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Category: IIT-JEE 2014 Physics Paper 2

Question 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.

Category: IIT-JEE 2014 Physics Paper 2

Question 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	always radially outwards.	B	always radially inwards.
C	radially outwards initially and radially inwards later.	D	radially inwards initially and radially outwards later.

Category: IIT-JEE 2014 Physics Paper 2

Question 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

A	60 ± 0.15Ω	B	135 ± 0.56Ω
C	60 ± 0.25Ω	D	135 ± 0.23Ω

Category: IIT-JEE 2014 Physics Paper 2

Question 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

A	$E_{1} > E_{2} > E_{3}$	B	$E_{3} > E_{1} > E_{2}$
C	$E_{2} > E_{1} > E_{3}$	D	$E_{3} > E_{2} > E_{1}$

Category: IIT-JEE 2014 Physics Paper 2

Question 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

A	1.21	B	1.30
C	1.36	D	1.42

Category: IIT-JEE 2014 Physics Paper 2

Question 6:

Parallel rays of light of intensity 𝐼𝐼=912 𝑊𝑊

m^{- 2}

are incident on a spherical black body kept in surroundings of temperature 300 K. Take Stefan-Boltzmann constant 𝜎𝜎=5.7 ×10−8𝑊𝑊𝑚𝑚−2𝐾𝐾−4 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

Category: IIT-JEE 2014 Physics Paper 2

Question 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

a r e u_{1}

and

u_{2}

,respectively. If the ratio

u_{1}

u_{2}

=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

Category: IIT-JEE 2014 Physics Paper 2

Question 8:

λ_{C u}

is the wavelength of

K_{a} X - r a y

ray line of copper (atomic number 29) and

λ_{M o}

is the wavelength of the

k_{a}

X-ray line of molybdenum (atomic number 42), then the ratio

λ_{c u}

λ_{M o}

is close to

A	1.99	B	2.14
C	0.50	D	0.48

Category: IIT-JEE 2014 Physics Paper 2

Question 9:

A planet of radius

R = \frac{1}{10} \times

(radius of Earth) has the same mass density as Earth. Scientists dig a well of depth

\frac{R}{5}

on it and lower a wire of the same length and of linear mass density

10^{- 3} K g m^{- 1}

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 \times 10^{6} m

and the acceleration due to gravity on Earth is 10

m s^{- 2}

)

A	96 N	B	108 N
C	120 N	D	150 N

Category: IIT-JEE 2014 Physics Paper 2

Question 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)

A	$\frac{2 S}{b p g} \cos (θ - α)$	B	$\frac{2 S}{b p g} \cos (θ + α)$
C	$\frac{2 S}{b p g} \cos (θ - α / 2)$	D	$\frac{2 S}{b p g} \cos (θ + α / 2)$

Category: IIT-JEE 2014 Physics Paper 2

Question 11:

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 capacities per mole of an ideal monatomic gas are

C_{V} = \frac{3}{2} R, C_{P} = \frac{5}{2} R,

and those for an ideal diatomic gas are

C_{V} = \frac{5}{2} R, C_{P} = \frac{7}{2} R .

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

Category: IIT-JEE 2014 Physics Paper 2

Question 12:

C_{V} = \frac{3}{2} R, C_{P} = \frac{5}{2} R,

and those for an ideal diatomic gas are

C_{V} = \frac{5}{2} R, C_{P} = \frac{7}{2} R .

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

Category: IIT-JEE 2014 Physics Paper 2

Question 13:

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.

If the piston is pushed at a speed of 5

m m s^{- 1}

, the air comes out of the nozzle with a speed of

A	$0.1 m s^{- 1}$	B	$1 m s^{- 1}$
C	$2 m s^{- 1}$	D	$8 m s^{- 1}$

Category: IIT-JEE 2014 Physics Paper 2

Question 14:

If the density of air is

P_{a}

and that of the liquid

p_{l}

, then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to

A	$\sqrt{\frac{P a}{p l}}$	B	$\sqrt{p a p l}$
C	$\sqrt{\frac{P l}{p a}}$	D	pl

Category: IIT-JEE 2014 Physics Paper 2

Question 15:

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.

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

Category: IIT-JEE 2014 Physics Paper 2

Question 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)

A	$\frac{μ_{0} I^{2} a^{2}}{d}$	B	$\frac{μ_{0} I^{2} a^{2}}{2 d}$
C	$\frac{\sqrt{3} μ_{0} I^{2} a^{2}}{d}$	D	$\frac{\sqrt{3} μ_{0} I^{2} a^{2}}{2 d}$

Category: IIT-JEE 2014 Physics Paper 2

Question 17:

Four charges

Q_{1} Q_{2} Q_{3} a n d Q_{4}

of same magnitude are fixed along the x axis at 𝑥=−2𝑎𝑎,−𝑎,+𝑎 and +2𝑎, respectively. A positive charge q is placed on the positive y axis at a distance 𝑏>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.

