JEE Advanced 2018 Paper 2

A particle of mass ݉ is initially at rest at the origin. It is subjected to a force and starts moving along the ݔ-axis. Its kinetic energy ܭ changes with time as ݐ݀/ܭ݀ ൌ ݐߛ , where ߛ is a positive constant of appropr iate dimensions. Which of the following statements is (are) true?

1. The force applied on the particle is constant
2. The speed of the particle is proportional to time
3. The distance of the particle from the origin increases line arly with time
4. The force is conservative

Consider a thin square plate floating on a viscous liquid in a large tank. The height ݄ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity ݑ଴. Which of the following statements is (are) true?

1. The resistive force of liqui d on the plate is inversely pro portional to ݄
2. The resistive force of liquid on the plate is independent o f the area of the plate
3. The tangential (shear) stress on the floor of the tank incr eases with ݑ଴
4. The tangential (shear) stress on the plate varies linearly with the viscosity ߟ of the liquid

An infinitely long thin non-conducting wire is parallel to the ݖ-axis and carries a uniform line charge density ߣ .It pierces a thin non-conducting spherical shell of radius ܴ in such a way that the arc ܳܲ subtends an angle 120° at the centre ܱ of the spherical shell, as shown in the figure. The permittivity of free space is ߳଴. Which of the following statements is (are) true?

1. The electric flux through the shell is √3߳/ߣܴ଴
2. The ݖ-component of the electric field is zero at all the points on t he surface of the shell
3. The electric flux through the shell is √2߳/ߣܴ଴
4. The electric field is normal to the surface of the shell at all points

A wire is bent in the shape of a right angled triangle and is p laced in front of a concave mirror of focal length ݂ ,as shown in the figure. Which of the figures shown in the fou r options qualitatively represent (s) the shape of the image of the bent wire? (These figures are no t to scale.)

In a radioactive decay chain, Thଽ଴ଶଷଶ nucleus decays to Pb଼ଶଶଵଶ nucleus. Let ܰఈ and ܰఉ be the number of ߙ and ߚି particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

1. ܰ ఈൌ5
2. ܰఈൌ6
3. ܰఉൌ2
4. ܰఉൌ4

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 ݖܪ is used. The length of the air column is varied by changing th e level of water in the resonance tube. Two successive resonances are hear d at air columns of length 50.7 ݉ܿ and 83.9 ݉ܿ . Which of the following st atements is (are) true?

1. The speed of sound determined from this experiment is 332ݏ݉ିଵ
2. The end correction in this experiment is 0.9݉ܿ
3. The wavelength of the sound wave is 66.4݉ܿ
4. The resonance at 50.7 ݉ܿ corresponds to the fundamental harmonic

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass ݉ൌ0 . 4 ݃݇ is at rest on this surface. An impulse of 1.0ݏܰ is applied to the block at time ݐൌ0 so that it starts moving along the ݔ-axis with a velocity ݒሺݐሻൌݒ଴݁ି௧/ఛ, where ݒ଴ is a constant and ߬ൌ4 ݏ . The displacement of the block, in ݏ݁ݎݐ݁݉ , at ݐൌ߬ is __________. Take ݁ିଵൌ0 . 3 7 .

Numerical answer type

A ball is projected from the ground at an angle of 45° with the horizontal surface. It reaches a maximum height of 120 ݉ and returns to the ground. Upon hitting the ground for the fir st time, it loses half of its kinetic energy. Immediately after th e bounce, the velocity of the ball makes an angle of 30° with the horizontal surface. The maximum height it reaches aft er the bounce, in ݏ݁ݎݐ݁݉ , is ___________.

Numerical answer type

A particle, of mass 10 ିଷ݃݇ and charge 1.0ܥ , is initially at rest. At time ݐൌ0 , the particle comes under the influence of an electric field ܧሬԦሺݐሻൌܧ଴sinݐ߱ଓ̂, where ܧ଴ൌ1 . 0 ܥܰିଵand ߱ൌ1 0ଷ ݏ ݀ܽݎିଵ. Consider the effect of only th e electrical force on the particle. Then the maximum speed, in ݏ ݉ିଵ, attained by the particle at subsequent times is ____________.

Numerical answer type

A moving coil galvanometer has 50 turns and each turn has an area 2ൈ10ିସ ݉ଶ. The magnetic field produced by the magnet inside the galvanometer i s 0.02 ܶ . The torsional constant of the suspension wire is 10ିସ ݀ܽݎ ݉ ܰିଵ. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotate s by 0.2 ݀ܽݎ . The resistance of the coil of the galvanometer is 50 ߗ . This galvanometer is to be converted into an ammeter capable of measuring current in the range 0െ1.0 ܣ . For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ݏ݄݉݋ , is __________.

