JEE Advanced 2015 Paper -2 with Solutions

The densities of two solid spheres A and B of the same radii R vary with radial distance r as ρ_A(r) =

, respectively, where k is a constant. The moments of inertia of the individual spheres about axes passing through their centres are I_A and I_B, respectively. If 

the value of n is

Four harmonic waves of equal frequencies and equal intensities I0 have phase angles 0, π/3, 2π/3 and π.
When they are superposed, the intensity of the resulting wave is nI₀. The value of n is

For a radioactive material, its activity A and rate of change of its activity R are defined as   and   , where N(t) is the number of nuclei at time t. Two radioactive sources P (mean life  ) and
Q(mean life  2) have the same activity at t = 0. Their rates of change of activities at t = 2 are RP and RQ,
respectively. If  , then the value of n is

A monochromatic beam of light is incident at 60⁰ on one face of an equilateral prism of refractive index n and
emerges from the opposite face making an angle θ(n) with the normal (see the figure). For n = √3 the value of θ is 60⁰ and . The value of m is

In the following circuit, the current through the resistor R (=2) is I Amperes. The value of I is

Section 2

Q. No. 9 -18 carry 4 marks each and 2 marks is deducted for every wrong answer.

Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct.

Q.

A fission reaction is given by  where x and y are two particles. Considering    to be at rest, the kinetic energies of the products are denoted by K_Xe, K_Sr, K_x(2MeV) and K_y(2MeV),

respectively. Let the binding energies per nucleon of  be 7.5 MeV, 8.5 MeV and 8.5 MeV respectively. Considering different conservation laws, the correct option(s) is(are)

Two spheres P and Q of equal radii have densities ρ₁ and ρ₂, respectively. The spheres are connected by a massless string and placed in liquids L₁ and L₂ of densities σ₁ and σ₂ and viscosities η₁ and
η₂, respectively. They float in equilibrium with the sphere P in L₁ and sphere Q in L₂ and the string being taut (see figure). If sphere P alone in L₂ has terminal velocity    and Q alone in L₁ has terminal velocity ,
ρ then

In terms of potential difference V, electric current I, permittivity ε₀, permeability μ₀ and speed of light c,
the dimensionally correct equation(s) is(are)

Consider a uniform spherical charge distribution of radius R1 centred at the origin O. In this distribution, a spherical cavity of radius R₂, centred at P with distance OP = a = R₁ – R₂ (see figure) is made. If the electric field inside the cavity at position   then the correct statement(s) is(are)

In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is(are)

A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own
gravity. If P(r) is the pressure at r(r < R), then the correct option(s) is(are)

parallel plate capacitor having plates of area S and plate separation d, has capacitance C₁ in air. When
two dielectrics of different relative permittivities (ε₁ = 2 and ε₂ = 4) are introduced between the two plates
as shown in the figure, the capacitance becomes C₂. The ratio C₂/C₁ is

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the
figure). Initially the gas is at temperature T₁, pressure P₁ and volume V₁ and the spring is in its relaxed state.
The gas is then heated very slowly to temperature T₂, pressure P₂ and volume V₂. During this process the
piston moves out by a distance x. Ignoring the friction between the piston and the cylinder, the correct
statement(s) is(are)

SECTION 3

Q. No.  17-20 carry  4 marks each and 2 marks is deducted for every wrong answer.

This section contains TWO paragraphs

Based on each paragraph, there will be TWO questions

Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct

PARAGRAPH 1

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index n₁ surrounded by a medium of lower refractive index n₂. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media n₁ and n₂ as shown in the figure. All rays with the angle of incidence i less than a particular value i_m are confined in the medium of refractive index n₁. The numerical aperture (NA) of the structure is defined as sin i_m.

Q.

For two structures namely S₁ with  n₁ =   and n₂₌ 3 / 2, and S₂ with n₁ = 8/5 and n₂ = 7/5 and
taking the refractive index of water to be 4/3 and that of air to be 1, the correct option(s) is(are)

PARAGRAPH 1

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index n₁ surrounded by a medium of lower refractive index n₂. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media n₁ and n₂ as shown in the figure. All rays with the angle of incidence i less than a particular value i_m are confined in the medium of refractive index n₁. The numerical aperture (NA) of the structure is defined as sin i_m.

Q.

If two structures of same cross-sectional area, but different numerical apertures NA₁ and
 NA₂ (NA₂ < NA₁ ) are joined longitudinally, the numerical aperture of the combined structure is

PARAGRAPH 2
In a thin rectangular metallic strip a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are , w and d, respectively. A uniform magnetic field  is applied on the strip along the positive y-direction. Due to this, the charge carriers experience a net deflection along the zdirection. This results in accumulation of charge carriers on the surface PQRS and appearance of equal and opposite charges on the face opposite to PQRS. A potential difference along the z-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.