List I	List II
p $Q_{1}, Q_{2,} Q_{3,} Q_{4}$ all positive	1.+𝑥
Q $Q_{1}, Q_{2}$ positive; $Q_{3} Q_{4}$ negative	2.−𝑥
R $Q_{1,} Q_{4} p o s i t i v e; Q_{2,} Q_{3} n e g a t i v e$	3.+𝑦
S $Q_{1,} Q_{3} p o s i t i v e; Q_{2,} Q_{4} n e g a t i v e$	4.−𝑦

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

Category: IIT-JEE 2014 Physics Paper 2

Question 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.

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

Category: IIT-JEE 2014 Physics Paper 2

Question 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]

List I	List II
P. θ = 5o	$�� m_{2} �� \sin �� $
Q.θ = 10o	$(m_{1} + m_{2}) �� \sin �� $
R.θ = 15o	$μ m_{2} g \cos θ$
S.θ = 20o	$μ (m_{1} + m_{2}) g \cos θ$

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

Category: IIT-JEE 2014 Physics Paper 2

Question 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. 𝑑=1.2 m
Q. Lift is accelerating vertically down with an acceleration less than the gravitational acceleration.	2. 𝑑>1.2 m
R. Lift is moving vertically up with constant speed.	3. 𝑑<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

Category: IIT-JEE 2014 Chemistry Paper 2

Question 21:

The acidic hydrolysis of ether (X) shown below is fastest when

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.

Category: IIT-JEE 2014 Chemistry Paper 2

Question 22:

Isomers of hexane, based on their branching, can be divided into three distinct classes as shown in the 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

Category: IIT-JEE 2014 Chemistry Paper 2

Question 23:

The major product in the following reaction is

Category: IIT-JEE 2014 Chemistry Paper 2

Question 24:

Hydrogen peroxide in its reaction with

K I O_{4}

and

N H_{2} O H

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

Category: IIT-JEE 2014 Chemistry Paper 2

Question 25:

The product formed in the reaction of

S O C I_{2}

with white phosphorous is

A	$P C l_{3}$	B	$S O_{2} C l_{2}$
C	$S C l_{2}$	D	$P O C l_{3}$

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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.

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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	C	3
B	2	D	1

Category: IIT-JEE 2014 Chemistry Paper 2

Question 29:

For the process

H_{2} O (I) \to H_{2} O (g)

at T = 100°C and 1 atmosphere pressure, the correct choice is

A	ΔSsystem > 0 and ΔSsurroundings > 0	B	ΔSsystem > 0 and ΔSsurroundings < 0
C	ΔSsystem < 0 and ΔSsurroundings > 0	D	ΔSsystem < 0 and ΔSsurroundings < 0

Category: IIT-JEE 2014 Chemistry Paper 2

Question 30:

Assuming 2s-2p mixing is NOT operative, the paramagnetic species among the following is

A	$B e_{2}$	B	$B_{2}$
C	$C_{2}$	D	$N_{2}$

Category: IIT-JEE 2014 Chemistry Paper 2

Question 31:

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.

The product X is

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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.

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.

Category: IIT-JEE 2014 Chemistry Paper 2

Question 33:

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:

M1, Q and R, respectively are

A	$Z n^{2 +}, K C N a n d H C l$	B	$N i^{2 +}, H C l a n d K C N$
C	$C d^{2 +}, K C N a n d H C l$	D	$C o^{2 +}, H C l, a n d K C N$

Category: IIT-JEE 2014 Chemistry Paper 2

Question 34:

Reagent S is

A	$K_{4} [F e {(C N)}_{6}]$	B	$N a_{2} H P O_{4}$
C	$K_{2} C r O_{4}$	D	KOH

Category: IIT-JEE 2014 Chemistry Paper 2

Question 35:

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.

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

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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.

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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.

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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.

{e n = H_{2} N C H_{2} C H_{2} N H_{2}; a t o m i c n y m b e r s : T i = 22; c r = 24; c o = 27; p t = 78}

Category: IIT-JEE 2014 Chemistry Paper 2

Question 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.

Category: IIT-JEE 2014 Math Paper 2

Question 41:

The function 𝑦𝑦=𝑓𝑓(𝑥𝑥) is the solution of the differential equation

\frac{d y}{d x} + \frac{x y}{x^{2} - 1} = \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}}

in (−1,1) satisfying 𝑓𝑓(0)= 0.

