**JEE Advanced 2018 Paper I**

Q. 1 The potential energy of a particle of mass m at a distance r from a fixed point 0 is given byV(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 0. If is the speed of the particle and 0 is the magnitude of its angular momentum about 0, which of the following statements is (are) true?

A. v=(k/2m)R

B. v=(k/m)R

C. L=(mk)R2

D. v=[(mk)/2]R2

Q. 2 Consider a body of mass 1.0 kg at rest at the origin at time t = 0. A force F=(t i+ j) is applied on the body, where = 1.0 N/s and = 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. = 1/3 N m

B. The torque is in the direction of the unit vector + k

C. The velocity of the body at t = 1 s is v = 1/2 (i + 2 j) m/s

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 o. The angle of contact between water and the wall of the capillary tube is 8. 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

B. For a given material of the capillary tube, ℎ is independent of

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 I(nax) at time t = v. Which of the following statements is (are) true?

A. I(nax) =V/2R

B. I(nax) =V/4R

C. v = L/R ln 2

D. v = 2L/R ln 2

Question 5 This section contains SIX (06) questions.

Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s). For each question, choose the correct option(s) to answer the question. Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If only (all) the correct option(s) is (are) chosen.

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen.

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen, both of which are correct options.

Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct option.

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered). Negative Marks : −2 In all other cases.

For Example: If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option) ,without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks.

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 +j true?

Direction. Which of the following statements regarding the magnetic ﬁeld B⃗⃗ is (are)

A. If I₁ = I₂, then B cannot be equal to zero at the origin (0, 0, 0)

B. If I₁ > 0 and I₂ < 0, then Bcan be equal to zero at the origin (0, 0, 0)

C. If I₁ < 0 and I₂ > 0, thenB 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 (−µ₀ I/2R)

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. 7 Two vectorsA and B are deﬁned as A = a îand B = a (cos mtî+ sin mtĵ), where a is a constant and m = n/6 rad/s. If |A + B| = √3|A − B | at time t = v for the ﬁrst time, the value of v, in seconds, is _______

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 n/s and the man behind walks at a speed 2.0 n/s. A third man is standing at a height 12 n 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 n/s. At the instant, when the moving men are 10 n 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 meters, is__________Take g = 10 n 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/n 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 n/s collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in meters, 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 cᵣ . 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 cᵣ = ________

Q. 12 In the xy-plane, the region y > 0 has a uniform magnetic ﬁeld B1k̂ and the region y < 0 has another uniform magnetic ﬁeld B₂k̂. A positively charged particle is projected from the origin along the positive y-axis with speed v₀ = n n/s 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 n/s, along the x- axis in the time interval T is

.

Q. 13 Sunlight of intensity 1.3 kW n⁻² is incident normally on a thin convex lens of focal length 20cm. 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 n⁻² , at a distance 22 cn 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 T₂ = 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 = .

Questions: 15 – 16

Short Summary

- 0.0.1 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 [c₀] and [µ₀] 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. (There are two questions based on PARAGRAPH “X”, the question given below is one of them)
- 0.0.2 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 z ± Δz = (x±Δx)/(y±Δy) =x/y (1 ± (Δx/x)) (1 ± (Δy/y))⁻¹.

- 0.1 Let S be the circle in the xy-plane defined by the equation x Let S be the circle in the xy-plane defined by the equation x² + y² = 4. (There are two questions based on PARAGRAPH “X”, the question given below is one of them)
- 1 There are five students S₁, S₂, S₃, S₄ and S₅ in a music class and for them there are five seats R₁, R₂, R₃, R₄ and R₅ arranged in a row, where initially the seat Rᵢ is allotted to the student Sᵢ, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

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]⁻

Q. 16 The relation between [c₀] and [µ₀] is

A. [µ₀] = [c₀] [L]²[T]⁻²

B. [µ₀] = [c₀] [L]⁻²[T]²

C. [µ₀] = [c₀]⁻¹[L]²[T]⁻²

D. [µ₀] = [c₀]⁻¹[L]⁻²[T]²

Questions: 17 – 18

The series expansion for (1 ± Δy/y)⁻¹, to first power in Δy/y, is 1 ∓ (Δy/y). The relative errors in independent variables are always added. So the error in z will be Δz = z (Δx/x + Δy/y). The above derivation makes the assumption that Δx/x ≪1, Δy/y ≪1. Therefore, the higher powers of these quantities are neglected.