Numerical answer type

A steel wire of diameter 0.5݉݉ and Young’s modulus 2ൈ10ଵଵ݉ ܰିଶ carries a load of mass ܯ .The length of the wire with the load is 1.0݉ . A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a r eference wire to which a main scale, of least count 1.0 ݉݉ , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier sc ale coincides with the zero of main scale. If the load on the steel wire is increased by 1.2 ݃݇ , the vernier scale division which coincides with a main scale division is __________. Take ݃ ൌ 10 ݏ ݉ିଶ and ߨൌ3 . 2 .

Numerical answer type

One mole of a monatomic ideal gas undergoes an adiabatic expans ion in which its volume becomes eight times its initial value. If the initial temperatu re of the gas is 100 ܭ and the universal gas constant ܴൌ8 . 0݈݋݉ܬ ିଵܭିଵ, the decrease in its internal energy, in ݈݁ݑ݋ܬ , is__________.

Numerical answer type

In a photoelectric experiment a parallel beam of monochromatic light with power of 200ܹ is incident on a perfectly absorbing cathode of work function 6.25ܸ݁ . The frequency of light is just above the threshold frequency so that the photoel ectrons are emitted with negligible kinetic energy. Assume that the photoelectron emissi on efficiency is 100% . A potential difference of 500 ܸ is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force ܨൌ݊ൈ1 0ିସܰ due to the impact of the electrons. The value of ݊ is __________. Mass of the electron ݉௘ൌ9ൈ1 0ିଷଵ݃݇ and 1.0ܸ݁ ൌ 1.6ൈ10ିଵଽܬ .

Numerical answer type

Consider a hydrogen-like ionized atom with atomic number ܼ with a single electron. In the emission spectrum of this atom, the photon emitted in the ݊ൌ2 t o ݊ൌ1 transition has energy 74.8 ܸ݁ higher than the photon emitted in the ݊ൌ3 t o ݊ൌ2 transition. The ionization energy of the hydrogen atom is 13.6ܸ݁ . The value of ܼ is __________.

Numerical answer type

The electric field ܧ is measured at a point ܲሺ0,0,݀ሻ generated due to various charge distributions and the dependence of ܧ on ݀ is found to be different for different charge distributions. List-I contains different relations between E and ݀ . List-II describes different electric charge distr ibutions, along with their locat ions. Match the functions in List-I with the related cha rge distributions in List-II. LIST–I P. ܧ is independent of ݀ Q. ܧ∝ଵ ௗ R. ܧ∝ଵ ௗమ S. ܧ∝ଵ ௗయ LIST–II 1. A point charge ܳ at the origin 2. A small dipole with point charges ܳ at ሺ0,0,݈ሻ and –ܳ at ሺ0,0,െ݈ሻ . Take 2݈ ≪ ݀ 3. An infinite line charge coincident with the ݔ-axis, with uniform linear charge density ߣ 4. Two infinite wires carrying uniform linear charge density parallel to the ݔ -axis. The one along ሺݕൌ0 ,ݖൌ݈ ሻ has a charge density ൅ߣ and the one along ሺݕൌ0 ,ݖൌെ ݈ ሻ has a charge density –ߣ .Take 2݈ ≪ ݀ 5. Infinite plane charge coincident with the ݕݔ-plane with uniform surface charge density

1. P → 5; Q → 3, 4; R → 1; S → 2
2. P → 5; Q → 3; R → 1, 4; S → 2
3. P → 5; Q → 3; R → 1, 2; S → 4
4. P → 4; Q → 2, 3; R → 1; S → 5

A planet of mass ܯ ,has two natural satellites with masses ݉ଵ and ݉ଶ. The radii of their circular orbits are ܴଵ a n d ܴଶ respectively. Ignore the gravitational force between the satellites. Define ݒଵ, ܮଵ, ܭଵ and ܶଵ to be, respectively , the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; an d ݒଶ, ܮଶ, ܭଶ and ܶଶ to be the corresponding quantities of satellite 2. Given ݉ଵ/݉ଶൌ2 and ܴଵ/ܴଶൌ1 / 4 , match the ratios in List-I to the numbers in List-II. LIST–I P. ௩భ ௩మ Q. ௅భ ௅మ R. ௄భ ௄మ S. ்భ ்మ LIST–II 1. ଵ ଼ 2. 1 3. 2 4. 8

