JEE Advanced 2018 Paper 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. T his 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?

1. 𝑣=√𝑘 2𝑚 𝑅
2. 𝑣=√𝑘 𝑚 𝑅
3. 𝐿=√𝑚𝑘 𝑅2
4. 𝐿=√𝑚𝑘 2 𝑅2

Consider a body of mass 1.0 𝑘𝑔 at rest at the origin at time 𝑡=0. A force 𝐹⃗=(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 𝜏⃗. Which of the following statements is (are) true?

1. |𝜏⃗|=1 3𝑁 𝑚
2. The torque 𝜏⃗ is in the direction of the unit vector + 𝑘̂
3. The velocity of the body at 𝑡=1 𝑠 is v⃗⃗=1 2(𝑖̂+2𝑗̂) 𝑚 𝑠−1
4. The magnitude of displacement of the body at 𝑡=1 𝑠 is 1 6 𝑚

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?

1. For a given material of the capillary tube, ℎ decreases with increase in 𝑟
2. For a given material of the capillary tube, ℎ is independent of 𝜎
3. If this experiment is performed in a lift going up with a constant acceleration, then ℎ decreases
4. ℎ is proportional to contact angle 𝜃

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?

1. 𝐼𝑚𝑎𝑥 =𝑉 2𝑅
2. 𝐼𝑚𝑎𝑥 =𝑉 4𝑅
3. 𝜏=𝐿 𝑅ln2
4. 𝜏=2𝐿 𝑅ln2

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 𝐵⃗⃗ is (are) true?

1. If 𝐼1=𝐼2, then 𝐵⃗⃗ cannot be equal to zero at the origin (0,0,0)
2. If 𝐼1>0 and 𝐼2<0, then 𝐵⃗⃗ can be equal to zero at the origin (0,0,0)
3. If 𝐼1<0 and 𝐼2>0, then 𝐵⃗⃗ can be equal to zero at the origin (0,0,0)
4. If 𝐼1=𝐼2, then the 𝑧-component of the magnetic field at the centre of the loop is (−𝜇0 𝐼 2𝑅)

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?

1. Process I is an isochoric process
2. In process II , gas absorbs heat
3. In process IV , gas releases heat
4. Processes I and III are not isobaric

Two vectors 𝐴⃗ and 𝐵⃗⃗ are defined as 𝐴⃗=𝑎 𝑖̂ and 𝐵⃗⃗=𝑎 (cos𝜔𝑡𝑖̂+sin𝜔𝑡𝑗̂), where 𝑎 is a constant and 𝜔=𝜋/6 𝑟𝑎𝑑 𝑠−1. If |𝐴⃗+𝐵⃗⃗|=√3|𝐴⃗−𝐵⃗⃗ | at time 𝑡=𝜏 for the first time, the value of 𝜏, in 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 , is __________.

Numerical answer type

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 __________.

Numerical answer type

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 t he height of the top of the inclined plane , in 𝑚𝑒𝑡𝑟𝑒𝑠 , is __________. Take 𝑔=10 𝑚 𝑠−2.

Numerical answer type

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 _________.

Numerical answer type

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 𝜖𝑟=____________.

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=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 __________ .

Numerical answer type

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 a ssume 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 __________.

Numerical answer type

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=__________. 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 [𝜖0] 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 )

The relation between [𝐸] and [𝐵] is

1. [𝐸]=[𝐵] [𝐿] [𝑇]
2. [𝐸]=[𝐵] [𝐿]−1 [𝑇]
3. [𝐸]=[𝐵] [𝐿] [𝑇]−1
4. [𝐸]=[𝐵] [𝐿]−1 [𝑇]−1 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 [𝜖0] 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 )

The relation between [𝜖0] and [𝜇0] is

1. [𝜇0]=[𝜖0] [𝐿]2 [𝑇]−2
2. [𝜇0]=[𝜖0] [𝐿]−2 [𝑇]2
3. [𝜇0]=[𝜖0]−1 [𝐿]2 [𝑇]−2
4. [𝜇0]=[𝜖0]−1 [𝐿]−2 [𝑇]2 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 examp le, 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 . 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 )

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?