\int_{\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f (x) d x

Then

A	$\frac{π}{3} - \frac{\sqrt{3}}{2}$	B	$\frac{π}{3} - \frac{\sqrt{3}}{4}$
C	$\frac{π}{6} - \frac{\sqrt{3}}{4}$	D	$\frac{π}{6} - \frac{\sqrt{3}}{2}$

Category: IIT-JEE 2014 Math Paper 2

Question 42:

The following integral

\int_{\frac{π}{4}}^{\frac{π}{2}} {(2 \cos e c x)}^{17} d x

is equal to

A	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} + e^{- u})}^{16} d u$	B	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} + e^{- u})}^{17} d u$
C	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} - e^{- u})}^{17} d u$	D	$\int_{0}^{\log (1 + \sqrt{2})} 2 {(e^{u} + e^{- u})}^{16} d u$

Category: IIT-JEE 2014 Math Paper 2

Question 43:

Coefficient of

x^{11}

in the expansion of

{(1 + x)}^{2} {(1 + x^{3})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12} i s

A	1051	B	1106
C	1113	D	1120

Category: IIT-JEE 2014 Math Paper 2

Question 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

F (x) = \int_{0}^{x^{2}} f (\sqrt{t}) d t

for 𝑥𝑥∈[0,2]. If 𝐹𝐹′(𝑥𝑥)= 𝑓𝑓′(𝑥𝑥) for all 𝑥𝑥 ∈(0,2), then 𝐹𝐹(2) equals

A	$e^{2} - 1$	B	$e^{4} - 1$
C	$e - 1$	D	$e^{4}$

Category: IIT-JEE 2014 Math Paper 2

Question 45:

The common tangents to the circle

x^{2} + y^{2} = 2

and the parabola

y^{2} = 8 x

touch the circle at the points 𝑃𝑃,𝑄𝑄 and the parabola at the points 𝑅,𝑆. Then the area of the quadrilateral 𝑃QRS is

A	3	B	6
C	9	D	15

Category: IIT-JEE 2014 Math Paper 2

Question 46:

For 𝑥𝑥∈(0,𝜋𝜋), the equation sin𝑥𝑥+2sin2𝑥𝑥−sin3𝑥𝑥=3 has

A	infinitely many solutions	B	three solutions
C	one solution	D	no solution

Category: IIT-JEE 2014 Math Paper 2

Question 47:

In a triangle the sum of two sides is 𝑥 and the product of the same two sides is 𝑦. If

x^{2} - c^{2} = y

where 𝑐𝑐 is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

A	$\frac{3 y}{z x (x + c)}$	B	$\frac{3 y}{z c (x + c)}$
C	$\frac{3 y}{4 x (x + c)}$	D	$\frac{3 y}{4 c (x + c)}$

Category: IIT-JEE 2014 Math Paper 2

Question 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

Category: IIT-JEE 2014 Math Paper 2

Question 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

A	$\frac{1}{2}$	B	$\frac{1}{3}$
C	$\frac{2}{3}$	D	$\frac{3}{4}$

Category: IIT-JEE 2014 Math Paper 2

Question 50:

The quadratic equation 𝑝𝑝(𝑥𝑥)= 0 with real coefficients has purely imaginary roots. Then the equation

p (p (x)) = 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

Category: IIT-JEE 2014 Math Paper 2

Question 51:

Let 𝑎,𝑟,𝑠,𝑡 be nonzero real numbers. Let 𝑃(𝑎

t^{2}

,2𝑎t ), 𝑄, 𝑅(

a r^{2}, 2 a r

) and 𝑆(

a s^{2}, 2 a s

) be distinct points on the parabola

y^{2} = 4 a x

. Suppose that 𝑃Q is the focal chord and lines 𝑄Rand 𝑃K are parallel, where 𝐾𝐾 is the point (2𝑎,0). The value of 𝑟𝑟 is Choices:

A	$- \frac{1}{t}$	B	$\frac{t^{2} + 1}{t}$
C	$\frac{1}{t}$	D	$\frac{t^{2} - 1}{t}$

Category: IIT-JEE 2014 Math Paper 2

Question 52:

If 𝑠 =1, then the tangent at 𝑃 and the normal at 𝑆 to the parabola meet at a point whose ordinate is

A	$\frac{{(t^{2} + 1)}^{2}}{2 t^{3}}$	B	$\frac{a {(t^{2} + 1)}^{2}}{2 t^{3}}$
C	$\frac{a {(t^{2} + 1)}^{2}}{t^{3}}$	D	$\frac{a {(t^{2} + 2)}^{2}}{t^{3}}$

Category: IIT-JEE 2014 Math Paper 2

Question 53:

Given that for each 𝑎𝑎∈(0,1),

\lim_{h \to 0^{+}} \int_{h}^{1 - h} t^{- a} {(1 - t)}^{a - 1} d t

exists. Let this limit be 𝑔 (𝑎𝑎). In addition, it is given that the function 𝑔 (𝑎𝑎) is differentiable on (0,1). 53. The value of 𝑔

(\frac{1}{2}) i s

A	$π$	B	2 $π$
C	$\frac{π}{2}$	D	$\frac{π}{4}$

Category: IIT-JEE 2014 Math Paper 2

Question 54:

Given that for each 𝑎𝑎∈(0,1),

\lim_{h \to 0^{+}} \int_{h}^{1 - h} t^{- a} {(1 - t)}^{a - 1} d t

exists. Let this limit be 𝑔 (𝑎𝑎). In addition, it is given that the function 𝑔 (𝑎𝑎) is differentiable on (0,1). The value of 𝑔 ′

(\frac{1}{2}) i s

A	$\frac{π}{2}$	B	$π$
C	$- \frac{π}{2}$	D	0

Category: IIT-JEE 2014 Math Paper 2

Question 55:

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 𝑥𝑖 be the number on the card drawn from the

i^{t h}

box, 𝑖=1,2,3. The probability that

x_{1} + x_{2} + x_{3}

is odd, is

A	$\frac{29}{105}$	B	$\frac{53}{105}$
C	$\frac{57}{105}$	D	$\frac{1}{2}$

Category: IIT-JEE 2014 Math Paper 2

Question 56:

i^{t h}

box, 𝑖=1,2,3. The probability that

x_{1,} x_{2}, x_{3}

are in an arithmetic progression, is

A	$\frac{9}{105}$	B	$\frac{10}{105}$
C	$\frac{11}{105}$	D	$\frac{7}{105}$

Category: IIT-JEE 2014 Math Paper 2

Question 57:

Let

Z_{K} = \cos (\frac{2 k π}{10}) + \sin (\frac{2 k π}{10}); k = 1, 2, . . . 9 .

LIST I	LIST II
P For each $z_{k}$ there exists a $z_{j}$ such that $z_{k} \cdot z_{j}$	1. True
Q. There exists a 𝑘 ∈ {1,2,…,9} such that $z_{1} \cdot z = z_{k}$ has no solution 𝑧𝑧 in the set of complex numbers.	2. False
R $\frac{\|1 - z_{1} \| \| 1 - z_{2}\| \cdot \cdot \|1 - z_{9}\|}{}$ equals	3. 1
S $1 - \sum_{K = 1}^{9} \cos (\frac{2 k π}{10}) e q u a l s$	4. 2

P Q R S

A	1 2 4 3	B	2 1 3 4
C	1 2 3 4	D	2 1 4 3

Category: IIT-JEE 2014 Math Paper 2

Question 58:

LIST I	LIST II
P The number of polynomials $f (x)$ with non-negative integer coefficients of degree $\leq 2$ , satisfying 𝑓(0)=0 and $\int_{0}^{1} (x) d = 1, i s$	1. 8
Q The number of points in the interval $[- \sqrt{13,} \sqrt{13}]$ at which 𝑓(𝑥)= $\sin {(x)}^{2} + \cos (x^{2})$ attains its maximum value, is	2. 2
R $\int_{- 2}^{2} \frac{3 x^{2}}{(1 + e^{x})} d x e q u a l s$	3. 4
S $\frac{(\int_{\frac{1}{2}}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) d x)}{(\int_{0}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) d x)} e q u a l$	4. 0

P Q R S

A	3 2 4 1	B	2 3 4 1
C	3 2 1 4	D	2 3 1 4

Category: IIT-JEE 2014 Math Paper 2

Question 59:

LIST I	LIST II
P Let y(x)=cos $(3 \cos^{- 1} x), x \in [- 1, 1], x \neq \pm \frac{\sqrt{3}}{2}$ Then $\frac{1}{y (x)} \{(x^{2} - 1)\} \frac{d^{2} y (x)}{d x^{2}} + x \begin{array}{r} d y (x) \\ d x \end{array}\} e q u a l s$	1. 1
Q Let $A_{1} A_{2} . . . A n (n > 2)$ be the vertices of a regular polygon of 𝑛𝑛 sides with its centre at the origin.Let $\vec{a_{k}}$ be the position vector of the point $A_{k}, K = 1, 2 . . ., n . I f \|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \times \vec{a_{k + 1}})\| \|=\| \|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \cdot \vec{a_{k + 1}})\|$ the minimum value of 𝑛𝑛 is	2. 2
R If the normal from the point 𝑃𝑃(ℎ,1) on the ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{3} = 1$ is perpendicular to the line 𝑥𝑥+𝑦𝑦=8, then the value of ℎ is	3. 8
S Number of positive solutions satisfying the equation $\tan^{- 1} (\frac{1}{2 x + 1}) + \tan^{- 1} (\frac{1}{4 x + 1}) = \tan^{- 1} (\frac{2}{x^{2}}) i s$	4. 9

PQRS

A	4 3 2 1	B	2 4 3 1
C	4 3 1 2	D	2 4 1 3

Category: IIT-JEE 2014 Math Paper 2

Question 60:

Let

f_{1}

:ℝ→ℝ ,

f_{2}

:[0,∞)→ℝ,

f_{3}

:ℝ→ℝ and

f_{4}

:ℝ→[0,∞) be defined by

f_{1} (x) = \{\begin{cases} |x| & i f x < 0, \\ e^{x} & i f x \geq 0; \end{cases}

f_{2} (x) = x^{2}; f_{3} (x) = \{\begin{cases} \sin x i f x < 0, \\ u i f x \geq 0 \end{cases} a n d f_{4} (x) = \{\begin{cases} f_{2} (f_{1} (x)) \\ f_{2} (f_{1} (x)) - 1 \end{cases} \frac{i f x < 0,}{i f x \geq 0 .}

LIST I	LIST II
P $f_{4} i s$	1. onto but not one-one
Q $f_{3} i s$	2. neither continuous nor one-one
R $f_{2} ° f_{1} i s$	3. differentiable but not one-one
S $f_{2} i s$	4. continuous and one-one

PQRS

A	3142	B	1342
C	3124	D	1324

⮜ Back Next ⮞

Not Visited	0	Not Answered
Answered	0	Marked for Review
Answered & Marked for Review (will be considered for evaluation)

Question 1:

Correct Answer : B

Question 2:

A	always radially outwards.	B	always radially inwards.
C	radially outwards initially and radially inwards later.	D	radially inwards initially and radially outwards later.

Correct Answer : D

Question 3:

A	60 ± 0.15Ω	B	135 ± 0.56Ω
C	60 ± 0.25Ω	D	135 ± 0.23Ω

Correct Answer : C

Question 4:

A	$E_{1} > E_{2} > E_{3}$	B	$E_{3} > E_{1} > E_{2}$
C	$E_{2} > E_{1} > E_{3}$	D	$E_{3} > E_{2} > E_{1}$

Correct Answer : C

Question 5:

A	1.21	B	1.30
C	1.36	D	1.42

Correct Answer : C

Question 6:

Parallel rays of light of intensity 𝐼𝐼=912 𝑊𝑊

m^{- 2}

A	330 K	B	660 K
C	990 K	D	1550 K

Correct Answer : A

Question 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

a r e u_{1}

and

u_{2}

,respectively. If the ratio

u_{1}

u_{2}

=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

Correct Answer : A

Question 8:

λ_{C u}

is the wavelength of

K_{a} X - r a y

ray line of copper (atomic number 29) and

λ_{M o}

is the wavelength of the

k_{a}

X-ray line of molybdenum (atomic number 42), then the ratio

λ_{c u}

λ_{M o}

is close to

A	1.99	B	2.14
C	0.50	D	0.48

Correct Answer : B

Question 9:

A planet of radius

R = \frac{1}{10} \times

(radius of Earth) has the same mass density as Earth. Scientists dig a well of depth

\frac{R}{5}

on it and lower a wire of the same length and of linear mass density

10^{- 3} K g m^{- 1}

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 \times 10^{6} m

and the acceleration due to gravity on Earth is 10

m s^{- 2}

)

A	96 N	B	108 N
C	120 N	D	150 N

Correct Answer : B

Question 10:

A	$\frac{2 S}{b p g} \cos (θ - α)$	B	$\frac{2 S}{b p g} \cos (θ + α)$
C	$\frac{2 S}{b p g} \cos (θ - α / 2)$	D	$\frac{2 S}{b p g} \cos (θ + α / 2)$

Correct Answer : D

Question 11:

C_{V} = \frac{3}{2} R, C_{P} = \frac{5}{2} R,

and those for an ideal diatomic gas are

C_{V} = \frac{5}{2} R, C_{P} = \frac{7}{2} R .

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

Correct Answer : D

Question 12:

C_{V} = \frac{3}{2} R, C_{P} = \frac{5}{2} R,

and those for an ideal diatomic gas are

C_{V} = \frac{5}{2} R, C_{P} = \frac{7}{2} R .

A	250 R	B	200 R
C	100 R	D	−100R

Correct Answer : D

Question 13:

If the piston is pushed at a speed of 5

m m s^{- 1}

, the air comes out of the nozzle with a speed of

A	$0.1 m s^{- 1}$	B	$1 m s^{- 1}$
C	$2 m s^{- 1}$	D	$8 m s^{- 1}$

Correct Answer : C

Question 14:

If the density of air is

P_{a}

and that of the liquid

p_{l}

, then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to

A	$\sqrt{\frac{P a}{p l}}$	B	$\sqrt{p a p l}$
C	$\sqrt{\frac{P l}{p a}}$	D	pl

Correct Answer : A

Question 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

Correct Answer : C

Question 16:

A	$\frac{μ_{0} I^{2} a^{2}}{d}$	B	$\frac{μ_{0} I^{2} a^{2}}{2 d}$
C	$\frac{\sqrt{3} μ_{0} I^{2} a^{2}}{d}$	D	$\frac{\sqrt{3} μ_{0} I^{2} a^{2}}{2 d}$

Correct Answer : B

Question 17:

Four charges

Q_{1} Q_{2} Q_{3} a n d Q_{4}

List I	List II
p $Q_{1}, Q_{2,} Q_{3,} Q_{4}$ all positive	1.+𝑥
Q $Q_{1}, Q_{2}$ positive; $Q_{3} Q_{4}$ negative	2.−𝑥
R $Q_{1,} Q_{4} p o s i t i v e; Q_{2,} Q_{3} n e g a t i v e$	3.+𝑦
S $Q_{1,} Q_{3} p o s i t i v e; Q_{2,} Q_{4} n e g a t i v e$	4.−𝑦

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

Correct Answer : A

Question 18:

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

Correct Answer : B

Question 19:

List I	List II
P. θ = 5o	$�� m_{2} �� \sin �� $
Q.θ = 10o	$(m_{1} + m_{2}) �� \sin �� $
R.θ = 15o	$μ m_{2} g \cos θ$
S.θ = 20o	$μ (m_{1} + m_{2}) g \cos θ$

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

Correct Answer : D

Question 20:

List I	List II
P. Lift is accelerating vertically up.	1. 𝑑=1.2 m
Q. Lift is accelerating vertically down with an acceleration less than the gravitational acceleration.	2. 𝑑>1.2 m
R. Lift is moving vertically up with constant speed.	3. 𝑑<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

Correct Answer : C

Question 21:

The acidic hydrolysis of ether (X) shown below is fastest when

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.

Correct Answer : C

Question 22:

Isomers of hexane, based on their branching, can be divided into three distinct classes as shown in the 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

Correct Answer : B

Question 23:

The major product in the following reaction is

Correct Answer : D

Question 24:

Hydrogen peroxide in its reaction with

K I O_{4}

and

N H_{2} O H

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

Correct Answer : A

Question 25:

The product formed in the reaction of

S O C I_{2}

with white phosphorous is

A	$P C l_{3}$	B	$S O_{2} C l_{2}$
C	$S C l_{2}$	D	$P O C l_{3}$

Correct Answer : A

Question 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

Correct Answer : C

Question 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.

Correct Answer : D

Question 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	C	3
B	2	D	1

Correct Answer : B

Question 29:

For the process

H_{2} O (I) \to H_{2} O (g)

at T = 100°C and 1 atmosphere pressure, the correct choice is

A	ΔSsystem > 0 and ΔSsurroundings > 0	B	ΔSsystem > 0 and ΔSsurroundings < 0
C	ΔSsystem < 0 and ΔSsurroundings > 0	D	ΔSsystem < 0 and ΔSsurroundings < 0

Correct Answer : B

Question 30:

Assuming 2s-2p mixing is NOT operative, the paramagnetic species among the following is

A	$B e_{2}$	B	$B_{2}$
C	$C_{2}$	D	$N_{2}$

Correct Answer : C

Question 31:

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.

The product X is

Correct Answer : A

Question 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.

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.

Correct Answer : C

Question 33:

M1, Q and R, respectively are

A	$Z n^{2 +}, K C N a n d H C l$	B	$N i^{2 +}, H C l a n d K C N$
C	$C d^{2 +}, K C N a n d H C l$	D	$C o^{2 +}, H C l, a n d K C N$

Correct Answer : B

Question 34:

Reagent S is

A	$K_{4} [F e {(C N)}_{6}]$	B	$N a_{2} H P O_{4}$
C	$K_{2} C r O_{4}$	D	KOH

Correct Answer : D

Question 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

Correct Answer : C

Question 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

Correct Answer : D

Question 37:

Correct Answer : A

Question 38:

Correct Answer : C

Question 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.

{e n = H_{2} N C H_{2} C H_{2} N H_{2}; a t o m i c n y m b e r s : T i = 22; c r = 24; c o = 27; p t = 78}

Correct Answer : B

Question 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.

Correct Answer : C

Question 41:

The function 𝑦𝑦=𝑓𝑓(𝑥𝑥) is the solution of the differential equation

\frac{d y}{d x} + \frac{x y}{x^{2} - 1} = \frac{x^{4} + 2 x}{\sqrt{1 - x^{2}}}

in (−1,1) satisfying 𝑓𝑓(0)= 0.

\int_{\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f (x) d x

Then

A	$\frac{π}{3} - \frac{\sqrt{3}}{2}$	B	$\frac{π}{3} - \frac{\sqrt{3}}{4}$
C	$\frac{π}{6} - \frac{\sqrt{3}}{4}$	D	$\frac{π}{6} - \frac{\sqrt{3}}{2}$

Correct Answer : B

Question 42:

The following integral

\int_{\frac{π}{4}}^{\frac{π}{2}} {(2 \cos e c x)}^{17} d x

is equal to

A	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} + e^{- u})}^{16} d u$	B	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} + e^{- u})}^{17} d u$
C	$\int_{0}^{\log (1 + \sqrt{2})} {(e^{u} - e^{- u})}^{17} d u$	D	$\int_{0}^{\log (1 + \sqrt{2})} 2 {(e^{u} + e^{- u})}^{16} d u$

