JEE Advanced 2013 Paper - 2 with Solutions

Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other.
Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f₁ . An
observer in the other vehicle hears the frequency of the whistle to be f₂ . The speed of sound in still air is
V. The correct statement(s) is (are)

Using the expression 2d sinθ = λ, one calculates the values of d by measuring the corresponding angles θ in
the range θ to 90°. The wavelength λ is exactly known and the error in θ is constant for all values of θ. As
θ increases from 0°,

Two non-conducting spheres of radii R1 and R2 and carrying uniform volume charge densities + ρ and –ρ, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

The figure shows the variation of specific heat capacity (C) of a solid as a function of temperature (T). The
temperature is increased continuously from 0 to 500 K at a constant rate. Ignoring any volume change, the
following statement(s) is (are) correct to a reasonable approximation.

The radius of the orbit of an electron in a Hydrogen-like atom is 4.5 a₀ where a₀ is the Bohr radius. Its
orbital angular momentum is  . It is given that h is Planck’s constant and R is Rydberg constant. The
possible wavelength(s), when the atom de-excites, is (are)

SECTION – 2 : (Paragraph Type)

This section contains 4 paragraphs each describing theory, experiment, date etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of paragraph has only one correct answer along the four choice (A), (B), (C) and (D).

Q. No. 9-16 carry 3 marks each and 1 mark is deducted for every wrong answer.

Paragraph for Questions 9 to 10

A small block of mass 1 kg is released from rest at the top of a rough track. The track is circular arc of radius 40 m. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point Q, as shown in the figure, below, is 150 J. (Take the acceleration due to gravity, g = 10 m/s^-2).

Q.

The speed of the block when it reaches the point Q is

A small block of mass 1 kg is released from rest at the top of a rough track. The track is circular arc of radius 40 m. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point Q, as shown in the figure, below, is 150 J. (Take the acceleration due to gravity, g = 10 m/s^-2).

Q.

The magnitude of the normal reaction that acts on the block at the point Q is

Paragraph for Questions 11 to 12

A thermal power plant produces electric power of 600 kW at 4000 V, which is to be transported to a place 20 km away from the power plant for consumers’ usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers’ end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with the power factor unity. All the currents and voltage mentioned are rms values.

Q. 

If the direct transmission method with a cable of resistance 0.4 km^-1 is used, the power dissipation (in %) during transmission is

Paragraph

A thermal power plant produces electric power of 600 kW at 4000 V, which is to be transported to a place 20 km away from the power plant for consumers’ usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers’ end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with the power factor unity. All the currents and voltage mentioned are rms values.

Q.

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in
the secondary in the step-up transformer is 1 : 10. If the power to the consumers has to be supplied at
200 V, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer
is

Paragraph for Questions 13 to 14

A point Q is moving in a circular orbit of radius R in the x-y plane with an angular velocity ω. This can be considered as equivalent to a loop carrying a steady current Qω/2π. A uniform magnetic field along the positive z-axis is now switched on, which increases at a constant rate from 0 to B in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant γ.

Q.

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change, is

Paragraph

A point Q is moving in a circular orbit of radius R in the x-y plane with an angular velocity ω. This can be considered as equivalent to a loop carrying a steady current Qω/2π. A uniform magnetic field along the positive z-axis is now switched on, which increases at a constant rate from 0 to B in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant γ.

Q.

The change in the magnetic dipole moment associated with the orbit, at the end of time interval of the
magnetic field change, is

Paragraph for Questions 15 to 16

The mass of nucleus is less than the sum of the masses of (A-Z) number of neutrons and Z number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass M can break into two light nuclei of mass m₁ and m₂ only if (m₁ + m₂) < M. Also  two light nuclei of masses m3 and m4 can undergo complete fusion and form a heavy nucleus of mass M' only if (m₃ + m₄) > M'. The masses of some neutral atoms are given in the table below:

Q.

The correct statement is

The kinetic energy (in keV) of the alpha particle, when the nucleus  at rest undergoes alpha decay, is

SECTION – 3 (Matching List Type)

This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Q. No. 17-20 carry 3 marks each and 1 mark is deducted for every wrong answer.

Q.

A right angled prism of refractive index μ₁ is placed in a rectangular block of refractive index μ₂, which is surrounded by a medium of refractive index μ₃, as shown in the figure. A ray of light ‘e’ enters the rectangular block at normal incidence. Depending upon the relationships between μ₁, μ₂ and μ₃, it takes one of the four possible paths ‘ef’, ‘eg’, ‘eh’, or ‘ei’. 