1. Δ𝑎 (1+𝑎)2
2. 2Δ𝑎 (1+𝑎)2
3. 2Δ𝑎 (1−𝑎2)
4. 2𝑎Δ𝑎 (1−𝑎2) 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 examp le, 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 . 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 )

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

1. 0.04
2. 0.03
3. 0.02
4. 0.01

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

1. NH 4NO 3
2. (NH 4)2Cr2O7
3. Ba(N 3)2
4. Mg 3N₂

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

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

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

1. Bi 2O5 is more basic than N 2O5
2. NF 3 is more covalent than BiF 3
3. PH 3 boils at lower temperature than NH 3
4. The N−N single bond is stronger than the P−P single bond

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

The reaction(s) leading to t he formation of 1,3,5 -trimethyl benzene 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)

1. 𝑞𝐴𝐶=∆𝑈𝐵𝐶 and 𝑤𝐴𝐵=𝑃2(𝑉2−𝑉1)
2. 𝑤𝐵𝐶=𝑃2(𝑉2−𝑉1) and 𝑞𝐵𝐶=∆𝐻𝐴𝐶
3. ∆𝐻𝐶𝐴<∆𝑈𝐶𝐴 and 𝑞𝐴𝐶=∆𝑈𝐵𝐶
4. 𝑞𝐵𝐶=∆𝐻𝐴𝐶 and ∆𝐻𝐶𝐴>∆𝑈𝐶𝐴

Among the species given below, t he total number of diamagnetic species is ___. H atom , NO 2 monomer , O₂− (superoxide) , dimeric sulphur in vapour phase , Mn 3O4, (NH 4)2[FeCl 4], (NH 4)2[NiCl 4], K2MnO 4, K2CrO 4

Numerical answer type

The a mmonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by NiCl 2.6H 2O to form a stable coordination compound. Assume that both the reactions are 100% complete . If 1584 g of ammonium sulphate and 952 g of NiCl 2.6H 2O are used in the preparation, the combined weight (in grams) of gypsum and the nickel - ammonia coordination compo und thus produced is ____. (Atomic weig hts in g mol⁻¹: H = 1, N = 14, O = 16, S = 32, Cl = 35.5 , Ca = 40, Ni = 59 )

Numerical answer type

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 centra l 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 ____.

Numerical answer type

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)

Numerical answer type

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 T orr. 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 T orr at temperature T )

Numerical answer type

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 (𝐾𝑠𝑝) = 2×1010 and the value of ionization constant of HB (𝐾𝑎) = 1×108)

Numerical answer type

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 mole s of a non-volatile solute S in equal a mount (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 dimeriza tion in solvent X is ____. PARAGRAPH “X” Treatment of benzene with CO/HCl in the presence of anhydrous AlCl 3/CuCl followed by reaction with Ac2O/NaOAc gives compound X as the major product. Compound X upon reaction with Br 2/Na 2CO 3, followed by heating at 473 K with moist KOH furnish es Y as the major product. Reaction of X with H 2/Pd-C, followed by H 3PO 4 treatment gives Z as the major product. (There are two questions based on PARAGRAPH “X”, the question given below is one of them )

Numerical answer type

The compound Y is

1. PARAGRAPH “X” Treatment of benzene with CO/HCl in the presence of anhydrous AlCl 3/CuCl followed by reaction with Ac 2O/NaOAc gives compound X as the major product. Compound X upon reaction with Br 2/Na 2CO 3, followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H 2/Pd-C, followed by H 3PO 4 treatment gives Z as the major product. (There are two questions b ased on PARAGRAPH “X”, the question given below is one of them )

The compound Z is

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

The compound R is

1. PARAGRAPH “A” An organic acid P (C11H12O₂) can easily be oxidized to a dibasic acid which reacts with ethyleneglycol 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 )

The compound S is

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?

1. arg(−1−𝑖)= 𝜋 4 , where 𝑖=√−1
2. 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
3. For any two non -zero complex numbers 𝑧1 and 𝑧2, arg(𝑧1 𝑧2)− arg( 𝑧1)+ arg( 𝑧2) is an integer multiple of 2𝜋
4. 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)) = 𝜋, lies on a straight line

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?

1. ∠𝑄𝑃𝑅 = 45°
2. The a rea of the triangle 𝑃𝑄𝑅 is 25√3 and ∠𝑄𝑅𝑃 =120°
3. The radius of the incircle of the triangle 𝑃𝑄𝑅 is 10√3−15
4. The area of the circumcircle of the triangle 𝑃𝑄𝑅 is 100 𝜋

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?

1. The line of intersection of 𝑃1 and 𝑃2 has direction ratios 1,2,−1
2. The line 3𝑥−4 9= 1−3𝑦 9=𝑧 3 is perpendicular to the line of intersection of 𝑃1 and 𝑃2
3. The acute angle between 𝑃1 and 𝑃2 is 60°
4. 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

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?