Correct Answer : A

Question 43:

Coefficient of

x^{11}

in the expansion of

{(1 + x)}^{2} {(1 + x^{3})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12} i s

A	1051	B	1106
C	1113	D	1120

Correct Answer : C

Question 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

F (x) = \int_{0}^{x^{2}} f (\sqrt{t}) d t

for 𝑥𝑥∈[0,2]. If 𝐹𝐹′(𝑥𝑥)= 𝑓𝑓′(𝑥𝑥) for all 𝑥𝑥 ∈(0,2), then 𝐹𝐹(2) equals

A	$e^{2} - 1$	B	$e^{4} - 1$
C	$e - 1$	D	$e^{4}$

Correct Answer : B

Question 45:

The common tangents to the circle

x^{2} + y^{2} = 2

and the parabola

y^{2} = 8 x

touch the circle at the points 𝑃𝑃,𝑄𝑄 and the parabola at the points 𝑅,𝑆. Then the area of the quadrilateral 𝑃QRS is

A	3	B	6
C	9	D	15

Correct Answer : D

Question 46:

For 𝑥𝑥∈(0,𝜋𝜋), the equation sin𝑥𝑥+2sin2𝑥𝑥−sin3𝑥𝑥=3 has

A	infinitely many solutions	B	three solutions
C	one solution	D	no solution

Correct Answer : D

Question 47:

In a triangle the sum of two sides is 𝑥 and the product of the same two sides is 𝑦. If

x^{2} - c^{2} = y

where 𝑐𝑐 is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

A	$\frac{3 y}{z x (x + c)}$	B	$\frac{3 y}{z c (x + c)}$
C	$\frac{3 y}{4 x (x + c)}$	D	$\frac{3 y}{4 c (x + c)}$

Correct Answer : B

Question 48:

A	264	B	265
C	53	D	67

Correct Answer : C

Question 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

A	$\frac{1}{2}$	B	$\frac{1}{3}$
C	$\frac{2}{3}$	D	$\frac{3}{4}$

Correct Answer : A

Question 50:

The quadratic equation 𝑝𝑝(𝑥𝑥)= 0 with real coefficients has purely imaginary roots. Then the equation

p (p (x)) = 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

Correct Answer : D

Question 51:

Let 𝑎,𝑟,𝑠,𝑡 be nonzero real numbers. Let 𝑃(𝑎

t^{2}

,2𝑎t ), 𝑄, 𝑅(

a r^{2}, 2 a r

) and 𝑆(

a s^{2}, 2 a s

) be distinct points on the parabola

y^{2} = 4 a x

. Suppose that 𝑃Q is the focal chord and lines 𝑄Rand 𝑃K are parallel, where 𝐾𝐾 is the point (2𝑎,0). The value of 𝑟𝑟 is Choices:

A	$- \frac{1}{t}$	B	$\frac{t^{2} + 1}{t}$
C	$\frac{1}{t}$	D	$\frac{t^{2} - 1}{t}$

Correct Answer : D

Question 52:

If 𝑠 =1, then the tangent at 𝑃 and the normal at 𝑆 to the parabola meet at a point whose ordinate is

A	$\frac{{(t^{2} + 1)}^{2}}{2 t^{3}}$	B	$\frac{a {(t^{2} + 1)}^{2}}{2 t^{3}}$
C	$\frac{a {(t^{2} + 1)}^{2}}{t^{3}}$	D	$\frac{a {(t^{2} + 2)}^{2}}{t^{3}}$

Correct Answer : B

Question 53:

Given that for each 𝑎𝑎∈(0,1),

\lim_{h \to 0^{+}} \int_{h}^{1 - h} t^{- a} {(1 - t)}^{a - 1} d t

exists. Let this limit be 𝑔 (𝑎𝑎). In addition, it is given that the function 𝑔 (𝑎𝑎) is differentiable on (0,1). 53. The value of 𝑔

(\frac{1}{2}) i s

A	$π$	B	2 $π$
C	$\frac{π}{2}$	D	$\frac{π}{4}$

Correct Answer : A

Question 54:

Given that for each 𝑎𝑎∈(0,1),

\lim_{h \to 0^{+}} \int_{h}^{1 - h} t^{- a} {(1 - t)}^{a - 1} d t

exists. Let this limit be 𝑔 (𝑎𝑎). In addition, it is given that the function 𝑔 (𝑎𝑎) is differentiable on (0,1). The value of 𝑔 ′

(\frac{1}{2}) i s

A	$\frac{π}{2}$	B	$π$
C	$- \frac{π}{2}$	D	0

Correct Answer : D

Question 55:

i^{t h}

box, 𝑖=1,2,3. The probability that

x_{1} + x_{2} + x_{3}

is odd, is

A	$\frac{29}{105}$	B	$\frac{53}{105}$
C	$\frac{57}{105}$	D	$\frac{1}{2}$

Correct Answer : B

Question 56:

i^{t h}

box, 𝑖=1,2,3. The probability that

x_{1,} x_{2}, x_{3}

are in an arithmetic progression, is

A	$\frac{9}{105}$	B	$\frac{10}{105}$
C	$\frac{11}{105}$	D	$\frac{7}{105}$

Correct Answer : C

Question 57:

Let

Z_{K} = \cos (\frac{2 k π}{10}) + \sin (\frac{2 k π}{10}); k = 1, 2, . . . 9 .

