JEE (Advanced) 2018 Paper - 1

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 . 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 + ˆj 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  are defined as , where a is a constant and  at time t = for the first time, the value of , 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 12 m 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 H_z. 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 H_z, 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 metres, is __________. Take g = 10 ms^–2.

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 N m⁻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 m s⁻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 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 . The cell electromotive force (emf) V₀ = 8V. 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  The value of =_________. 

In the xy-plane, the region y > 0 has a uniform magnetic field B₁ kˆ and the region y < 0 has another uniform magnetic field B₂ kˆ. A positively charged particle is projected from the origin along the positive y-axis with speed  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 kWm^–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 / K₂ =__________.

The relation between [E] and [B] is

The relation between [ ] and [ ] is

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 ?

In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed in the first 1.0s. For |x| ≪ 1, ln(1 + x) = x up to first power in x . The error , in the determination of the decay constant 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)

B ased 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, O₂ ⁻ (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)
The value of in Z is _______ .

F or the electrochemical cell,
Mg(s) | Mg²⁺ (aq, 1 M) || Cu²⁺ (aq, 1 M) | Cu(s) 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, = 11500 K V⁻¹, 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)

The solubility of a salt of weak acid (AB) at pH 3 is Y × 10^–3 mol L⁻¹. The value of Y is ____.
(Given that the value of solubility product of AB (K_sp) = 2 × 10^–10 and the value of ionization constant of HB (K_a) = 1 × 10^–8)

The plot given below shows P—T curves (where P is the pressure and T is the temperature) for two
solvents X and Y and isomolal solutions of NaCl in these solvents. NaCl completely dissociates in both the
solvents.

On addition of equal number of moles of a non-volatile solute S in equal amount (in kg) of these solvents, the elevation of boiling point of solvent X is three times that of solvent Y. Solute S is known to undergo dimerization in these solvents. If the degree of dimerization is 0.7 in solvent Y, the degree of dimerization in solvent X is _________.

The compound Y is

The compound Z is

The compound P undergoes the following reactions and forms the compound R. The compound R is: 

The compound S is

For a non-zero complex number z, let arg(z) denote the principal argument with < arg(z) Then, which of the following statement(s) is (are) FALSE ?

In a triangle PQR, let ∠PQR = 30° and the sides PQ and QR have lengths 10√3 and 10, respectively.
Then, which of the following statement(s) is (are) TRUE ?

Let P₁ : 2x + y – z = 3 and P₂ : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is (are) TRUE ?

For every twice differentiable function f : R → [–2, 2] with (f(0))² + (f '(0))² = 85, which of the following
statement(s) is (are) TRUE ?

Let f : R → R and g : R → R be two non-constant differentiable functions. If

and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE ?

The value of  is ______ .

Let f : [0, ∝) → R be a continuous function such that

for all x ∈[0, ∝). Then, which of the following statement(s) is (are) TRUE ?

The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the
repetition of digits is allowed, is ______ .

Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ….. , and Y be the set consisting of the first 2018 terms of arithmetic progression 9, 16, 23, ….. . Then, the number of
elements in the set X ∪ Y is _____ .

The number of real solutions of the equation

lying in the interval 
(Here, the inverse trigonometric functions sin^–1x and cos^–1x assume values in respectively.)

For each positive integer n, let .
For x ∈ R, let [x] be the greatest integer less than or equal to x. If  ,  then the value of [L] is_____ .

Let  and   be two unit vectors such that . For some x, y ∈ R, let  and the vector c  is inclined at the same angle α to both and , then the value of 8cos² α is _____ .

Let a, b, c be three non - zero real numbers such that the equation

 has two distinct real roots α and ß with . Then, the value of is ____ .

A farmer F₁ has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring f armer F₂ takes away the region which lies between the side PQ and a curve of the form y = xⁿ (n > 1). If the area of the region taken away by the farmer F₂ is exactly 30% of the area of ΔPQR, then the value of n is _____ .

Let E₁E₂ and F₁F₂ be the chords of S passing through the point P₀(1, 1) and parallel to the x-axis and the yaxis, respectively. Let G₁G₂ be the chord of S passing through P0 and having slope –1. Let the tangents to S at E₁ and E₂ meet at E₃, the tangents to S at F₁ and F₂ meet at F₃, and the tangents to S at G₁ and G₂ meet at G₃. Then, the points E₃, F₃, and G₃ lie on the curve

Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect thecoordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

The probability that, on the examination day, the student S₁ gets the previously allotted seat R₁ , and
NONE of the remaining students gets the seat previously allotted to him/her is

For i = 1, 2, 3, 4, let Ti denote the event that the students S_i and S_i+1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T₁ ∩ T₂ ∩ T₃ ∩ T₄ is