JEE Advanced 2018 Question Paper with Solutions (20th May - Morning) Attempt

The potential energy of a particle of mass m at a distance r from a fixed point O is given by (r)=kr²/2, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

Consider a body of mass 1.0 ka at rest at the origin at time t =0. A force is applied on the body, where α= 1.0 Ns^-1 and β=1.0 N. The torque acting on the body about the origin at time t = 1.0 s is . Which of the following statements is (are) true?

A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water. The water rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of water is σ. The angle of contact between water and the wall of the capillary tube is θ. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

In the figure below, the switches S₁ and S₂ are closed simultaneously at t = 0 and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude I_max at time t =τ. Which of the following statements is (are) true?

Two infinitely long straight wires lie in the xy-plane along the lines x = ±R. The wire located at x = +R carries a constant current I₁ and the wire located at x = -R carries a constant current I₂. A circular loop of radius R is suspended with its centre at (0, 0,√3R) and in a plane parallel to the xy-plane. This loop carries a constant current I in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the direction. Which of the following statements regarding the magnetic field is (are) true?

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?

Two vectors and are defined as and , where a is a constant and
ω = π/6 rad s^-1. If at time t = τ for the first time, the value of τ, in seconds, is  _________  .

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed 1.0 ms^-1 and the man behind walks at a speed 2.0 ms^-1. A third man is standing at a height 12m above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430 Hz. The speed of sound in air is 330 ms^-1. At the instant, when the moving men are 10 m apart, the stationary man is equidistant from them. The frequency of beats in Hz, heard by the stationary man at this instant, is_________. 

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is  then the height of the top of the inclined plane, in meters, is __________. Take g = 10m s^-2.

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 Nm^-1 and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition.
Another block of mass 1.0 kg moving with a speed of 2.0 ms^-1 collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is ____________ .

Three identical capacitors C₁, C₂ and C₃ have a capacitance of 1.0 μF each and they are uncharged initially. They are connected in a circuit as shown in the figure and C₁ is then filled completely with a dielectric material of relative permittivity ∈_r. The cell electromotive force (emf) V_o = 8 V. First the switch S₁ is closed while the switch S₂ is kept open. When the capacitor C₃ is fully charged, S₁ is opened and S₂ is closed simultaneously. When all the capacitors reach equilibrium, the charge on C₃  is found to be 5 μC.  The value of  ∈_r = ______________.

In the xy-plane, the region y > 0 has a uniform magnetic field and the region y < 0 has another uniform magnetic field  A positively charged particle is projected from the origin along the positive y-axis with speed V_o=πms^-1 at t = 0, as shown in the figure. Neglect gravity in this problem. Let t = T be the time when the particle crosses the x-axis from below for the first time. If B₂ = 4B₁, the average speed of the particle, in ms^-1, along the x-axis in the time interval T is .

Sunlight of intensity 1.3 kW m^-2 is incident normally on a thin convex lens of focal length 20 cm. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in kW m^-2, at a distance 22 cm from the lens on the other side is__________.

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures T₁ = 300 K and T₂ = 100 K, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are K₁ and K₂ respectively. If the temperature at the junction of the two cylinders in the steady state is 200 K, then K₁/K_{2 = ________________.}

PARAGRAPH “X”
In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [E] and [B] stand for dimensions of electric and magnetic fields respectively, while [∈O] and [μO] stand for dimensions of the permittivity and permeability of free space respectively. [L] and [T] are dimensions of length and time respectively. All the quantities are given in SI units.
Q. The relation between [E] and [B] is

PARAGRAPH “X”
In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [E] and [B] stand for dimensions of electric and magnetic fields respectively, while [∈_O] and [μ_O] stand for dimensions of the permittivity and permeability of free space respectively. [L] and [T] are dimensions of length and time respectively. All the quantities are given in SI units.
Q. The relation between [∈_O] and [μ_O] is

PARAGRAPH “A”
If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are Δx, Δy and Δz, respectively, then
. The series expansion for  to first power in Δy/y is   The relative errors in independent variables are always added. So the error in z will be

The above derivation makes the assumption that Δx/x << 1, Δy/y << 1,  Therefore, the higher powers of these quantities are neglected.
Q. Consider the ratio  to be determined by measuring a dimensionless quantity a. If the error in the measurement of a is  then what is the error Δr in determining r?

PARAGRAPH “A”
If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are Δx, Δy and Δz, respectively, then

The series expansion for  to first power in Δy/y, is  The relative errors in independent variables are always added. So the error in z will be

The above derivation makes the assumption that Δx/x <<1, Δy/y << 1. Therefore, the higher powers of these quantities are neglected.
​Q. In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed in the first 1.0 s. For up to first power in x. The error Δλ, in the determination of the decay constant λ, in s^-1, is

The compound(s) which generate(s) N₂ gas upon thermal decomposition below 300 °C is (are)

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: Fe = 26, Ni = 28 )

Based on the compounds of group 15 elements, the correct statement(s) is (are)

In the following reaction sequence, the correct structure(s) of X is (are)

The reaction(s) leading to the formation of 1,3,5-trimethylbenzene is (are)

A reversible cyclic process for an ideal gas is shown below. Here, P, V, and T are pressure, volume, and temperature, respectively. The thermodynamic parameters q, w, H, and U are heat, work, enthalpy and internal energy, respectively.

The correct option(s) is (are)

Among the species given below, the total number of diamagnetic species is ______. H atom, NO₂ monomer, O^2- (superoxide), dimeric sulphur in vapour phase, Mn₃O₄, (NH₄)₂[FeCl₄], (NH₄)₂[NiCl₄], K₂MnO₄, K₂CrO₄

The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by NiCl₂.6H₂O to form a stable coordination compound. Assume that both the reactions are 100% complete. If 1584 g of ammonium sulphate and 952 g of NiCl₂.6H₂O are used in the preparation, the combined weight (in grams) of gypsum and the nickel-ammonia coordination compound thus produced is______.
(Atomic weights in g mol^-1: H = 1, N = 14, O = 16, S = 32, Cl = 35.5, Ca = 40, Ni = 59)

Consider an ionic solid MX with NaCl structure. Construct a new structure (Z) whose unit cell is constructed from the unit cell of MX following the sequential instructions given below. Neglect the charge balance.
(i) Remove all the anions (X) except the central one
(ii) Replace all the face centered cations (M) by anions (X)
(iii) Remove all the corner cations (M)
(iv) Replace the central anion (X) with cation (M)

For the electrochemical cell,

the standard emf of the cell is 2.70 V at 300 K. When the concentration of Mg²⁺ is changed to x M, the cell potential changes to 2.67 V at 300 K. The value of x is ____. (given, where F is the Faraday constant and R is the gas constant, ln(10) = 2.30)

A closed tank has two compartments A and B, both filled with oxygen (assumed to be ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Figure 1). If the old partition is replaced by a new partition which can slide and conduct heat but does NOT allow the gas to leak across (Figure 2), the volume (in m³) of the compartment A after the system attains equilibrium is ____.

Liquids A and B form ideal solution over the entire range of composition. At temperature T, equimolar binary solution of liquids A and B has vapour pressure 45 Torr. At the same temperature, a new solution of A and B having mole fractions x_a and x_b, respectively, has vapour pressure of 22.5 Torr. The value of x_a/x_b in the new solution is______. (given that the vapour pressure of pure liquid A is 20 Torr at temperature T)