1. P → 4; Q → 2; R → 1; S → 3
2. P → 3; Q → 2; R → 4; S → 1
3. P → 2; Q → 3; R → 1; S → 4
4. P → 2; Q → 3; R → 4; S → 1

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the ܸܲ-diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adi abatic. Match the processes mentioned in List-1 with the corresponding statements in List-II. LIST–I P. In process I Q. In process II R. In process III S. In process IV LIST–II 1. Work done by the gas is zero 2. Temperature of the gas remains unchanged 3. No heat is exchanged between the gas and its surroundings 4. Work done by the gas is 6ܲ଴ܸ଴

1. P → 4; Q → 3; R → 1; S → 2
2. P → 1; Q → 3; R → 2; S → 4
3. P → 3; Q → 4; R → 1; S → 2
4. P → 3; Q → 4; R → 2; S → 1

In the List-I below, four differe nt paths of a particle are giv en as functions of time. In these functions, ߙ and ߚ are positive constants of appropriate dimensions and ߙ്ߚ . In each case, the force acting on the particle is either zero or conservative . In List-II, five physical quantities of the particle are mentioned: ݌Ԧ is the linear momentum, ܮሬԦ is the angular momentum about the origin, ܭ is the kinetic energy, ܷ is the potential energy and ܧ is the total energy. Match each path in List-I with those quantities i n List-II, which are conserved for that path . LIST–I P. ݎԦሺݐሻൌ ݐߙ ଓ̂൅ ݐߚଔ̂ Q. ݎԦሺݐሻൌߙc o s ݐ߱ ଓ ̂൅ߚsin ݐ߱ଔ ̂ R. ݎԦሺݐሻൌߙሺ c o s ݐ߱ ଓ ̂൅sin ݐ߱ଔ ̂ሻ S. ݎԦሺݐሻൌ ݐߙ ଓ̂൅ఉ ଶݐଶ ଔ̂ LIST–II 1. ݌Ԧ 2. ܮሬԦ 3. ܭ 4. U 5. E

1. P → 1, 2, 3, 4, 5; Q → 2, 5; R → 2, 3, 4, 5; S → 5
2. P → 1, 2, 3, 4, 5; Q → 3, 5; R → 2, 3, 4, 5; S → 2, 5
3. P → 2, 3, 4; Q → 5; R → 1, 2, 4; S → 2, 5
4. P → 1, 2, 3, 5; Q → 2, 5; R →2, 3, 4, 5; S → 2, 5 JEE (ADVANCED) 2018 PAPER 2 PART II-CHEMISTRY  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 resul t in +1 marks. Selecting any in correct option(s) (second option in this case), with or without selection of an y correct option(s) will result in ‐2 marks.

The correct option(s) regarding the complex [Co(en)(NH 3)3(H₂O)]3+ (en = H 2NCH 2CH 2NH 2) is (are)

1. It has two geometrical isomers
2. It will have three geometrical isomers if bidentate ‘en’ is replaced by two cyanide ligands
3. It is paramagnetic
4. It absorbs light at longer wavelength as compared to [Co(en )(NH 3)4]3+

The correct option(s) to distinguish nitrate salts of Mn²⁺ and Cu²⁺ taken separately is (are)

1. Mn²⁺ shows the characteristic green colour in the flame test
2. Only Cu²⁺ shows the formation of precipitate by passing H 2S in acidic medium
3. Only Mn²⁺ shows the formation of precipitate by passing H 2S in faintly basic medium
4. Cu²⁺/Cu has higher reduction potential than Mn²⁺/Mn (measured under similar conditions)

Aniline reacts with mixed acid (conc. HNO 3 and conc. H 2SO 4) at 288 K to give P (51 %), Q (47%) and R (2%). The major product(s) of the following reaction sequence is (are)

The Fischer presentation of D-glucose is given below. The correct structure(s) of β-L-glucopyranose is (are)

For a first order reaction A(g) → 2B(g) + C(g) at constant volume and 300 K, the total pressure at the beginning ( ݐ = 0) and at time ݐare ܲ଴ and ܲ௧, respectively. Initially, only A is present with concentration [A] 0, and ݐ1/3 is the time required for the partial pressure of A to reach 1/3rd of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)

For a reaction, A ⇌ P, the plots of [A] and [P] with time at temperatures T1 and T2 are given below. If ܶଶ > ܶଵ, the correct statement(s) is (are) (Assume ∆ܪƟ and ∆ܵƟ are independent of temperature and ratio of ln K at ܶଵ to ln K at ܶଶ is greater than ܶଶܶଵൗ. Here H, S, G and K are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively .)

