JEE Advanced 2020 Paper 1

A football of radius R is kept on a hole of radius r (𝑟<𝑅) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle 𝜃 from the horizontal as shown in the figure below . The maximum value of 𝜃 so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

1. sin𝜃=𝑟 𝑅
2. tan𝜃=𝑟 𝑅
3. sin𝜃=𝑟 2𝑅
4. cos𝜃=𝑟 2𝑅

A light disc made of aluminium (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure . A strong magnet is held vertically at a point above the disc away from its axis . On revolving the magnet about the axis of the disc , the disc will (figure is schematic and not drawn to scale)

1. rotate in the direction opposite to the direction of magnet’s motion
2. rotate in the same direction as the direction of magnet’s motion
3. not rotate and its temperature will remain unchanged
4. not rotate but its temperature will slowly rise

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle , and the roller starts rolling without slipping. After t he roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)

A circular coil of radius R and N turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge Q through a switch. The coil is in a horizontal uniform magnetic field Bo parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)

1. 𝜋 2𝑁𝑄𝐵𝑜𝑅2
2. 𝜋𝑁𝑄𝐵𝑜𝑅2
3. 2𝜋𝑁𝑄𝐵𝑜𝑅2
4. 4𝜋𝑁𝑄𝐵𝑜𝑅2

A parallel beam of light strikes a piece of transparent glass having cross section as shown in the figure below . Correct shape of the emergent wavefront will be (figures are schematic and not drawn to scale)

An open -ended U -tube of uniform cross -sectional area contains water (density 103kg m−3). Initially the water level stands at 0.29 m from the bottom in each arm. Kerosene oil ( a water -immiscible liquid) of density 800 kg m−3 is added to the left arm until its length is 0.1 m, as shown in the schematic figure below. The ratio (ℎ1 ℎ2) of the heights of the liquid in the two arms is

1. 15 14
2. 35 33
3. 7 6
4. 5 4

A particle of mass m moves in circular orbits with potential energy 𝑉(𝑟)=𝐹𝑟, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle ’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively , then for the nth orbit (here h is the Planck’s constant )

1. 𝑅∝𝑛13⁄ and v∝𝑛23⁄
2. 𝑅∝𝑛23⁄ and v∝𝑛13⁄
3. 𝐸=3 2(𝑛2ℎ2𝐹2 4𝜋2𝑚)13⁄
4. 𝐸=2(𝑛2ℎ2𝐹2 4𝜋2𝑚)13⁄

The filament of a light bulb has surface area 64 mm². The filament can be considered as a black body at temperature 2500 K emitting radiation like a point sourc e when viewed from far . At night the light bulb is observed from a distance of 100 m. Assume the pupil of the eyes of the observer to be circular with radius 3 mm. Then (Take Stefan -Boltzmann constant = 5.67×10−8 Wm−2K−4, Wien’s displacement constant = 2.90×10−3 m-K, Planck’s constant = 6.63×10−34 Js, speed of light in vacuum = 3.00× 108 ms⁻¹)

1. power radiated by the filament is in the range 642 W to 645 W
2. radiated power entering into one eye of the observer is in the range 3.15×10−8 W to 3.25×10−8 W
3. the wavelength corresponding to the maximum intensity of light is 1160 nm
4. taking the average wavelength of emitted radiation to be 1740 nm, the total number of photons entering per second into one eye of the observer is in the range 2.75×1011 to 2.85×1011

Sometimes it is convenien t to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of diff erent quantities are given in terms of a quantity X as follows: [position ] = [𝑋𝛼]; [speed ] = [𝑋𝛽]; [acceleration ] =[𝑋𝑝]; [linear momentum ] = [𝑋𝑞]; [force ] = [𝑋𝑟]. Then

