IIT JEE 2018 Paper 1

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JEE (ADVANCED) 2018 PAPER 1
PART-I PHYSICS

Q.1 The potential energy of a particle of mass 𝑚 at a distance 𝑟 from a fixed point 𝑂 is given by 𝑉(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?

Q.2 Consider a body of mass 1.0 𝑘𝑔 at rest at the origin at time 𝑡 = 0. A force 𝐹 ⃗ = (𝛼𝑡 𝑖̂ + 𝛽 𝑗̂) 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 𝜏 ⃗. Which of the following statements is (are) true?

(A) |𝜏 ⃗| = 1/3 𝑁 𝑚
(B) The torque 𝜏 ⃗ is in the direction of the unit vector + 𝑘 ̂
(C) The velocity of the body at 𝑡 = 1 𝑠 is v ⃗⃗ = 1/2 (𝑖̂ + 2𝑗̂) 𝑚 𝑠⁻¹
(D) The magnitude of displacement of the body at 𝑡 = 1 𝑠 is 1/6 𝑚

Q.3 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?

(A) For a given material of the capillary tube, ℎ decreases with increase in 𝑟
(B) For a given material of the capillary tube, ℎ is independent of 𝜎
(C) If this experiment is performed in a lift going up with a constant acceleration, then ℎ decreases
(D) ℎ is proportional to contact angle 𝜃

Q.4 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 𝐼𝑚𝑎𝑥 at time 𝑡 = 𝜏. Which of the following statements is (are) true?

Q.5 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 +𝑗̂ direction. Which of the following statements regarding the magnetic field 𝐵 ⃗⃗ is (are) true?

(A) If 𝐼1 = 𝐼2, then 𝐵 ⃗⃗ cannot be equal to zero at the origin (0, 0, 0) (C) If 𝐼1 < 0 and 𝐼2 > 0, then 𝐵 ⃗⃗ can be equal to zero at the origin (0, 0, 0) (D) If 𝐼1 = 𝐼2, then the 𝑧-component of the magnetic field at the centre of the loop is (− 𝜇 2 0 𝑅𝐼)
(B) If 𝐼1 > 0 and 𝐼2 < 0, then 𝐵 ⃗⃗ can be equal to zero at the origin (0, 0, 0)

Q.6 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?

(A) Process I is an isochoric process
(B) In process II, gas absorbs heat
(C) In process IV, gas releases heat
(D) Processes I and III are not isobaric

Q.8 Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed 1.0 𝑚 𝑠⁻¹ and the man behind walks at a speed 2.0 𝑚 𝑠⁻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 𝐻𝑧. The speed of sound in air is 330 𝑚 𝑠⁻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 __________.

Q.9 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 (2 − √3) /√10 𝑠, then the height of the top of the inclined plane, in 𝑚𝑒𝑡𝑟𝑒𝑠, is __________. Take 𝑔 = 10 𝑚 𝑠⁻².

Q.10 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⁻¹collides 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 _________.

Q.11 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 𝜖𝑟. The cell electromotive force (emf) 𝑉0 = 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 𝜖
𝑟 =____________.

Q.12 In the 𝑥𝑦-plane, the region 𝑦 > 0 has a uniform magnetic field 𝐵₁𝑘 ̂ and the region 𝑦 < 0 has another uniform magnetic field 𝐵₂𝑘 ̂. A positively charged particle is projected from the origin along the positive 𝑦-axis with speed 𝑣₀ = 𝜋 𝑚 𝑠⁻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 𝑚 𝑠⁻¹, along the 𝑥-axis in the time interval 𝑇 is __________.

Q.13 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 __________.

Q.14 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 =__________.

Q.15 The relation between [𝐸] and [𝐵] is

(A) [𝐸] = [𝐵] [𝐿] [𝑇]
(B) [𝐸] = [𝐵] [𝐿]⁻¹ [𝑇]
(C) [𝐸] = [𝐵] [𝐿] [𝑇]⁻¹
(D) [𝐸] = [𝐵] [𝐿]⁻¹ [𝑇]⁻¹

Q.16 The relation between [𝜖0] and [𝜇0] is

(A) [𝜇₀] = [𝜖₀] [𝐿]² [𝑇]⁻²
(B) [𝜇₀] = [𝜖₀] [𝐿]⁻² [𝑇]²
(C) [𝜇₀] = [𝜖₀]⁻¹ [𝐿]² [𝑇]⁻²
(D) [𝜇₀] = [𝜖₀]⁻¹ [𝐿]⁻² [𝑇]²

Q.17 Consider the ratio 𝑟 = (1−𝑎)/(1+𝑎) to be determined by measuring a dimensionless quantity 𝑎. If the error in the measurement of 𝑎 is Δ𝑎 (Δ𝑎/𝑎 ≪ 1), then what is the error Δ𝑟 in determining 𝑟?