1. ∆ܪƟ < 0, ∆ܵƟ < 0
2. ∆ܩƟ < 0, ∆ܪƟ > 0
3. ∆ܩƟ < 0, ∆ܵƟ < 0
4. ∆ܩƟ < 0, ∆ܵƟ > 0

The total number of compounds having at least one bridging oxo group among the molecules given below is ____. N2O3, N 2O5, P 4O6, P 4O7, H 4P2O5, H 5P3O10, H 2S2O3, H 2S2O5

Numerical answer type

Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the passage of air is stopped, but the heating is co ntinued in a closed furnace such that the contents undergo sel f-reduction. The weight (in kg) o f Pb produced per kg of O 2 consumed is ____. (Atomic weights in g mol1: O = 16, S = 32, Pb = 207)

Numerical answer type

To measure the quantity of MnCl 2 dissolved in an aqueous solution, it was completely converted to KMnO 4 using the reaction, MnCl 2 + K 2S2O8 + H 2O  KMnO 4 + H 2SO 4 + HCl (equation not balanced). Few drops of concentrated HCl were added to this solution and g ently warmed. Further, oxalic acid (225 mg) was added in portions till the colour of t he permanganate ion disappeared. The quantity of MnCl 2 (in mg) present in the initial solution is ____. (Atomic weights in g mol⁻¹: Mn = 55, Cl = 35.5)

Numerical answer type

For the given compound X, the total number of optically active stereoisomers is ____.

Numerical answer type

In the following reaction sequence, the amount of D (in g) formed from 10 moles of acetophenone is ____. (Atomic weights in g mol1: H = 1, C = 12, N = 14, O = 16, Br = 80. The yield (%) corresponding to the product in each step is given in the paren thesis)

Numerical answer type

The surface of copper gets tarnished by the formation of copper oxide. N 2 gas was passed to prevent the oxide formation during heating of copper at 1250 K. However, the N 2 g a s contains 1 mole % of water vapour as impurity. The water vapour oxidises copper as per the reaction given below: 2Cu(s) + H 2O(g)  Cu 2O(s) + H 2(g) ݌ୌమis the minimum partial pressure of H 2 (in bar) needed to prevent the oxidation at 1250 K. The value of ln ሺ݌ୌమሻ is ____. (Given: total pressure = 1 bar, R (universal gas constant) = 8 J K⁻¹ mol⁻¹, ln(10) = 2.3. Cu(s) and Cu 2O(s) are mutually immiscible. At 1250 K: 2Cu(s) + ½ O 2(g)  Cu 2O(s); ∆ܩƟ = − 78,000 J mol⁻¹ H₂(g) + ½ O 2(g)  H₂O(g); ∆ܩƟ = − 1,78,000 J mol⁻¹; G is the Gibbs energy)

Numerical answer type

Consider the following reversible reaction, Aሺgሻ൅Bሺgሻ⇌A B ሺ g ሻ . The activation energy of the backward reaction exceeds that of the forward reaction by 2ܴܶ (in J mol⁻¹). If the pre-exponential factor of the forward reaction is 4 times that of the reverse reaction, the absolute value of ∆ܩƟ (in J mol⁻¹) for the reaction at 300 K is ____. (Given; ln(2) = 0.7, ܴܶ = 2500 J mol⁻¹ at 300 K and G is the Gibbs energy )

Numerical answer type

Consider an electrochemical cell: A(s) | An+(aq, 2 M) || B2n+(aq, 1 M) | B(s) . The value of ∆ܪƟ for the cell reaction is twice that of ∆ܩƟ at 300 K. If the emf of the cell is zero, the ∆ܵƟ (in J K⁻¹ mol⁻¹) of the cell reaction per mole of B formed at 300 K is ____. (Given: ln(2) = 0.7, ܴ( universal gas constant ) = 8.3 J K⁻¹ mol⁻¹. H, S and G are enthalpy, entropy and Gibbs energy, respectively.)