(There are two questions based on PARAGRAPH “A”, the question given below is one of them)

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?

A. Δa/(1+a)²

B. 2Δa/(1+a)²

C. 2Δa/(1−a²)

D. 2aΔa/(1−a²)

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 Δh, in the determination of the decay constant h, in s⁻¹, is

A. 0.04

B. 0.03

C. 0.02

D. 0.01

Q. 19 The compound(s) which generate(s) N₂ gas upon thermal decomposition below 300⁰C is (are)

A. NH₄NO₃

B. (NH₄)₂Cr₂O₇

C. Ba(N₃)₂

D. Mg₃N₂

Q. 20 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. 21 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. 22 In the following reaction sequence, the correct structure(s) of X is (are)

A. A

B. B

C. C

D. D

Q. 23 The reaction(s) leading to the formation of 1,3,5-trimethylbenzene is (are)

A. A

B. B

C. C

D. D

Q. 24 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. A

B. B

C. C

D. D

Q. 25 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. 26 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 : H = 1, N = 14, O = 16, S = 32, Cl = 35.5, Ca = 40, Ni = 59)

Q. 27 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. 28 For the electrochemical cell, Mg(s) I Mg⁺² (aq, 1 M) II Cu⁺² (aq, 1 M) I Cu(s), the standard emf of the cell is 2.70 V at 300 K. When the concentration of Mg⁺² 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. 29 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. 30 Liquids u and v form ideal solution over the entire range of composition. At temperature T, equimolar binary solution of liquids u and v has vapour pressure 45 Torr. At the same temperature, a new solution of u and v having mole fractions xᵤ and xᵥ, respectively, has vapour pressure of 22.5 Torr. The value of xᵤ/xᵥ in the new solution is (given that the vapour pressure of pure liquid u is 20 Torr at temperature T)

Q. 31 The solubility of a salt of weak acid (AB) at pH 3 is Y×10⁻³ mol/L. The value of Y is .(Given that the value of solubility product of AB (Ksp) = 2×10⁻¹⁰ and the value of ionization constant of HB (Kₐ) = 1×10⁻⁸)

Q. 32 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 isomolar 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 .

Questions: 33 – 34

Treatment of benzene with CO/HCl in the presence of anhydrous AlCl₃/CuCl followed by reaction with Ac₂O/NaOAc gives compound X as the major product. Compound X upon reaction with Br₂/Na₂CO₃, followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H₂/Pd-C, followed by H₃PO₄ treatment gives Z as the major product. (There are two questions based on PARAGRAPH “X”, the question given below is one of them)

Q. 33 The compound Y is

A. A

B. B

C. C

D. D

Q. 34 The compound Z is

A. A

B. B

C. C

D. D

Questions: 35 – 36

An organic acid P (C₁₁H₁₂O₂) can easily be oxidized to a dibasic acid which reacts with ethylene glycol to produce a polymer dacron. Upon ozonolysis, P gives an aliphatic ketone as one of the products. P undergoes the following reaction sequences to furnish R via Q. The compound P also undergoes another set of reactions to produce S.(There are two questions based on PARAGRAPH “A”, the question given below is one of them)

Q. 35 The compound R is

A. A

B. B

C. C

D. D

Q. 36 The compound S is

A. A

B. B

C. C

D. D

Q. 37 For a non-zero complex number z, let arg(z) denote the principal argument with − n < arg(z) ≤ n. Then, which of the following statement(s) is (are) FALSE?

A. arg(−1 − i) =n/4, where i = √−1

B. The function ƒ: ℝ → (−n, n], defined by ƒ(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, arg (z₁/z₂) − arg( z₁) + arg( z₂) is an integer multiple of 2n

D. For any three given distinct complex numbers z₁, z₂ and z₃, the locus of the point z

satisfying the condition arg ((z−z₁) (z₂−z₃)/(z−z₃) (z₂−z₁)) = n,lies on a straight line

Q. 38 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

Q. 39 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 (3x − 4)/9=(1 − 3y)/9= z/3 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. 40 For every twice differentiable function ƒ: ℝ → [−2, 2] with (ƒ(0))² + (ƒ′(0))² = 85, which of the following statement(s) is (are) TRUE?