1. There exist 𝑟,𝑠∈ℝ, where 𝑟<𝑠, such that 𝑓 is one -one on the open interval (𝑟,𝑠)
2. There exist s 𝑥0∈(−4,0) such that |𝑓′(𝑥0)|≤1
3. lim 𝑥→∞𝑓(𝑥)=1
4. There exist s 𝛼∈(−4,4) such that 𝑓(𝛼)+𝑓′′(𝛼) = 0 and 𝑓′(𝛼)≠0

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?

1. 𝑓(2)<1−log e2
2. 𝑓(2)>1−log e2
3. 𝑔(1)>1−log e2
4. 𝑔(1)<1−log e2

Let 𝑓:[0,∞)→ℝ be a continuous function such that 𝑓(𝑥)=1−2𝑥+∫𝑒𝑥−𝑡𝑓(𝑡)𝑑𝑡𝑥 0 for all 𝑥∈[0,∞). Then , which of the following statement(s) is (are) TRUE?

1. The curve 𝑦=𝑓(𝑥) passes through the point (1,2)
2. The curve 𝑦=𝑓(𝑥) passes through the point (2,−1)
3. The area of the region {(𝑥,𝑦)∈[0,1]×ℝ∶𝑓(𝑥)≤𝑦≤√1−𝑥2 } is 𝜋−2 4
4. The area of the region {(𝑥,𝑦)∈[0,1]×ℝ∶𝑓(𝑥)≤𝑦≤√1−𝑥2 } is 𝜋−1 4

The value of ((log2 9)2)1 log2 (log2 9)×(√7)1 log47 is ______ .

Numerical answer type

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 _____ .

Numerical answer type

Let 𝑋 be the set cons isting of the first 2018 term s 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 _____ .

Numerical answer type

The number of real solutions o f the equation sin−1(∑ 𝑥𝑖+1∞ 𝑖=1−𝑥∑(𝑥 2)𝑖∞ 𝑖=1)= 𝜋 2−cos⁻¹(∑(−𝑥 2)𝑖 −∞ 𝑖=1∑(−𝑥)𝑖∞ 𝑖=1) lying in the interval (−1 2,1 2) is _____ . (Here, the inverse trigonometric functions sin−1𝑥 and cos⁻¹𝑥 assume values in [−𝜋 2,𝜋 2] and [0,𝜋], respectively.)

Numerical answer type

For each positive integer 𝑛, let 𝑦𝑛= 1 𝑛((𝑛+1)(𝑛+2)⋯(𝑛+𝑛))1 𝑛 . For 𝑥∈ℝ, let [𝑥] be the greatest integer less than or equal to 𝑥. If lim 𝑛→∞𝑦𝑛= 𝐿, then the value of [𝐿] is _____ .

Numerical answer type

Let 𝑎⃗ and 𝑏⃗⃗ be two unit vectors such that 𝑎⃗⋅𝑏⃗⃗=0. For some 𝑥,𝑦∈ℝ, let 𝑐⃗= 𝑥 𝑎⃗+𝑦 𝑏⃗⃗+(𝑎⃗×𝑏⃗⃗). If |𝑐⃗|=2 and the vector 𝑐⃗ is inclined at the same angle 𝛼 to both 𝑎⃗ and 𝑏⃗⃗, then the value of 8cos2𝛼 is _____ .

Numerical answer type

Let 𝑎,𝑏,𝑐 be three non -zero real numbers such that the equation √3 𝑎cos𝑥+2 𝑏sin𝑥=𝑐, 𝑥∈[−𝜋 2,𝜋 2], has two distinct real roots 𝛼 and 𝛽 with 𝛼+𝛽= 𝜋 3. Then , the value of 𝑏 𝑎 is _____ .

Numerical answer type

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 neighbo uring farmer 𝐹2 takes a way 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 _____ . PARAGRAPH “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 )

Numerical answer type

Let 𝐸1𝐸2 and 𝐹1𝐹2 be the chord s 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

1. 𝑥+𝑦=4
2. (𝑥−4)2+ (𝑦−4)2=16
3. (𝑥−4)(𝑦−4)=4
4. 𝑥𝑦=4 PARAGRAPH “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 )

Let 𝑃 be a point o n 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

1. (𝑥+𝑦)2=3𝑥𝑦
2. 𝑥2/3+ 𝑦2/3= 24/3
3. 𝑥2+ 𝑦2=2𝑥𝑦
4. 𝑥2+ 𝑦2=𝑥2 𝑦2

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

1. 3 40
2. 1 8
3. 7 40
4. 1 5 PARAGRAPH “A” 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. (There are two questions based on PARAGRAPH “A”, the question given below is one of them)

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

1. 1 15
2. 1 10
3. 7 60
4. 1 5