1. 𝛼+𝑝=2𝛽
2. 𝑝+𝑞−𝑟=𝛽
3. 𝑝−𝑞+𝑟=𝛼
4. 𝑝+𝑞+𝑟=𝛽

A uniform electric field, 𝐸⃗ =−400√3ŷ NC−1 is applied in a region. A charge d particle of mass m carrying positive charge q is projected in this region with an initial speed of 2√10×106 ms⁻¹. This particle is aimed to hit a target T, which is 5 m away from its en try point into the field as shown schematically in the figure. Take 𝑞 𝑚 =1010 Ckg−1. Then

1. the particle will hit T if projected at an angle 45o from the horizontal
2. the particle will hit T if projected either at an angle 30o or 60o from the horizontal
3. time taken by the particle to hit T could be √5 6 μs as well as √5 2 μs
4. time taken by the particle to hit T is √5 3 μs

Shown in the figure is a semicircular metallic strip that has thickness t and resistivity  . Its inner radius is R1 and outer radius is R2. If a voltage V0 is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage ∆𝑉 develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current) . Then (figure is schematic and not drawn to scale)

1. 𝐼=𝑉0𝑡 𝜋𝜌ln(𝑅2 𝑅1)
2. the outer surface is at a higher voltage than the inner surface
3. the outer surface is at a lower voltage than the inner surface
4. ∆𝑉∝𝐼2

As shown schematically in the figure, t wo vessels contain water solutions (at temperature 𝑇) of potassium permanganate (KMnO 4) of different concentrations 𝑛1 and 𝑛2 (𝑛1>𝑛2) molecules per unit volume with ∆𝑛=(𝑛1−𝑛2)≪𝑛1. When the y are connected by a tube of small length 𝑙 and cross -section al area S, KMnO 4 starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gas es and the difference in the ir partial pressure in the two vessels causing the diffusion. The speed 𝑣 of the molecule s is limited by the viscous force −𝛽𝑣 on eac h molecule , where 𝛽 is a constant. Neglect ing all terms of the order (∆𝑛)2, which of the following is/are correct? ( 𝑘𝐵 is the Boltzmann constant)

1. the force causing the molecule s to move across the tube is ∆𝑛𝑘𝐵𝑇𝑆
2. force balance implies 𝑛1𝛽𝑣𝑙=∆𝑛𝑘𝐵𝑇
3. total number of molecules going across the tube per sec is (∆𝑛 𝑙)(𝑘𝐵𝑇 𝛽)𝑆
4. rate of molecules getting transferred through the tube does not change with time

Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping . Then the right finger stops at a distance 𝑥𝑅 from the center (50.00 cm) of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of 𝑥𝑅 (in cm) is ______.

Numerical answer type

When water is filled carefully in a glass, one can fill it to a height h above the rim of the glass due to the surface tension of water. To calculate h just before water starts flowing , model the shape of the water above the rim as a disc of thickness h having semicircular edges , as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are 103kg m−3, 0.07 Nm−1 and 10 ms−2, respectively, the value of h (in mm) is _________.

Numerical answer type

One end of a spring of negligible unstretched length and spring constant k is fixed at the origin (0,0) . A point particle of mass m carrying a positive charge q is attached at its other end. The entire system is kept on a smooth horizontal surface . When a point dipole 𝑝 pointing towards the charge q is fixed at the origin, the spring gets stretched to a length l and attains a new equilibrium position (see figur e below). If the point mass is now displaced slightly by ∆𝑙≪𝑙 from its equilibrium position and released, it is found to oscillate at frequency 1 𝛿√𝑘 𝑚. The value of 𝛿 is ______.

Numerical answer type

Consider one mole of helium gas enclosed in a container at initial pressure 𝑃1 and volume 𝑉1. It expands isothermally to volume 4𝑉1. After this , the gas expands adiabatically and its volume becomes 32𝑉1. The work done by the gas during isothermal and adiabatic expansion processes are 𝑊𝑖𝑠𝑜 and 𝑊𝑎𝑑𝑖𝑎, respectively. If the ratio 𝑊𝑖𝑠𝑜 𝑊𝑎𝑑𝑖𝑎=𝑓 ln2, then 𝑓 is ________.

Numerical answer type

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of 2 ms⁻¹ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is 320 ms⁻¹, the smallest value of the percentage change required in the length of the pipe is ____________.