A. There exist r, s ∈ ℝ, where r < s, such that ƒ is one-one on the open interval (r, s)

B. There exists x0 ∈ (−4, 0) such that |ƒ′(x₀)| ≤ 1

C. limx→∞ ƒ(x) = 1

D. There exists α ∈ (−4, 4) such that ƒ(α) + ƒ′′(α) = 0 and ƒ′(α) ≠ 0

Q. 41 Let ƒ: ℝ → ℝ and g: ℝ → ℝ be two non-constant differentiable functions. If ƒ′(x) = (e(ƒ(x) −g(x)))g′(x) for all x ∈ ℝ, and ƒ(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?

A. A

B. B

C. C

D. D

Q. 42 Let ƒ:[0, ∞) → ℝ be a continuous function such that x0 for all x ∈ [0, ∞). Then, which of the following statement(s) is (are) TRUE?

A. The curve y = ƒ(x) passes through the point (1, 2)

B. The curve y = ƒ(x) passes through the point (2, −1)

C. The area of the region {(x, y) ∈ [0, 1] × ℝ ∶ ƒ(x) ≤ y ≤ √1 − x₂} is (n−2)/4

D. The area of the region {(x, y) ∈ [0,1] × ℝ ∶ ƒ(x) ≤ y ≤ √1 − x₂} is (n−1)/4

Q. 43 The value of the given equation is ______ .

Q. 44 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. 45 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. 46 The number of real solutions of the equation lying in the interval (-1/2,1/2).(Here, the inverse trigonometric functions sin−1x and cos−1x assume values in [−n/2,n/2] and [0, n], respectively.)

Q. 47 For each positive integer n, let For x ∈ ℝ, let [x] be the greatest integer less than or equal to x. If the limit is given then the value of [L] is ________.

Q. 48 Let a and b be two unit vectors such thata ⋅ b = 0. For some x, y ∈ ℝ, let c = x a + y b + (a × b ) . If |c = 2 and the vector c⃗ is inclined at the same angle α to both a and b , then the value

of 8 cos² α is

Q. 49 Let a, b, c be three non-zero real numbers such that the equation √3 a cos x + 2 b sin x = c, x ∈ [−n/2,n/2], has two distinct real roots α and þ with α + þ = n/3. Then, the value of ba is_____ .

Q. 50 4 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 = xⁿ (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 .

Questions: 51 – 52

Q. 51 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 G₀ and G₂ meet at G₃. Then, the points E₃, F₂, and G₃ lie on the curve

(A) (B)

A. x + y = 4

B. (x − 4)² + (y − 4)²= 16 (C) (D)

C. (x − 4)(y − 4) = 4

D. xy = 4

Q. 52 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

Questions: 53 – 54

(There are two questions based on PARAGRAPH “A”, the question given below is one of them)

Q. 53 The probability that, on the examination day, the student S₁ gets the previously allotted seat Rᵢ, and NONE of the remaining students gets the seat previously allotted to him/her is

A. 3/40

B. 1/8

C. 7/40

D. 1/5

Q. 54 For i = 1, 2, 3, 4, let Tᵢ denote the event that the students Sᵢ and Sᵢ₊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

A. 1/15

B. 1/10

C. 7/60

D. 1

Answer Sheet |
||||||||||

Question |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |

Answer | ABD | BCD | CD | ABD | ABD | BCD | 2.00 | 5.00 | 2.00 | 2.09 |

Question |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |

Answer | 1.50 | 2.00 | 130.00 | 4.00 | C | D | B | C | BC | BC |

Question |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |
30 |

Answer | ABC | B | ABD | BC1 | 1 | 2992 | 3 | 10 | 2.22 | 19 |

Question |
31 |
32 |
33 |
34 |
35 |
36 |
37 |
38 |
39 |
40 |

Answer | 4.47 | 0.05 | C | A | A | B | ABD | BCD | CD | ABD |

Question |
41 |
42 |
43 |
44 |
45 |
46 |
47 |
48 |
49 |
50 |

Answer | BC | BC | 8 | 625 | 3748 | 2 | 1 | 3 | 0.5 | 4 |

Question |
51 |
52 |
53 |
54 |
||||||

Answer | A | D | A | C |

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