Q.18 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 |𝑥| ≪ 1, ln(1 + 𝑥) = 𝑥 up to first power in 𝑥. The error Δ𝜆, in the determination of the decay constant 𝜆, in 𝑠−1, is

(A) 0.04
(B) 0.03
(C) 0.02
(D) 0.01

JEE (ADVANCED) 2018 PAPER 1
PART II-CHEMISTRY

Q.1 The compound(s) which generate(s) N₂ gas upon thermal decomposition below 300^oC is (are)

(A) NH₄NO₃
(B) (NH₄)₂Cr₂O₇
(C) Ba(N₃)₂
(D) Mg₃N₂

Q.2 The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: Fe = 26, Ni = 28)

(A) Total number of valence shell electrons at metal centre in Fe(CO)₅ or Ni(CO)₄ is 16
(B) These are predominantly low spin in nature
(C) Metal–carbon bond strengthens when the oxidation state of the metal is lowered
(D) The carbonyl C−O bond weakens when the oxidation state of the metal is increased

Q.3 Based on the compounds of group 15 elements, the correct statement(s) is (are)

(A) Bi₂O₅ is more basic than N₂O₅
(B) NF₃ is more covalent than BiF₃
(C) PH₃ boils at lower temperature than NH₃
(D) The N−N single bond is stronger than the P−P single bond

Q.4 In the following reaction sequence, the correct structure(s) of X is (are)

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

Q.6 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)

(A) 𝑞_𝐴𝐶 = ∆𝑈_𝐵𝐶 and 𝑤_𝐴𝐵 = 𝑃₂(𝑉₂ − 𝑉₁)
(B) 𝑤_𝐵𝐶 = 𝑃₂(𝑉₂ − 𝑉₁) and 𝑞_𝐵𝐶 = ∆𝐻_𝐴𝐶
(C) ∆𝐻_𝐶𝐴 < ∆𝑈_𝐶𝐴 and 𝑞_𝐴𝐶 = ∆𝑈_𝐵𝐶
(D) 𝑞_𝐵𝐶 = ∆𝐻_𝐴𝐶 and ∆𝐻_𝐶𝐴 > ∆𝑈_𝐶𝐴

Q.7 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

Q.8 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 nickelammonia 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)

Q.9 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 (number of anions/ number of cations) in Z is ____.

Q.10 For 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, 𝐹/𝑅 = 11500 K V⁻¹, where 𝐹 is the Faraday constant and 𝑅 is the gas constant, ln(10) = 2.30)

Q.11 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 ____.

Q.12 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 𝑥𝐴 and 𝑥𝐵, respectively, has vapour pressure of 22.5 Torr. The value of 𝑥𝐴/𝑥𝐵 in the new solution is ____. (given that the vapour pressure of pure liquid A is 20 Torr at temperature T)

Q.13 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)

Q.14 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 ____.

Q.15 The compound Y is

Q.16 The compound Z is

Q.17 The compound R is

Q.18 The compound S is

JEE (ADVANCED) 2018 PAPER 1
PART-III MATHEMATICS

Q.1 For a non-zero complex number 𝑧, let arg(𝑧) denote the principal argument with − 𝜋 < arg(𝑧) ≤ 𝜋. Then, which of the following statement(s) is (are) FALSE?

(A)
(B) The function 𝑓: ℝ → (−𝜋, 𝜋], defined by 𝑓(𝑡) = arg(−1 + 𝑖𝑡) for all 𝑡 ∈ ℝ, is continuous at all points of ℝ, where 𝑖 = √−1
(C) For any two non-zero complex numbers 𝑧1 and 𝑧2,

is an integer multiple of 2𝜋
(D) For any three given distinct complex numbers 𝑧1, 𝑧2 and 𝑧3, the locus of the point 𝑧 satisfying the condition

lies on a straight line

Q.2 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?