Numerical answer type

A circular disc of radius 𝑅 carries surface charge density 𝜎(𝑟)=𝜎0(1−𝑟 𝑅), where 𝜎0 is a constant and 𝑟 is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is 𝜙0. Electric flux through another spherical surface of radius 𝑅 4 and concentric with the disc is 𝜙. Then the rati o 𝜙0 𝜙 is_________.

Numerical answer type

If the distribution of molecular speeds of a gas is as per the figure shown below, then the ratio of the most probable, the average, and the root mean square speeds , respectively, is

1. 1 : 1 : 1
2. 1 : 1 : 1.224
3. 1 : 1.128 : 1.224
4. 1 : 1.128 : 1

Which of the following liberates O 2 upon hydrolysis?

1. Pb3O4
2. KO 2
3. Na2O₂
4. Li2O₂

A colorless aqueous solution contains nitrates of two metals, X and Y. When it was added to an aqueous solution of NaCl , a white precipitate was formed. This precipitate was found to be partly soluble in hot water to give a residue P and a solution Q. The residue P was soluble in aq. NH 3 and also in excess sodium thiosulfate. The hot solution Q gave a yellow precipitate with KI. The metals X and Y, respectively, are

1. Ag and Pb
2. Ag and Cd
3. Cd and Pb
4. Cd and Zn

Newman projections P, Q, R and S are shown below : Which one of the following options represent s identical molecules?

1. P and Q
2. Q and S
3. Q and R
4. R and S

Which one of the following structure s has the IUPAC name 3-ethynyl-2-hydroxy -4-methylhex -3-en-5-ynoic acid ?

The Fischer projection of D -erythrose is shown below. D-Erythrose and its isomers are listed as P, Q, R, and S in Colum n-I. Choose the correct relationship of P, Q, R, and S with D-erythrose from Column II .

1. P2, Q3, R2, S2
2. P3, Q1, R1, S2
3. P2, Q1, R1, S3
4. P2, Q3, R3, S1

In thermodynamics, the 𝑃−𝑉 work done is given by 𝑤=−∫𝑑𝑉 𝑃ext . For a system undergoing a particular process, the work done is , 𝑤=−∫𝑑𝑉 (𝑅𝑇 𝑉−𝑏− 𝑎 𝑉2) . This equation is applicable to a

1. system that satisfies the van der Waals equation of state.
2. process that is reversible and isothermal.
3. process that is reversible and adiabatic.
4. process that is irreversible and at constant pressure.

With respect to the compounds I-V, choose the correct statement(s).

1. The a cidity of compound I is due to delocalization in the conjugate base.
2. The conjugate base of compound IV is aromatic.
3. Compound II becomes more acidic , when it has a -NO 2 substituent .
4. The a cidity of compounds follows the order I > IV > V > II > III.

In the reaction scheme shown below, Q, R, and S are the major products. The correct structure of

1. S is
2. Q is
3. R is
4. S is

Choose the correct statement(s) among the following:

1. [FeCl 4]− has tetrahedral geometry.
2. [Co(en)(NH₃)2Cl₂]+ has 2 geometrical isomers.
3. [FeCl 4]− has higher spin -only magnetic moment than [Co(en)(NH₃)2Cl₂]+.
4. The cobalt ion in [Co(en)(NH₃)2Cl₂]+ has s𝑝3𝑑2 hybridization.

With respect to hypochlorite, chlorate and perchlorate ions, choose the correct statement(s).

1. The hypochlorite ion is the strongest conjugate base.
2. The molecular shape of only chlorate ion is influenced by the lone pair of electrons of Cl.
3. The hypochlorite and chlorate ions disproportionate to give rise to identical set of ions.
4. The hypochlorite ion oxidizes the sulfite ion.

The cubic unit cell structure of a compound containing cation M and anion X is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).

1. The empirical formula of the compound is MX.
2. The cation M and anion X have different coordination geometries.
3. The ratio of M -X bond length to the cubic unit cell edge length is 0.866.
4. The ratio of the ionic radii of cation M to anion X is 0.414.