Q.

Consider two different metallic strips (1 and 2) of the same material. Their lengths are the same, widths are
w₁ and w₂ and thicknesses are d₁ and d₂, respectively. Two points K and M are symmetrically located on
the opposite faces parallel to the x-y plane (see figure). V₁ and V₂ are the potential differences between K
and M in strips 1 and 2, respectively. Then, for a given current I flowing through them in a given magnetic
field strength B, the correct statement(s) is(are)

PARAGRAPH 2
In a thin rectangular metallic strip a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are , w and d, respectively. A uniform magnetic field  is applied on the strip along the positive y-direction. Due to this, the charge carriers experience a net deflection along the zdirection. This results in accumulation of charge carriers on the surface PQRS and appearance of equal and opposite charges on the face opposite to PQRS. A potential difference along the z-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.

Q.

Consider two different metallic strips (1 and 2) of same dimensions (lengths , width w and thickness d)
with carrier densities n₁ and n₂, respectively. Strip 1 is placed in magnetic field B₁ and strip 2 is placed in
magnetic field B₂, both along positive y-directions. Then V₁ and V₂ are the potential differences developed
between K and M in strips 1 and 2, respectively. Assuming that the current I is the same for both the strips,
the correct option(s) is(are)

SECTION 1

Q. No. 21 -28 carry 4 marks each.

The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive

Q.

In dilute aqueous H₂SO₄, the complex diaquodioxalatoferrate(II) is oxidized by . For this reaction, the ratio of the rate of change of [H⁺] to the rate of change of  is

The number of hydroxyl group(s) in Q is

Among the following, the number of reaction(s) that produce(s) benzaldehyde is

In the complex acetylbromidodicarbonylbis(triethylphosphine)iron(II), the number of Fe–C bond(s) is

Among the complex ions,   

 number of complex
ion(s) that show(s) cis-trans isomerism is

Three moles of B₂H₆ are completely reacted with methanol. The number of moles of boron containing
product formed is

The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar
conductivity of a solution of a weak acid HY (0.10 M). If  the difference in their pK_a values,
pK_a (HX) - pK_a (HY), is (consider degree of ionization of both acids to be << 1)

closed vessel with rigid walls contains 1 mol of  and 1 mol of air at 298 K. Considering complete
decay of  , the ratio of the final pressure to the initial pressure of the system at 298 K is

SECTION 2

Q. No. 29-36 carry 4 marks each and 2 marks is deducted for every wrong answer.

Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct

Q.

One mole of a monoatomic real gas satisfies the equation p(V – b) = RT where b is a constant. The
relationship of interatomic potential V(r) and interatomic distance r for the gas is given by

In the following reactions, the product S is

The major product U in the following reactions is

In the following reactions, the major product W is

The correct statement(s) regarding, (i) HClO, (ii) HClO₂, (iii) HClO₃ and (iv) HClO₄, is (are)

The pair(s) of ions where BOTH the ions are precipitated upon passing H₂S gas in presence of dilute HCl,
is(are)

Under hydrolytic conditions, the compounds used for preparation of linear polymer and for chain
termination, respectively, are

When O₂  is adsorbed on a metallic surface, electron transfer occurs from the metal to O₂. The TRUE
statement(s) regarding this adsorption is(are)

SECTION 3

Q. No 37 - 40 carry 4 marks each and 2 marks is deducted for every wrong answer

This section contains TWO paragraphs
Based on each paragraph, there will be TWO questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct

PARAGRAPH 1

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of 5.7^oC was measured for the beaker and its contents (Expt. 1). Because the enthalpy of neutralization of a strong acid with a strong base is a constant (-57.0 kJ mol^-1), this experiment could be used to measure the calorimeter constant. In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid (K_a = 2.0 × 10^-5) was mixed with 100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperature rise of 5.6^oC  as measured.

(Consider heat capacity of all solutions as 4.2 J g^-1 K^-1 and density of all solutions as 1.0 g mL^-1)

Q.

Enthalpy of dissociation (in kJ mol-1) of acetic acid obtained from the Expt. 2 is

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of 5.7^oC was measured for the beaker and its contents (Expt. 1). Because the enthalpy of neutralization of a strong acid with a strong base is a constant (-57.0 kJ mol^-1), this experiment could be used to measure the calorimeter constant. In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid (K_a = 2.0 × 10^-5) was mixed with 100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperature rise of 5.6^oC  as measured.