Numerical answer type

Match each set of hybrid orbital s from LIST–I with complex(es) given in LIST–II. LIST–I P. dsp2 Q. sp3 R. sp3d2 S. d2sp3 LIST–II 1. [FeF 6]4− 2. [Ti(H 2O)3Cl3] 3. [Cr(NH 3)6]3+ 4. [FeCl 4]2− 5. Ni(CO) 4 6. [Ni(CN) 4]2− The correct option is

1. P → 5; Q → 4,6; R → 2,3; S → 1
2. P → 5,6; Q → 4; R → 3; S → 1,2
3. P → 6; Q → 4,5; R → 1; S → 2,3
4. P → 4,6; Q → 5,6; R → 1,2; S → 3

The desired product X can be prepared by reacting the major product of the reactions in LIST-I with one or more appropriate reagents in LIST-II. (given, order of migratory aptitude: aryl > alkyl > hydrogen) The correct option is

1. P → 1; Q → 2,3; R → 1,4; S → 2,4
2. P → 1,5; Q → 3,4; R → 4,5; S → 3
3. P → 1,5; Q → 3,4; R → 5; S → 2,4
4. P → 1,5; Q → 2,3; R → 1,5; S → 2,3

LIST-I contains reactions and LI ST-II contains major products. Match each reaction in LIST-I with one or more products in LIST -II and choose the correct option.

1. P → 1,5; Q → 2; R → 3; S → 4
2. P → 1,4; Q → 2; R → 4; S → 3
3. P → 1,4; Q → 1,2; R → 3,4; S → 4
4. P → 4,5; Q → 4; R → 4; S → 3,4

Dilution processes of different aqueous solutions, with water, are given in LIST-I. The effects of dilution of the solutions on [H+] are given in LIST-II. (Note: Degree of dissociation ( ) of weak acid and weak base is << 1; degree of hydrolysis of salt <<1; [H+] represents the concentration of H+ ions) LIST–I P. (10 mL of 0.1 M NaOH + 20 mL of 0.1 M acetic acid) diluted to 60 mL Q. (20 mL of 0.1 M NaOH + 20 mL of 0.1 M acetic acid) diluted to 80 mL R. (20 mL of 0.1 M HCl + 20 mL of 0.1 M ammonia solution) diluted to 80 mL S. 10 mL saturated solution of Ni(OH) 2 i n equilibrium with excess solid Ni(OH) 2 is diluted to 20 mL (solid Ni(OH) 2 is still present after dilution). LIST–II 1. the value of [H+] does not change on dilution 2. the value of [H+] changes to half of its initial value on dilution 3. the value of [H+] changes to two times of its initial value on dilution 4. the value of [H+] changes to ଵ √ଶ times of its initial value on dilution 5. the value of [H+] changes to √2 times of its initial value on dilution Match each process given in LIST -I with one or more effect(s) i n LIST-II. The correct option is

1. P → 4; Q → 2; R → 3; S → 1
2. P → 4; Q → 3; R → 2; S → 3
3. P → 1; Q → 4; R → 5; S → 3
4. P → 1; Q → 5; R → 4; S → 1 JEE (ADVANCED) 2018 PAPER 2 PART-III MATHEMATICS  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 resul t in +1 marks. Selecting any in correct option(s) (second option in this case), with or without selection of an y correct option(s) will result in ‐2 marks.

For any positive integer ݊ ,define ݂௡:ሺ0,∞ሻ→Թ as ݂௡ሺݔሻൌ ∑tanିଵቀଵ ଵାሺ௫ା௝ሻሺ௫ା௝ିଵሻቁ݊ ݆ൌ1 for all ݔ∈ሺ0,∞ሻ. ቀHere,the inverse trigonometric function tanିଵ ݔassumes values in ቀെ గ ଶ,గ ଶቁ.ቁ Then, which of the following statement(s) is (are) TRUE?

1. ∑tanଶሺ݂௝ሺ0ሻሻൌ5 5ହ ௝ୀଵ
2. ∑൫1൅݂௝ᇱሺ0ሻ൯secଶሺ݂௝ሺ0ሻሻൌ1 0ଵ଴ ௝ୀଵ
3. For any fixed positive integer ݊ ,lim ௫→ஶtanሺ݂௡ሺݔሻሻൌଵ ௡
4. For any fixed positive integer ݊ ,lim ௫→ஶsecଶሺ݂௡ሺݔሻሻൌ1

Let ܶ be the line passing through the points ܲሺെ2,7ሻ and ܳሺ2,െ5ሻ . Let ܨଵ be the set of all pairs of circles ሺܵଵ,ܵଶሻ such that ܶ is tangent to ܵଵ at ܲ and tangent to ܵଶ at ܳ ,and also such that ܵଵ and ܵଶ touch each other at a point, say, ܯ .Let ܧଵ be the set representing the locus of ܯ as the pair ሺܵଵ,ܵଶሻ varies in ܨଵ. Let the set of all straight line segments joining a pair of distinct points of ܧଵ and passing through the point ܴሺ1,1ሻ be ܨଶ. Let ܧଶ be the set of the mid-points of the line segments in the set ܨଶ. Then, which of the following statement(s) is (are) TRUE?

