JEE Main 2011 Paper - 1 with Solutions

5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial temperature to be
T₁, the work done in the process is

A 2 μF capacitor is charged as shown in the figure. The percentage of its stored energy dissipated after the switch S is turned to position 2 is

A meter bridge is set-up as shown, to determine an unknown resistance ‘X’ using a standard 10 ohm resistor.
The galvanometer shows null point when tapping-key is at 52 cm mark. The end-corrections are 1 cm and 2 cm respectively for the ends A and B. The determined value of ‘X’ is

A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is

SECTION – II (Total Marks : 16)

(Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of
which ONE OR MORE may be correct.

Q. A metal rod of length ‘L’ and mass ‘m’ is pivoted at one end. A thin disk of mass ‘M’ and radius ‘R’ (<L) is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A). The disc is not free to rotate about its centre and (case B) the disc is free to rotate about its centre. The rod-disc system
performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

A spherical metal shell A of radius R_A and a solid metal sphere B of radius R_B (<R_A) are kept far apart and
each is given charge ‘+Q’. Now they are connected by a thin metal wire. Then

A composite block is made of slabs A, B, C, D and E of different thermal conductivities (given in terms of a
constant K) and sizes (given in terms of length, L) as shown in the figure. All slabs are of same width. Heat
‘Q’ flows only from left to right through the blocks. Then in steady state

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semiinfinite
region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is / are true?

SECTION-III (Total Marls : 15)

(Paragraph Type)

This section contains 2 paragraphs. Based upon one of paragraphs 2 multiple choice questions and based on the other paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question Nos. 12 and 13

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let ‘N’ be the number density of free electrons, each of mass ‘m’. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency ‘wp’ which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency w, where a part of the energy is absorbed and a part of it is reflected. As w approaches wp all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

Q. Taking the electronic charge as ‘e’ and the permittivity as ‘e₀’. Use dimensional analysis to determine the
correct expression for ω_p.

Paragraph for Question Nos. 12 and 13

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let ‘N’ be the number density of free electrons, each of mass ‘m’. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency ‘ω_p’ which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency ω, where a part of the energy is absorbed and a part of it is reflected. As ω approaches ω_p all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

Q. Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons
 where these quantities are in proper SI units.

Paragraph for Question Nos. 14 to 16

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momenum are changed. Here we consider some simple dynamical systems in onedimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which positon or momentum upwards (or to right) is positive and downwards (or to left) is negative. 

Q. The phase space diagram for a ball thrown vertically up from ground is

The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and E₁ and E₂ are the total mechanical energies respectively. Then

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

SECTION-IV (Total Marks : 28)

(Integer Answer Type)

This section contains 7 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9.

Enter only the numerical value in the space provided below.

Q. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2N on the ring and rolls it without slipping with an acceleration of 0.3 m/s². The coefficient of
friction between the ground and the ring is large enough that rolling always occurs and the coefficeint of friction between the stick and the ring is (P/10). The value of P is

Steel wire of lenght ‘L' at 40°C is suspended from the ceiling and then a mass ‘m’ is hung from its free end. The wire is cooled down from 40°C to 30°C to regain its original length ‘L’. The coefficient of linear thermal expansion of the steel is 10^-5/°C, Young’s modulus of steel is 10¹¹ N/m² and radius of the wire is 1 mm. Assume that L diameter of the wire. Then the value of ‘m’ in kg is nearly

Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap film of side ‘a’.
The surface tension of the soap film is γ. The system of charges and planar film are in equilibrium, and
  where ‘k’ is a constant. Then N is

A block is moving on an inclined plane making an angle 45° with the horizontal and the coefficient of friction is m. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define N = 10 μ, then N is

The activity of a freshly prepared radioactive sample is 10¹⁰ disintegrations per second, whose mean life is
10⁹ s. The mass of an atom of this radioisotope is 10^-25 kg. The mass (in mg) of the radioactive sample is

A long circular tube of length 10 m and radius 0.3 m carries a current I along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as I = I₀cos(300 t) where I₀ is constant. If the magnetic moment of the loop is Nm₀I₀sin(300 t), then ‘N’ is

Four solid spheres each of diameter cm and mass 0.5 kg are placed with their centers at the corners of
a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N x 10^-4 kgm²,
then N is

SECTION – I (Total Marks : 21)

(Single Correct Answer Type)

This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of
which ONLY ONE is correct.