(Consider heat capacity of all solutions as 4.2 J g^-1 K^-1 and density of all solutions as 1.0 g mL^-1)

Q.

The pH of the solution after Expt. 2 is

PARAGRAPH 2
In the following reactions

Q.

Compound X is

In the following reactions

Q.

The major compound Y is

Section 1

Q. No. 41- 48 carry 4 marks each.

The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive

Q.

Suppose that   are three non-coplanar vectors in  R³. Let the components of a vector along  be 4, 3 and 5, respectively. If the components of this vector  and  are x, y and z, respectively, then the value of 2x + y + z is

For any integer k, let  . The value of the expression

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of
the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130
and 140, then the common difference of this A.P. is

The coefficient of x⁹ in the expansion of  (1 + x) (1 + x²) (1 + x³) ….. (1 + x¹⁰⁰) is

Suppose that the foci of the ellipse  are (f₁, 0) and (f₂, 0) where f₁ > 0 and f₂ < 0. Let P₁ and P₂
be two parabolas with a common vertex at (0, 0) and with foci at (f₁, 0) and (2f₂, 0), respectively. Let T₁ be
a tangent to P₁ which passes through (2f₂, 0) and T₂ be a tangent to P₂ which passes through (f1, 0). The m1
is the slope of T₁ and m₂ is the slope of T₂, then the value of 

Let m and n be two positive integers greater than 1. If 

If 

where tan^-1x takes only principal values, then the value of 

Let   be a continuous odd function, which vanishes exactly at one point and f(1) =1/2.  Suppose
that  and  then the value of f(1/2) is

Section 2

Q. No. 49 - 56 carry 4 marks eah and 2 mark is deducted forr every wrong answer.

Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.

Q.

Let   for all  , then the possible values of m and M are

Let S be the set of all non-zero real numbers α such that the quadratic equation αx² - x + α = 0 has two
distinct real roots x¹ and x² satisfying the inequality x¹ - x² < 1. Which of the following intervals is(are) a
subset(s) of S ?

If  , where the inverse trigonometric functions take only the principal
values, then the correct option(s) is(are)

Let E₁ and E₂ be two ellipses whose centers are at the origin. The major axes of E₁ and E₂ lie along the
x-axis and the y-axis, respectively. Let S be the circle x² + (y - 1)² = 2. The straight line x + y = 3 touches the curves S, E₁ and E₂ at P, Q and R, respectively. Suppose that PQ = PR =  . If e₁ and e₂ are the
eccentricities of E₁ and E₂, respectively, then the correct expression(s) is(are)

Consider the hyperbola H : x² - y² = 1 and a circle S with center N(x₂, 0). Suppose that H and S touch each
other at a point P(x₁, y₁) with x₁ > 1 and y₁ > 0. The common tangent to H and S at P intersects the x-axis at
point M. If (l, m) is the centroid of the triangle ΔPMN, then the correct expression(s) is(are)

The option(s) with the values of a and L that satisfy the following equation is(are)

Let   be continuous functions which are twice differentiable on the interval (-1, 2). Let the
values of f and g at the points -1, 0 and 2 be as given in the following table:

In each of the intervals (-1, 0) and (0, 2) the function (f - 3g)" never vanishes. Then the correct
statement(s) is(are)

Let f (x) = 7tan⁸x + 7tan⁶x - 3tan⁴x - 3tan²x for all  Then the correct expression(s) is(are)

SECTION 3

Q. No. 57 - 60 carry  4 marks each and 2 mark is deducted for every wrong answer.

This section contains TWO paragraphs.
Based on each paragraph, there will be TWO questions
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct

PARAGRAPH 1

Let   be a thrice differentiable function. Suppose that F(1) = 0, F(3) = -4 and F'(x) < 0 for all x  (1/2, 3). Let f (x) = xF(x) for all  

Q.

The correct statement(s) is(are)

PARAGRAPH 

Let   be a thrice differentiable function. Suppose that F(1) = 0, F(3) = -4 and F'(x) < 0 for all x  (1/2, 3). Let f (x) = xF(x) for all  

Q.

If  then the correct expression(s) is(are)

PARAGRAPH 2

Let n₁ and n₂ be the number of red and black balls, respectively, in box I. Let n₃ and n₄ be the number of red and black balls, respectively, in box II.

Q.

One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is 1/3 then the correct option(s) with the possible values of n₁, n₂, n₃ and n₄ is(are)

PARAGRAPH 2

Let n₁ and n₂ be the number of red and black balls, respectively, in box I. Let n₃ and n₄ be the number of red and black balls, respectively, in box II.

Q.

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from
box I, after this transfer, is 1/3,  then the correct option(s) with the possible values of n₁ and n₂ is(are)