5.00 mL of 0.10 M oxalic acid solution taken in a conical flask is titrated against NaOH from a burette using phenolphthalein indicator. The volume of NaOH required for the appearance of permanent faint pink color is tabulated below for five experiments . What is the concentration, in molarity, of the NaOH solution? Exp. No. Vol. of NaOH (mL ) 1 12.5 2 10.5 3 9.0 4 9.0 5 9.0

Consider the reaction A⇌B at 1000 K. At time 𝑡’, the temperature of the system was increased to 2000 K and the system was allowed to reach equilibrium. Throughout this experiment the partial pressure of A was maintained at 1 bar. Given below is the plot of the partial pressure of B with time. What is the ratio of the standard Gibbs energy of the reaction at 1000 K to that at 2000 K?

Consider a 70% efficient hydrogen -oxygen fuel cell working under standard conditions at 1 bar and 298 K. Its cell reaction is H₂(𝑔)+1 2O₂ (𝑔)→H₂O (𝑙) . The work derived from the cell on the consumption of 1.0×10−3 mol of H₂(𝑔) is used to compress 1.00 mol of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in K) of the ideal gas? The standard reduction potentials for the two half -cells are given below. O₂(𝑔)+4H+(𝑎𝑞)+4𝑒−→2 H 2O (𝑙), 𝐸0=1.23 V, 2H+(𝑎𝑞)+2𝑒−→H₂ (𝑔), 𝐸0=0.00 V . Use 𝐹=96500 C mol⁻¹,𝑅=8.314 J mol⁻¹ K−1.

Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas in liters (L) produced at 300 K and 1 .0 atm pressure, when 5.4 g of aluminium and 50.0 mL of 5.0 M sulfuric acid are combined for the reaction? (Use molar mass of aluminium as 27.0 g mol⁻¹, 𝑅=0.082 atm L mol⁻¹ K−1)

U92238 is known to undergo radioactive decay to form Pb82206 by emitting alpha and beta particles. A rock initially contained 68×10−6 g of U92238. If the number of alpha particles that it would emit during its radioactive decay of U92238 to Pb82206 in three half-lives is 𝑍×1018, then what is the value of 𝑍?

In the following reaction, compound Q is obtained from compound P via an ionic intermediate. What is the degree of unsaturation of Q?

Suppose 𝑎,𝑏 denote the distinct real roots of the quadratic polynomial 𝑥2+20𝑥−2020 and suppose 𝑐,𝑑 denote the distinct complex roots of the quadratic polynomial 𝑥2−20𝑥+2020 . Then the value of 𝑎𝑐(𝑎−𝑐)+𝑎𝑑(𝑎−𝑑)+𝑏𝑐(𝑏−𝑐)+𝑏𝑑(𝑏−𝑑) is

1. 0
2. 8000
3. 8080
4. 16000

If the function 𝑓:ℝ⟶ℝ is defined by 𝑓(𝑥)=|𝑥|(𝑥−sin𝑥), then which of the following statements is TRUE ?

1. 𝑓 is one -one, but NOT onto
2. 𝑓 is onto, but NOT one-one
3. 𝑓 is BOTH one-one and onto
4. 𝑓 is NEITHER one-one NOR onto

Let the functions 𝑓:ℝ⟶ℝ and 𝑔:ℝ⟶ℝ be defined by 𝑓(𝑥)= 𝑒𝑥−1−𝑒−|𝑥−1| and 𝑔(𝑥)= 1 2(𝑒𝑥−1+𝑒1−𝑥). Then t he area of the region in the first quadrant bounded by the curves 𝑦=𝑓(𝑥), 𝑦=𝑔(𝑥) and 𝑥=0 is

1. (2−√3)+1 2(𝑒−𝑒−1)
2. (2+√3)+1 2(𝑒−𝑒−1)
3. (2−√3)+1 2(𝑒+𝑒−1)
4. (2+√3)+1 2(𝑒+𝑒−1)