Match the paths in List I with conditions of refractive indices in List II and select the correct answer using
the codes given below the lists:

Match List I with List II and select the correct answer using the codes given below the lists

One mole of mono-atomic ideal gas is taken along two cyclic processes E→F→G→E and E→F→H→E as shown in the PV diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic.

Match the paths in List I with the magnitudes of the work done in List II and select the correct answer using
the codes given below the lists.

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of
each process and then select the correct answer using the codes given below the lists:

SECTION –1

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

Q.No. 1-8 carry 3 marks each and 1 marks is deducted for every worng answer.

Q.

The K_sp of Ag₂CrO₄ is 1.1 x 10^–12 at 298K. The solubility (in mol/L) of Ag₂CrO₄ in a 0.1M AgNO₃ solution
is

In the following reaction, the product(s) formed is(are)

The major product(s) of the following reaction is (are)

After completion of the reactions (I and II), the organic compound(s) in the reaction mixtures is(are)

The correct statement(s) about O₃ is(are)

In the nuclear transmutation

(X, Y) is (are)

The carbon–based reduction method is NOT used for the extraction of

The thermal dissociation equilibrium of CaCO₃(s) is studied under different conditions

For this equilibrium, the correct statement(s) is(are)

SECTION-2 (Paragraph Type)

This section contains 4 paragraphs each describing theory, experiment, data etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of a paragraph has only one correct answer among the four choices (A), (B), (C) and (D).

Q.No. 29-36 carry 3 marks each and 1 mark is deducted for every wrong answer.

Paragraph for Question Nos. 29 and 30

An aqueous solution of a mixture of two inorganic salts, when treated with dilute HCl, gave a precipitate (P) and a filtrate (Q). The precipitate P was found to dissolve in hot water. The filtrate (Q) remained unchanged, when treated with H₂S in a dilute mineral acid medium. However, it gave a precipitate (R) with H₂S in an ammoniacal medium. The precipitate R gave a coloured solution (S), when treated with H₂O₂ in an aqueous NaOH medium

Q.

The precipitate P contains

Paragraph

An aqueous solution of a mixture of two inorganic salts, when treated with dilute HCl, gave a precipitate (P) and a filtrate (Q). The precipitate P was found to dissolve in hot water. The filtrate (Q) remained unchanged, when treated with H₂S in a dilute mineral acid medium. However, it gave a precipitate (R) with H₂S in an ammoniacal medium. The precipitate R gave a coloured solution (S), when treated with H₂O₂ in an aqueous NaOH medium

Q.

The coloured solution S contains

Paragraph for Question Nos. 31 to 32

P and Q are isomers of dicarboxylic acid C₄H₄O₄. Both decolorize Br₂/H₂O. On heating, P forms the cyclic anhydride. Upon treatment with dilute alkaline KMnO₄, P as well as Q could produce one or more than one from S, T and U.

Q.

Compounds formed from P and Q are, respectively

P and Q are isomers of dicarboxylic acid C₄H₄O₄. Both decolorize Br₂/H₂O. On heating, P forms the cyclic anhydride. Upon treatment with dilute alkaline KMnO₄, P as well as Q could produce one or more than one from S, T and U.

Q.

In the following reaction sequences V and W are, respectively

Paragraph for Question Nos. 33 to 34

A fixed mass ‘m’ of a gas is subjected to transformation of states from K to L to M to N and back to K as shown in the figure

Q.

The succeeding operations that enable this transformation of states are

A fixed mass ‘m’ of a gas is subjected to transformation of states from K to L to M to N and back to K as shown in the figure

Q.

The pair of isochoric processes among the transformation of states is

Paragraph for Question Nos. 35 to 36

The reactions of Cl₂ gas with cold-dilute and hot-concentrated NaOH in water give sodium salts of two (different) oxoacids of chlorine, P and Q, respectively. The Cl₂ gas reacts with SO₂ gas, in presence of charcoal, to give a product R. R reacts with white phosphorus to give a compound S. On hydrolysis, S gives an oxoacid of phosphorus, T

Q. 

 P and Q, respectively, are the sodium salts of

The reactions of Cl₂ gas with cold-dilute and hot-concentrated NaOH in water give sodium salts of two (different) oxoacids of chlorine, P and Q, respectively. The Cl₂ gas reacts with SO₂ gas, in presence of charcoal, to give a product R. R reacts with white phosphorus to give a compound S. On hydrolysis, S gives an oxoacid of phosphorus, T

Q.