1. The point ሺെ2,7ሻ lies in ܧଵ
2. The point ቀସ ହ,଻ହቁ does NOT lie in ܧଶ
3. The point ቀଵ ଶ,1 ቁ lies in ܧଶ
4. The point ቀ0,ଷ ଶቁ does NOT lie in ܧଵ

Let ܵ be the set of all column matrices ൥ܾଵ ܾଶ ܾଷ൩ such that ܾଵ,ܾଶ,ܾଷ∈Թ and the system of equations (in real variables) െݔ ൅2ݕ൅5ݖ ൌ ܾ ଵ 2ݔ െ4ݕ൅3ݖ ൌ ܾ ଶ ݔ െ2ݕ൅2ݖ ൌ ܾ ଷ has at least one solution. Then, which of the following system( s) (in real variables) has (have) at least one solution for each ൥ܾଵ ܾଶ ܾଷ൩∈ܵ ?

1. ݔ൅2 ݕ൅3 ݖൌܾ ଵ,4 ݕ ൅ 5 ݖൌܾ ଶ and ݔ൅2 ݕ൅6 ݖൌܾ ଷ
2. ݔ൅ݕ൅3 ݖൌܾ ଵ,5 ݔ൅ 2 ݕ ൅ 6 ݖൌܾ ଶ and െ2ݔ െݕെ3ݖ ൌ ܾ ଷ
3. െݔ ൅2ݕെ5ݖ ൌ ܾ ଵ,2 ݔെ 4 ݕ ൅ 1 0 ݖൌܾ ଶ and ݔെ2 ݕ൅5 ݖൌܾ ଷ
4. ݔ൅2 ݕ൅5 ݖൌܾ ଵ,2 ݔ൅ 3 ݖൌܾ ଶ and ݔ൅4 ݕെ5 ݖൌܾ ଷ

Consider two straight lines, each of which is tangent to both t he circle ݔଶ൅ݕଶൌଵ ଶ and the parabola ݕଶൌ4 ݔ . Let these lines intersect at the point ܳ .Consider the ellipse whose center is at the origin ܱሺ0,0ሻ and whose semi-major axis is ܱܳ .If the length of the minor axis of this ellipse is √2 , then which of the following statement(s) is (are) TRUE?

1. For the ellipse, the eccentricity is ଵ √ଶ and the length of the latus rectum is 1
2. For the ellipse, the eccentricity is ଵ ଶ and the length of the latus rectum is ଵଶ
3. The area of the region bounded by the ellipse between the l ines ݔൌଵ √ଶ and ݔൌ1 is ଵ ସ√ଶሺߨെ2ሻ
4. The area of the region bounded by the ellipse between the l ines ݔൌଵ √ଶ and ݔൌ1 is ଵ ଵ଺ሺߨെ2ሻ

Let ݎ,ݐ,ݏ be non-zero complex numbers and ܮ be the set of solutions ݖൌݔ൅ݕ݅ ൫ݕ,ݔ ∈Թ,݅ൌ√െ1൯ of the equation ݖݏ൅ݖݐ̅൅ݎൌ0 , where ݖ̅ൌݔെݕ݅ .Then, which of the following statement(s) is (are) TRUE?

1. If ܮ has exactly one element, then |ݏ|്| ݐ |
2. If |ݏ|ൌ| ݐ | , then ܮ has infinitely many elements
3. The number of elements in ܮ∩ሼݖ∶|ݖെ1൅݅ |ൌ5ሽ is at most 2
4. If ܮ has more than one element, then ܮ has infinitely many elements

Let ݂:ሺ0,ߨሻ→Թ be a twice differentiable function such that lim ௧→௫ ௙ሺ௫ሻ ୱ୧୬௧ି௙ሺ௧ሻୱ୧୬௫ ௧ି௫ൌs i nଶݔ for all ݔ∈ሺ0,ߨሻ. If ݂ቀగ ଺ቁൌ െగ ଵଶ , then which of the following statement(s) is (are) TRUE?

1. ݂ቀగ ସቁൌ గ ସ√ଶ
2. ݂ሺݔሻ൏ ௫ర ଺െݔଶ for all ݔ∈ሺ 0 ,ߨ ሻ
3. There exists ߙ∈ሺ0,ߨሻ such that ݂′ሺߙሻൌ0
4. ݂ᇱᇱቀగ ଶ ቁ൅݂ቀగ ଶ ቁൌ0 SECTION 2 (Maxim um Marks: 24)  This section contains EIGHT (08) questions. The answer to each question is a NUMERICAL VALUE.  For each question, enter the co rrect numerical value (in decima l notation, truncated/rounded‐off to the second decimal place ; e.g. 6.25, 7.00, ‐0.33, ‐.30, 30.27, ‐127.30) using the mouse and the on‐ screen virtual numeric keypad in the place designated to enter the answer.  Answer to each question will be evaluated according to the following marking scheme: 0 In all other cases.