Q. Extra pure N₂ can be obtained by heating

Dissolving 120 g of urea (mol. wt. 60) in 1000 g of water gave a solution of density 1.15 g/mL. The molarity of the solution is

Bombardment of aluminium by a-particle leads to its artificial disintegration in two ways, (i) and (ii) as shown. Products X, Y and Z respectively are,

Geometrical shapes of the complexes formed by the reaction of Ni²⁺ with Cl^– , CN^– and H₂O, respectively,
are

The major product of the following reaction is

Among the following compounds, the most acidic is

AgNO₃(aq.) was added to an aqueous KCl solution gradually and the conductivity of the solution was
measured. The plot of conductance (L) versus the volume of AgNO₃ is

SECTION – II (Total Marks : 16)

(Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of
which ONE OR MORE may be correct.

Q. Amongst the given options, the compound(s) in which all the atoms are in one plane in all the possible
conformations (if any), is (are)

Extraction of metal from the ore cassiterite involves

According to kinetic theory of gases

The correct statement(s) pertaining to the adsorption of a gas on a solid surface is (are)

SECTION-III (Total Marls : 15)

(Paragraph Type)

This section contains 2 paragraphs. Based upon one of paragraphs 2 multiple choice questions and based on the other paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question Nos. 35 and 36

An acyclic hydrocarbon P, having molecular formula C₆H₁₀, gave acetone as the only organic product through the following sequence of reaction, in which Q is an intermediate organic compound.

Q. The structure of compound P is

Paragraph for Question Nos. 35 and 36

An acyclic hydrocarbon P, having molecular formula C₆H₁₀, gave acetone as the only organic product through the following sequence of reaction, in which Q is an intermediate organic compound.

Q. The structure of the compound Q is

Paragraph for Question Nos. 37 to 39

When a metal rod M is dipped into an aqueous colourless concentrated solution of compound N, the solution turns light blue. Addition of aqueous NaCl to the blue solution gives a white precipitate O. Addition of aqueous NH₃ dissolves O and gives an intense blue solution.

Q. The metal rod M is

Paragraph for Question Nos. 37 to 39

When a metal rod M is dipped into an aqueous colourless concentrated solution of compound N, the solution turns light blue. Addition of aqueous NaCl to the blue solution gives a white precipitate O. Addition of aqueous NH₃ dissolves O and gives an intense blue solution.

Q. The compound N is

Paragraph for Question Nos. 37 to 39

When a metal rod M is dipped into an aqueous colourless concentrated solution of compound N, the solution turns light blue. Addition of aqueous NaCl to the blue solution gives a white precipitate O. Addition of aqueous NH₃ dissolves O and gives an intense blue solution.

Q. The final solution contains

SECTION-IV (Total Marks : 28)

(Integer Answer Type)
This section contains 7 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9.

Enter the numerical value only in the space provided below.

Q. The difference in the oxidation numbers of the two types of sulphur atoms in Na₂S₄O₆ is

A decapeptide (Mol. Wt. 796) on complete hydrolysis gives glycine (Mol. Wt. 75), alanine and phenylalanine. Glycine contributes 47.0% to the total weight of the hydrolysed products. The number of glycine units present in the decapeptide is

The work function (φ) of some metals is listed below. The number of metals which will show photoelectric effect when light of 300 nm wavelength falls on the metal is

The maximum number of electrons that can have principal quantum number, n = 3, and spin quantum
number,  is

Reaction of Br₂ with Na₂CO₃ in aqueous solution gives sodium bromide and sodium bromate with evolution of CO₂ gas. The number of sodium bromide molecules involved in the balanced chemical equation is

To an evacuated vessel with movable piston under external pressure of 1 atm, 0.1 mol of He and 1.0 mol of
an unknown compound (vapour pressure 0.68 atm. at 0^oC) are introduced. Considering the ideal gas behaviour, the total volume (in litre) of the gases at 0^oC is close to

The total number of alkenes possible by dehydrobromination of 3-bromo-3-cyclopentylhexane using alcoholic KOH is

SECTION – I (Total Marks : 21)

(Single Correct Answer Type)

This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of
which ONLY ONE is correct.

