cc)
3333
(D)
4444(xx+cc)
Answer (B)
48. Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so
that each envelope contains exactly one card and no card is placed in the envelope bearing the same
number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number
of ways it can be done is
(A) 264
(B) 265
(C) 53
(D) 67
Answer (C)
Text of page 32
49. Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every
girl is at least one more than the number of girls ahead of her, is
11 11 22 33
(A) (B) (C) (D)
22 33 33 44
Answer (A)
50. The quadratic equation pp(xx) = 0 with real coefficients has purely imaginary roots. Then the equation
pp�pp(xx)� = 0
has
(A) only purely imaginary roots
(B) all real roots
(C) two real and two purely imaginary roots
(D) neither real nor purely imaginary roots
Answer (D)
Text of page 33
Section -2 Comprehension Type
This section contains 3 paragraphs each describing theory, experiments, data etc. Six questions
relate to the three paragraphs with two questions on each paragraph. Each question has only one
correct answer among the four given options (A), (B), (C) and (D).
Paragraph For Questions 51 and 52
Let aa, rr, ss, tt be nonzero real numbers. Let PP(aatt 2 , 2aaaa), QQ, RR(aarr 2 , 2aaaa) and SS(aass 2 , 2aaaa) be distinct points on
the parabola yy 2 = 4aaaa. Suppose that PPPP is the focal chord and lines QQQQ and PPPP are parallel, where KK is
the point (2aa, 0).
51. The value of rr is
Choices:
1
(A) −
tt
tt 2 +1
(B)
tt
1
(C)
tt
tt 2 −1
(D)
tt
Answer (D)
52. If ssss = 1, then the tangent at PP and the normal at SS to the parabola meet at a point whose ordinate is
(tt 2 +1 )2
(A)
2tt 3
aa(tt 2 +1 )2
(B)
2tt 3
aa(tt 2 +1 )2
(C)
tt 3
aa(tt 2 +2 )2
(D)
tt 3
Answer (B)
Text of page 34
Paragraph For Question 53 and 54
Given that for each aa ∈ (0, 1),
1−ℎ
lim � tt −aa (1 − tt)aa−1 dddd
ℎ→ 0 +
ℎ
exists. Let this limit be gg(aa). In addition, it is given that the function gg(aa) is differentiable on (0, 1).
1
53. The value of gg � � is
2
(A) ππ
(B) 2ππ
ππ
(C)
2
ππ
(D)
4
Answer (A)
1
54. The value of gg′ � � is
2
ππ
(A)
2
(B) ππ
ππ
(C) −2
(D) 0
Answer (D)
Text of page 35
Paragraph For Questions 55 and 56
Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5;
and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes.
Let xxii be the number on the card drawn from the ii ttℎ box, ii = 1,2,3.
55. The probability that xx1 + xx2 + xx3 is odd, is
29
(A)
105
53
(B)
105
57
(C)
105
1
(D)
2
Answer (B)
56. The probability that xx1 , xx2 , xx3 are in an arithmetic progression, is
9
(A)
105
10
(B)
105
11
(C)
105
7
(D)
105
Answer (C)
Text of page 36
Section – 3 (Matching List Type)
This section contains four questions, each having two matching lists. Choices for the correct
combination of elements from List-I and List-II are given as options (A), (B), (C) and (D), out of
which one is correct.
2kπ 2kkkk
57. Let zzkk = cos � � + ii sin � �; kk = 1, 2, … ,9.
10 10
List I List II
P. For each zzkk there exists a zzjj such that zzkk ∙ zzjj 1. True
Q. There exists a kk ∈ {1,2, … , 9} such that zz1 ∙ zz = 2. False
zzkk has no solution zz in the set of complex numbers.
|1−zz 1 ||1−zz 2 | ⋯|1−zz 9 | 3. 1
R. equals
10
2kkkk
S. 1 − ∑9kk=1 cos � � equals 4. 2
10
PQRS
(A) 1 2 4 3
(B) 2 1 3 4
(C) 1 2 3 4
(D) 2 1 4 3
Answer (C)
58.
List I List II
P. The number of polynomials ff(xx) with non-negative integer coefficients 1. 8
1
of degree ≤ 2, satisfying ff(0) = 0 and ∫0 ff(xx)ddxx = 1, is
Q. The number of points in the interval [−√13, �13] at which ff(xx) = 2. 2
sin(xx 2 ) + cos(xx 2 ) attains its maximum value, is
2 3xx 2 3. 4
R. ∫−2 (1+ee xx ) dddd equals
1 4. 0
1+xx
�∫2 1 cos 2xx log � � dddd �
− 1−xx
2
S. 1 equals
2 1+xx
�∫0 cos 2xx log � � dddd �
1−xx
P QR S
(A) 3 2 4 1
(B) 2 3 4 1
(C) 3 2 1 4
(D) 2 3 1 4
Answer (D)
Text of page 37
59.
List I List II
−1 √3 1 2 1. 1
P. Let yy(xx) = cos(3 cos xx) , xx ∈ [−1,1], xx ≠ ± . Then �(xx −
2 yy (xx)
dd 2 yy(xx) dddd (xx)
1) + xx � equals
ddxx 2 dddd
Q. Let AA1 , AA2 , … , AAnn (nn > 2) be the vertices of a regular polygon of nn sides 2. 2
with its centre at the origin. Let ����⃗ aakk be the position vector of the point
AAkk , kk = 1,2, … , nn. If |∑nn−1
kk=1 ( aa
����⃗
kk × aa
���������⃗ nn−1
kk+1 )| = |∑kk=1 ( ����⃗
aakk ∙ ���������⃗
aakk+1 )|, then the
minimum value of nn is
xx 2 yy 2 3. 8
R. If the normal from the point PP(ℎ, 1) on the ellipse + = 1 is
6 3
perpendicular to the line xx + yy = 8, then the value of ℎ is
S. Number of positive solutions satisfying the equation 4. 9
1 1 2
tan−1 � � + tan−1 � � = tan−1 � 2 � is
2xx+1 4xx+1 xx
PQ RS
(A) 4 3 2 1
(B) 2 4 3 1
(C) 4 3 1 2
(D) 2 4 1 3
Answer (A)
Text of page 38
60. Let ff1 : ℝ → ℝ , ff2 : [0, ∞) → ℝ, ff3 : ℝ → ℝ and ff4 : ℝ → [0, ∞) be defined by
|xx| if xx < 0,
ff1 (xx) = � xx
ee if xx ≥ 0;
ff2 (xx) = xx 2 ;
sin xx if xx < 0,
ff3 (xx) = �
xx if xx ≥ 0
and
ff2 �ff1 (xx)� if xx < 0,
ff4 (xx) = �
ff2 �ff1 (xx)� − 1 if xx ≥ 0.
List II List I
P. ff4 is 1. onto but not one-one
Q. ff3 is 2. neither continuous nor one-one
R. ff2 ooff1 is 3. differentiable but not one-one
S. ff2 is 4. continuous and one-one
P Q RS
(A) 3 1 4 2
(B) 1 3 4 2
(C) 3 1 2 4
(D) 1 3 2 4
Answer (D)
Official key not published for this paper.
Page 1 of 38
Source: Organising IIT for the year. Question paper published by Organising IIT for the year.