(A) ∠𝑄𝑃𝑅 = 45°
(B) The area of the triangle 𝑃𝑄𝑅 is 25√3 and ∠𝑄𝑅𝑃 = 120°
(C) The radius of the incircle of the triangle 𝑃𝑄𝑅 is 10√3 − 15
(D) The area of the circumcircle of the triangle 𝑃𝑄𝑅 is 100 𝜋

Q.3 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?

(A) The line of intersection of 𝑃1 and 𝑃2 has direction ratios 1, 2, −1
(B) The line

is perpendicular to the line of intersection of 𝑃1 and 𝑃2
(C) The acute angle between 𝑃1 and 𝑃2 is 60°
(D) If 𝑃₃ is the plane passing through the point (4, 2, −2) and perpendicular to the line of intersection of 𝑃₁ and 𝑃₂, then the distance of the point (2, 1, 1) from the plane 𝑃3 is 2/ √3

Q.4 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?

(A) There exist 𝑟, 𝑠 ∈ ℝ, where 𝑟 < 𝑠, such that 𝑓 is one-one on the open interval (𝑟, 𝑠)
(B) There exists 𝑥0 ∈ (−4, 0) such that |𝑓′(𝑥0)| ≤ 1
(C)
(D) There exists 𝛼 ∈ (−4, 4) such that 𝑓(𝛼) + 𝑓′′(𝛼) = 0 and 𝑓′(𝛼) ≠ 0

Q.5 Let 𝑓: ℝ → ℝ and 𝑔: ℝ → ℝ be two non-constant differentiable functions. If

𝑓′(𝑥) = (𝑒(𝑓(𝑥)−𝑔(𝑥)))𝑔′(𝑥) for all 𝑥 ∈ ℝ,

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

(A) 𝑓(2) < 1 − log_e 2
(B) 𝑓(2) > 1 − log_e 2
(C) 𝑔(1) > 1 − log_e 2
(D) 𝑔(1) < 1 − log_e 2

Q.6 Let 𝑓: [0, ∞) → ℝ be a continuous function such that

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

(A) The curve 𝑦 = 𝑓(𝑥) passes through the point (1, 2)
(B) The curve 𝑦 = 𝑓(𝑥) passes through the point (2, −1)
(C) The area of the region
(D) The area of the region

Q.7 The value of

is ______ .

Q.8 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 _____ .

Q.9 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 the arithmetic progression 9, 16, 23, … . Then, the number of elements in the set 𝑋 ∪ 𝑌 is _____.

Q.10 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 and [0, 𝜋], respectively.)

Q.11 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 _____ .

Q.12 Let be two unit vectors such that . For some 𝑥, 𝑦 ∈ ℝ, let If |𝑐 ⃗| = 2 and the vector 𝑐 ⃗ is inclined at the same angle 𝛼 to both , then the value of 8 cos² 𝛼 is _____ .

Q.13 Let 𝑎, 𝑏, 𝑐 be three non-zero real numbers such that the equation

has two distinct real roots 𝛼 and 𝛽 with

Q.14 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 farmer 𝐹₂ takes away the region which lies between the side 𝑃𝑄 and a curve of the form 𝑦 = 𝑥^𝑛 (𝑛 > 1). If the area of the region taken away by the farmer 𝐹₂ is exactly 30% of the area of ∆𝑃𝑄𝑅, then the value of 𝑛 is _____ .

Q.15 Let 𝐸₁𝐸₂ and 𝐹₁𝐹₂ be the chords of 𝑆 passing through the point 𝑃₀ (1, 1) and parallel to the x-axis and the y-axis, 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

(A) 𝑥 + 𝑦 = 4
(B) (𝑥 − 4)² + (𝑦 − 4)² = 16
(C) (𝑥 − 4)(𝑦 − 4) = 4
(D) 𝑥𝑦 = 4

Q.16 Let 𝑃 be a point on the circle 𝑆 with both coordinates being positive. Let the tangent to 𝑆 at 𝑃 intersect the coordinate axes at the points 𝑀 and 𝑁. Then, the mid-point of the line segment 𝑀𝑁 must lie on the curve.

(A) (𝑥 + 𝑦)² = 3𝑥𝑦
(B) 𝑥^2/3 + 𝑦^2/3 = 2^4/3
(C) 𝑥² + 𝑦² = 2𝑥𝑦
(D) 𝑥² + 𝑦² = 𝑥² 𝑦²

Q.17 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

Q.18 For 𝑖 = 1, 2, 3, 4, let 𝑇_𝑖 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