The value of the integral න1൅√3 ሺሺݔ൅1ሻଶሺ1െݔሻ଺ሻభ రభమ ଴ ݔ݀ is _____ .

Numerical answer type

Let ܲ be a matrix of order 3ൈ3 such that all the entries in ܲ are from the set ሼെ1,0,1ሽ. Then, the maximum possible value of the determinant of ܲ is _____ .

Numerical answer type

Let ܺ be a set with exactly 5 elements and ܻ be a set with exactly 7 elements. If ߙ is the number of one-one functions from ܺ to ܻ and ߚ is the number of onto functions from ܻ to ܺ ,then the value of ଵ ହ !ሺߚെߙሻ is _____ .

Numerical answer type

Let ݂ :Թ→Թ be a differentiable function with ݂ሺ0ሻൌ0. If ݕൌ݂ሺݔሻ satisfies the differential equation ௗ௬ ௗ௫ൌ ሺ2൅5ݕሻሺ5ݕെ2ሻ, then the value of lim ௫→ ିஶ ݂ሺݔሻ is _____.

Numerical answer type

Let ݂ :Թ→Թ be a differentiable function with ݂ሺ0ሻൌ1 and satisfying the equation ݂ሺݔ൅ݕሻൌ݂ሺݔሻ݂ᇱሺݕሻ൅݂ᇱሺݔሻ݂ሺݕሻ for all ݕ,ݔ ∈Թ. Then, the value of log௘ሺ݂ሺ4ሻሻ is _____.

Numerical answer type

Let ܲ be a point in the first octant, whose image ܳin the plane ݔ൅ݕൌ3 (that is, the line segment ܳܲ is perpendicular to the plane ݔ൅ݕൌ3 and the mid-point of ܳܲ lies in the plane ݔ൅ݕൌ3 ) lies on the ݖ-axis. Let the distance of ܲ from the ݔ-axis be 5. If ܴ is the image of ܲ in the ݕݔ-plane, then the length of ܴܲ is _____ .

Numerical answer type

Consider the cube in the first octant with sides ܱܲ ,ܱܳ and ܴܱ of length 1, along the ݔ-axis, ݕ-axis and ݖ-axis, respectively, where ܱሺ0,0,0ሻ is the origin. Let ܵቀଵ ଶ,ଵଶ,ଵଶቁ be the centre of the cube and ܶ be the vertex of the cube opposite to the origin ܱ such that ܵ lies on the diagonal ܱܶ .If ݌Ԧൌܲܵ ሬሬሬሬԦ, ݍԦൌ,ܳܵ ሬሬሬሬሬሬԦ ݎԦൌ ܴܵ ሬሬሬሬሬԦ and ݐԦൌ ܶܵሬሬሬሬԦ, then the value of หሺ݌ԦൈݍԦሻൈ൫ݎԦൈݐԦ൯ห is _____ .

Numerical answer type

Let ܺൌ൫ C110൯ଶ൅2൫ C210൯ଶ൅3൫ C310൯ଶ൅⋯൅10൫ C1010൯ଶ, where ,10Cr ݎ∈ ሼ1,2,⋯,10 ሽ denote binomial coefficients. Then, the value of ଵ ଵସଷ଴ ܺ is _____ .

Numerical answer type

Let ܧଵൌ ቄݔ∈Թ∶ݔ്1 a n d௫ ௫ିଵ൐0ቅ and ܧଶൌ ൜ݔ∈ܧ ଵ∶ sinିଵ൬log௘ቀ௫ ௫ିଵቁ൰is a real number ൠ . ቀHere,the inverse trigonometric function sinିଵ ݔassumes values in ቂെ ߨ 2,ߨ 2ቃ.ቁ Let :݂ ܧଵ→Թ be the function defined by ݂ሺݔሻൌl o g௘ቀ௫ ௫ିଵቁ and :݃ ܧଶ→Թ be the function defined by ݃ሺݔሻൌs i nିଵ൬log௘ቀ௫ ௫ିଵቁ൰. LIST–I P. The range of ݂ is Q. The range of ݃ contains R. The domain of ݂ contains S. The domain of ݃ is LIST–II 1. ቀെ∞,ଵ ଵି௘ቃ ∪ቂ௘ ௘ିଵ,∞ቁ 2. ሺ0,1ሻ 3. ቂെଵ ଶ,ଵଶቃ 4. ሺെ∞, 0ሻ∪ሺ0,∞ሻ 5. ቀെ∞,௘ ௘ିଵቃ 6. ሺെ∞, 0ሻ∪ቀଵ ଶ,௘ ௘ିଵቃ The correct option is:

1. P → 4; Q → 2; R → 1; S → 1
2. P → 3; Q → 3; R → 6; S → 5
3. P → 4; Q → 2; R → 1; S → 6
4. P → 4; Q → 3; R → 6; S → 5

In a high school, a committee has to be formed from a group of 6 boys ܯଵ, ܯଶ, ܯଷ, ܯସ, ܯହ, ܯ଺ and 5 girls ܩଵ,ܩଶ,ܩଷ,ܩସ,ܩହ. (i) Let ߙଵ be the total number of ways in which the committee can be form ed such that the committee has 5 members, having exactly 3 boys and 2 girls. (ii) Let ߙଶ be the total number of ways in which the committee can be form ed such that the committee has at least 2 members, and having an equal number of boys and girls. (iii) Let ߙଷ be the total number of ways in which the committee can be form ed such that the committee has 5 members, at least 2 of them being girls. (iv) Let ߙସ be the total number of ways in which the committee can be form ed such that the committee has 4 members, having at least 2 girls and such that both ܯଵ and ܩଵ are NOT in the committee together. LIST–I P. The value of ߙଵ is Q. The value of ߙଶ is R. The value of ߙଷ is S. The value of ߙସ is LIST–II 1. 136 2. 189 3. 192 4. 200 5. 381 6. 461 The correct option is:

1. P → 4; Q → 6; R → 2; S → 1
2. P → 1; Q → 4; R → 2; S → 3
3. P → 4; Q → 6; R → 5; S → 2
4. P → 4; Q → 2; R → 3 ; S → 1

Let :ܪ௫మ ௔మെ௬మ ௕మൌ1 , where ܽ൐ܾ൐0 , be a hyperbola in the ݕݔ-plane whose conjugate axis ܯܮ subtends an angle of 60଴ at one of its vertices ܰ .Let the area of the triangle ܰܯܮ be 4√3. LIST–I P. The length of the conjugate axis of ܪ is Q. The eccentricity of ܪ is R. The distance between the foci of ܪ is S. The length of the latus rectum of ܪ is LIST–II 1. 8 2. ସ √ଷ 3. ଶ √ଷ 4. 4 The correct option is:

1. P → 4; Q → 2; R → 1; S → 3
2. P → 4 ; Q → 3 ; R → 1; S → 2
3. P → 4 ; Q → 1; R → 3; S → 2
4. P → 3; Q → 4; R → 2; S → 1

Let ݂ଵ: Թ→Թ, ݂ଶ: ቀെగ ଶ,గ ଶቁ→Թ, ݂ଷ:ቀെ1, ݁ഏ మെ2ቁ→Թ and ݂ସ:Թ→Թ be functions defined by (i) ݂ଵሺݔሻൌ sinቀඥ1െ݁ି௫మቁ, (ii) ݂ଶሺݔሻൌቊ|ୱ୧୬௫| ୲ୟ୬షభ௫ if ݔ ് 0 1 if ݔ ൌ 0 , where the inverse trigonometric function tanିଵݔ assumes values in ቀെ గ ଶ,గ ଶቁ, (iii) ݂ଷሺݔሻൌሾ sinሺlog௘ሺݔ ൅2ሻሻሿ, where, for ݐ∈Թ , ሾݐሿ denotes the greatest integer less than or equal to ݐ , (iv) ݂ସሺݔሻൌቊݔଶsinቀଵ ௫ቁifݔ്0 0 i fݔൌ0. LIST–I P. The function ݂ଵ is Q. The function ݂ଶ is R. The function ݂ଷ is S. The function ݂ସ is LIST–II 1. NOT continuous at ݔൌ0 2. continuous at ݔൌ0 and NOT differentiable at ݔൌ0 3. differentiable at ݔൌ0 and its derivative is NOT continuous at ݔൌ0 4. differentiable at ݔൌ0 and its derivative is continuous at ݔൌ0 The correct option is:

1. P → 2; Q → 3; R → 1; S → 4
2. P → 4 ; Q → 1 ; R → 2 ; S → 3
3. P → 4; Q → 2; R → 1; S → 3
4. P → 2; Q → 1; R → 4; S → 3