Let 𝑎,𝑏 and 𝜆 be positive real numbers. Suppose 𝑃 is an end point of the latus rectum of the parabola 𝑦2=4𝜆𝑥, and suppose the ellipse 𝑥2 𝑎2+𝑦2 𝑏2=1 passes through the point 𝑃. If the tangents to the parabola and the ellipse at the point 𝑃 are perpendicular to each other, then the eccentricity of the ellipse is

1. 1 √2
2. 1 2
3. 1 3
4. 2 5

Let 𝐶1 and 𝐶2 be two biased coins such that the probabilities of getting head in a single toss are 2 3 and 1 3 , respectively. Suppose 𝛼 is the number of heads that appear when 𝐶1 is tossed twice, independently, and suppose 𝛽 is the number of heads that appear when 𝐶2 is tossed twice, independently. Then the probability that the roots of the quadratic polynomial 𝑥2−𝛼𝑥+𝛽 are real and equal , is

1. 40 81
2. 20 81
3. 1 2
4. 1 4

Consider all rectangles lying in the region {(𝑥,𝑦)∈ℝ×ℝ∶0≤𝑥≤𝜋 2 and 0≤𝑦≤2sin(2𝑥)} and having one side on the 𝑥-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

1. 3𝜋 2
2. 𝜋
3. 𝜋 2√3
4. 𝜋√3 2

Let the function 𝑓:ℝ→ℝ be defined by 𝑓(𝑥)=𝑥3−𝑥2+(𝑥−1)sin𝑥 and let 𝑔:ℝ→ℝ be an arbitrary function. Let 𝑓𝑔: ℝ→ℝ be the product function defined by (𝑓𝑔)(𝑥)=𝑓(𝑥)𝑔(𝑥). Then which of the following statements is/are TRUE?

1. If 𝑔 is continuous at 𝑥=1, then 𝑓𝑔 is differentiable at 𝑥=1
2. If 𝑓𝑔 is differentiable at 𝑥=1, then 𝑔 is continuous at 𝑥=1
3. If 𝑔 is differentiable at 𝑥=1, then 𝑓𝑔 is differentiable at 𝑥=1
4. If 𝑓𝑔 is differentiable at 𝑥=1, then 𝑔 is differentiable at 𝑥=1

Let 𝑀 be a 3×3 invertible matrix with real entries and let 𝐼 denote the 3×3 identity matrix. If 𝑀−1=adj (adj 𝑀), then which of the following statements is/are ALWAYS TRUE?

1. 𝑀=𝐼
2. det𝑀=1
3. 𝑀2=𝐼
4. (adj 𝑀)2=𝐼

Let 𝑆 be the set of all complex numbers 𝑧 satisfying |𝑧2+𝑧+1|=1. Then which of the following statements is/are TRUE?

1. |𝑧+1 2|≤1 2 for all 𝑧∈𝑆
2. |𝑧|≤2 for all 𝑧∈𝑆
3. |𝑧+1 2|≥1 2 for all 𝑧∈𝑆
4. The set 𝑆 has exactly four elements

Let 𝑥,𝑦 and 𝑧 be positive real numbers. Suppose 𝑥,𝑦 and 𝑧 are the lengths of the sides of a triangle opposite to its angles 𝑋,𝑌 and 𝑍, respectively. If tan𝑋 2+tan𝑍 2=2𝑦 𝑥+𝑦+𝑧 , then which of the following statements is/are TRUE?

1. 2𝑌=𝑋+𝑍
2. 𝑌=𝑋+𝑍
3. tanX 2=𝑥 𝑦+𝑧
4. 𝑥2+𝑧2−𝑦2=𝑥𝑧

Let 𝐿1 and 𝐿2 be the following straight lines . 𝐿1: 𝑥−1 1=𝑦 −1=𝑧−1 3 and 𝐿2: 𝑥−1 −3=𝑦 −1=𝑧−1 1 . Suppose the straight line 𝐿: 𝑥−𝛼 𝑙=𝑦−1 𝑚=𝑧−𝛾 −2 lies in the plane containing 𝐿1 and 𝐿2, and passes through the point of intersection of 𝐿1 and 𝐿2. If the line 𝐿 bisects the acute angle between the lines 𝐿1 and 𝐿2, then which of the following statements is/are TRUE?