Q. Let (x₀, y₀) be solution of the following equations

Then x₀ is

Let P = {θ : sinθ - cosθ = cosθ} and Q = {θ : sin θ + cosθ = sin θ} be two sets. Then

Let  be three vectors. A vector , whose projection on , is given by

The value of 

A straight line L through the point (3, -2) is inclined at an angle 60° to the line . If L also
intersects the x-axis, then the equation of L is

Let α and β be the roots of x² – 6x – 2 = 0, with a > b. If a_n = α_n – β_n for n > 1, then the value of is

Let the straight line x = b divides the area enclosed by y = (1 - x)², y = 0 and x = 0 into two parts R₁(0 < x < b) and R₂(b < x < 1) such that R₁ - R₂ = 1/4. Then b equals

SECTION – II (Total Marks : 16)

(Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of
which ONE or MORE may be correct.

Q. Let f: R → R be a function such that If f(x) is differentiable at x = 0, then

The vector(s) which is/are coplanar with vectors and perpendicular to the vector  is/are

Let the eccentricity of the hyperbola  be reciprocal to that of the ellipse x² + 4y² = 4. If the
hyperbola passes through a focus of the ellipse, then

Let M and N be two 3 x 3 non-singular skew symmetric matrices such that MN = NM. If P^T denotes the
transpose of P, then M²N² (M^TN)–1 (MN^–1)^T is equal to

SECTION-III (Total Marls : 15)

(Paragraph Type)

This section contains 2 paragraphs. Based upon one of paragraphs 2 multiple choice questions and based on the other paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question Nos. 58 to 59

Let U₁ and U₂ be two urns such that U1 contains 3 white and 2 red balls, and U₂ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U₁ and put into U₂. However, if tail appears then 2 balls are drawn at random from U₁ and put into U₂. Now 1 ball is drawn at random from U₂.

Q. The probability of the drawn ball from U₂ being white is

Paragraph for Question Nos. 58 to 59

Let U₁ and U₂ be two urns such that U1 contains 3 white and 2 red balls, and U₂ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U₁ and put into U₂. However, if tail appears then 2 balls are drawn at random from U₁ and put into U₂. Now 1 ball is drawn at random from U₂.

Q. Given that the drawn ball from U₂ is white, the probability that head appeared on the coin is

Paragraph for Question Nos. 60 to 62

Let a, b and c be three real numbers satisfying 

Q. If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

Paragraph for Question Nos. 60 to 62

Let a, b and c be three real numbers satisfying 

Q. Let w be a solution of x³ – 1 = 0 with I_m(ω) > 0. If a = 2 with b and c satisfying (E), then the value of
 is equal to

Paragraph for Question Nos. 60 to 62

Let a, b and c be three real numbers satisfying 

Q. Let b = 6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax² + bx + c = 0,
then 

SECTION-IV (Total Marks : 28)

(Integer Answer Type)

This section contains 7 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9.

Enter only the numerical value in space provided below.

Q. Consider the parabola y² = 8x. Let Δ₁ be the area of the triangle formed by the end points of its latus rectum and the point  on the parabola, and Δ₂ be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum. Then  is

Let f: [1, ∞) → [2, ∞) be a differentiable function such that f(1) = 2. If  = 3xf(x) - x³ for all x >1, then the value of f(2) is

The positive integer value of n > 3 satisfying the equation 

Let a₁, a₂, a₃, …, a₁₀₀ be an arithmetic progression with a₁ = 3 and S_p = , 1 < p < 100. For any integer
n with 1 < n < 20, let m = 5n. If  does not depend on n, then a₂ is

If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| is

The minimum value of the sum of real numbers a^–5, a^–4, 3a^–3, 1, a⁸ and a¹⁰ with a > 0 is