LIST I	LIST II
P For each $z_{k}$ there exists a $z_{j}$ such that $z_{k} \cdot z_{j}$	1. True
Q. There exists a 𝑘 ∈ {1,2,…,9} such that $z_{1} \cdot z = z_{k}$ has no solution 𝑧𝑧 in the set of complex numbers.	2. False
R $\frac{\|1 - z_{1} \| \| 1 - z_{2}\| \cdot \cdot \|1 - z_{9}\|}{}$ equals	3. 1
S $1 - \sum_{K = 1}^{9} \cos (\frac{2 k π}{10}) e q u a l s$	4. 2

P Q R S

A	1 2 4 3	B	2 1 3 4
C	1 2 3 4	D	2 1 4 3

Correct Answer : C

Question 58:

LIST I	LIST II
P The number of polynomials $f (x)$ with non-negative integer coefficients of degree $\leq 2$ , satisfying 𝑓(0)=0 and $\int_{0}^{1} (x) d = 1, i s$	1. 8
Q The number of points in the interval $[- \sqrt{13,} \sqrt{13}]$ at which 𝑓(𝑥)= $\sin {(x)}^{2} + \cos (x^{2})$ attains its maximum value, is	2. 2
R $\int_{- 2}^{2} \frac{3 x^{2}}{(1 + e^{x})} d x e q u a l s$	3. 4
S $\frac{(\int_{\frac{1}{2}}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) d x)}{(\int_{0}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) d x)} e q u a l$	4. 0

P Q R S

A	3 2 4 1	B	2 3 4 1
C	3 2 1 4	D	2 3 1 4

Correct Answer : D

Question 59:

LIST I	LIST II
P Let y(x)=cos $(3 \cos^{- 1} x), x \in [- 1, 1], x \neq \pm \frac{\sqrt{3}}{2}$ Then $\frac{1}{y (x)} \{(x^{2} - 1)\} \frac{d^{2} y (x)}{d x^{2}} + x \begin{array}{r} d y (x) \\ d x \end{array}\} e q u a l s$	1. 1
Q Let $A_{1} A_{2} . . . A n (n > 2)$ be the vertices of a regular polygon of 𝑛𝑛 sides with its centre at the origin.Let $\vec{a_{k}}$ be the position vector of the point $A_{k}, K = 1, 2 . . ., n . I f \|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \times \vec{a_{k + 1}})\| \|=\| \|\sum_{k = 1}^{n - 1} (\vec{a_{k}} \cdot \vec{a_{k + 1}})\|$ the minimum value of 𝑛𝑛 is	2. 2
R If the normal from the point 𝑃𝑃(ℎ,1) on the ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{3} = 1$ is perpendicular to the line 𝑥𝑥+𝑦𝑦=8, then the value of ℎ is	3. 8
S Number of positive solutions satisfying the equation $\tan^{- 1} (\frac{1}{2 x + 1}) + \tan^{- 1} (\frac{1}{4 x + 1}) = \tan^{- 1} (\frac{2}{x^{2}}) i s$	4. 9

PQRS

A	4 3 2 1	B	2 4 3 1
C	4 3 1 2	D	2 4 1 3

Correct Answer : A

Question 60:

Let

f_{1}

:ℝ→ℝ ,

f_{2}

:[0,∞)→ℝ,

f_{3}

:ℝ→ℝ and

f_{4}

:ℝ→[0,∞) be defined by

f_{1} (x) = \{\begin{cases} |x| & i f x < 0, \\ e^{x} & i f x \geq 0; \end{cases}

f_{2} (x) = x^{2}; f_{3} (x) = \{\begin{cases} \sin x i f x < 0, \\ u i f x \geq 0 \end{cases} a n d f_{4} (x) = \{\begin{cases} f_{2} (f_{1} (x)) \\ f_{2} (f_{1} (x)) - 1 \end{cases} \frac{i f x < 0,}{i f x \geq 0 .}

LIST I	LIST II
P $f_{4} i s$	1. onto but not one-one
Q $f_{3} i s$	2. neither continuous nor one-one
R $f_{2} ° f_{1} i s$	3. differentiable but not one-one
S $f_{2} i s$	4. continuous and one-one

PQRS

A	3142	B	1342
C	3124	D	1324

Correct Answer : D

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