R, S and T, respectively, are

SECTION – 3: (Matching List Type)

This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Q.No.  37 - 40 carry 3 marks each and 1 mark is deducted for every wrong answer

Q.

The unbalanced chemical reactions given in List – I show missing reagent or condition (?) which are
provided in List – II. Match List – I with List – II and select the correct answer using the code given below
the lists:

Match the chemical conversions in List – I with appropriate reagents in List – II and select the correct
answer using the code given below the lists:

An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List – I. The variation in
conductivity of these reactions in List – II. Match List – I with List – II and select the correct answer using
the code given below the lists:

The standard reduction potential data at 25^oC is given below :

Match E⁰ of the redox pair in List – I with the values given in List – II and select the correct answer using
the code given below the lists:

SECTION - 1 : (One or more option correct Type)

Q. No. 1 -8 carry 3 marks each and  1  mark is deducted for every wrong answer

This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

Q.

For a  R (the set of all real numbers) 

Then a =

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 2√7 on y-axis is
(are)

Two lines  are coplanar. Then α can take value(s)

In a triangle PQR, P is the largest angle and cos P=1/3 . Further the incircle of the triangle touches the sides
PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even
integers. Then possible length(s) of the side(s) of the triangle is (are)

Let   and H₂ =  where C is the set of all complex numbers.   and O represents the origin, then z₁ Oz₂ =

If 3^x = 4^x-1, then x =

Let ω be a complex cube root of unity with ω 1 and P = [p_ij] be a n x n matrix with p_ij = ω^i+j. Then P² 0,
when n =

The function   has a local minimum or a local maximum at x =

SECTION - 2 : (Paragraph Type)

Q. No. 49 - 54 carry 3 marks each and 1 mark is deducted for every wrong answer.

This section contains 6 multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct

Paragraph for Questions 49 and 50

Let f : [0, 1] → R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0) =  f(1) = 0 and satisfies 

Q.

Which of the following is true for 0 < x < 1 ?

Let f : [0, 1] → R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0) =  f(1) = 0 and satisfies 

Q.

If the function e^-x f(x) assumes its minimum in the interval [0, 1] at x = 1/4 which of the following is true ?

Paragraph for Questions 51 and 52

Let PQ be a focal chord of the parabola y² = 4ax. The tangents to the parabola at P and Q meet at a point lying on the line y = 2x + a, a > 0.

Q.

Length of chord PQ is

Let PQ be a focal chord of the parabola y² = 4ax. The tangents to the parabola at P and Q meet at a point lying on the line y = 2x + a, a > 0.

Q.

If chord PQ subtends an angle θ at the vertex of y² = 4ax, then tanθ =

Paragraph for Questions 53 and 54

Let  where

Q.

Area of S =

Let  where

Q.

Paragraph for Questions 55 and 56

A box B₁ contains 1 white ball, 3 red balls and 2 black balls. Another box B₂ contains 2 white balls, 3 red balls and 4 black balls. A third box B₃ contains 3 white balls, 4 red balls and 5 black balls.

Q.

If 1 ball is drawn from each of the boxes B₁, B₂ and B₃, the probability that all 3 drawn balls are of the
same colour is

A box B₁ contains 1 white ball, 3 red balls and 2 black balls. Another box B₂ contains 2 white balls, 3 red balls and 4 black balls. A third box B₃ contains 3 white balls, 4 red balls and 5 black balls

Q.

If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box B₂ is

SECTION - 3 : (Matching list Type)

Q. No. 57 - 60 carry 3 marks each and 1 mark is deducted for every wrong answer

This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Q.

Match List I with List II and select the correct answer using the code given below the lists :

A line L : y = mx + 3 meets y-axis at E(0, 3) and the arc of the parabola y2 = 16x, 0 y 6 at the point
F(x₀, y₀). The tangent to the parabola at F(x₀, y₀) intersects the y-axis at G(0, y₁). The slope m of the line L
is chosen such that the area of the triangle EFG has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists

Match List I with List II and select the correct answer using the code given below the lists :

Consider the lines  and the planes P₁ : 7x + y + 2z = 3, P2: 3x + 5y - 6z = 4. Let ax + by + cz = d be the equation of the plane passing through the point of
intersection of lines L₁ and L₂, and perpendicular to planes P₁ and P₂.
Match List I with List II and select the correct answer using the code given below the lists :