1. 𝛼−𝛾=3
2. 𝑙+𝑚=2
3. 𝛼−𝛾=1
4. 𝑙+𝑚=0

Which of the following inequalities is/are TRUE?

1. ∫𝑥cos𝑥 𝑑𝑥≥3 81 0
2. ∫𝑥sin𝑥 𝑑𝑥≥3 101 0
3. ∫𝑥2cos𝑥 𝑑𝑥≥1 21 0
4. ∫𝑥2sin𝑥 𝑑𝑥≥2 91 0

Let 𝑚 be the minimum possible value of log 3(3𝑦1+3𝑦2+3𝑦3), where 𝑦1, 𝑦2, 𝑦3 are real numbers for which 𝑦1+𝑦2+𝑦3=9. Let 𝑀 be the maximum possible value of (log 3𝑥1+log 3𝑥2+log 3𝑥3), where 𝑥1, 𝑥2,𝑥3 are positive real numbers for which 𝑥1+𝑥2+𝑥3=9. Then the value of log 2(𝑚3)+log 3(𝑀2) is _____

Numerical answer type

Let 𝑎1,𝑎2,𝑎3,… be a sequence of positive integers in arithmetic progression with common difference 2. Also, let 𝑏1,𝑏2,𝑏3,… be a sequence of positive integers in geometric progression with common ratio 2. If 𝑎1=𝑏1=𝑐, then the number of all possible values of 𝑐, for which the equality 2(𝑎1+ 𝑎2+⋯+𝑎𝑛)=𝑏1+ 𝑏2+⋯+𝑏𝑛 holds for some positive integer 𝑛, is _____

Numerical answer type

Let 𝑓:[0,2]⟶ℝ be the function defined by 𝑓(𝑥)=( 3−sin(2𝜋𝑥))sin(𝜋𝑥−𝜋 4)−sin(3𝜋𝑥+𝜋 4). If 𝛼,𝛽∈[0,2] are such that {𝑥∈[0,2]∶ 𝑓(𝑥)≥0}=[𝛼,𝛽], then the value of 𝛽−𝛼 is _____

Numerical answer type

In a triangle 𝑃𝑄𝑅 , let 𝑎⃗=𝑄𝑅⃗⃗⃗⃗⃗⃗,𝑏⃗⃗=𝑅𝑃⃗⃗⃗⃗⃗⃗ and 𝑐⃗=𝑃𝑄⃗⃗⃗⃗⃗⃗. If |𝑎⃗|=3, |𝑏⃗⃗|=4 and 𝑎⃗⋅(𝑐⃗−𝑏⃗⃗) 𝑐⃗⋅(𝑎⃗−𝑏⃗⃗)=|𝑎⃗| |𝑎⃗|+|𝑏⃗⃗| , then the value of |𝑎⃗×𝑏⃗⃗|2 is _____

Numerical answer type

For a polynomial 𝑔(𝑥) with real coefficients, let 𝑚𝑔 denote the number of distinct real roots of 𝑔(𝑥). Suppose 𝑆 is the set of polynomials with real coefficients def ined by 𝑆={(𝑥2−1)2(𝑎0+𝑎1𝑥+𝑎2𝑥2+𝑎3𝑥3)∶ 𝑎0,𝑎1,𝑎2,𝑎3∈ℝ}. For a polynomial 𝑓, let 𝑓′and 𝑓′′ denote its first and second order derivatives, respectively. Then the minimum possible value of (𝑚𝑓′+𝑚𝑓′′), where 𝑓∈𝑆, is _____

Numerical answer type

Let 𝑒 denote the base of the natural logarithm. The value of the real number 𝑎 for which the right hand limit lim 𝑥→0+(1−𝑥)1 𝑥−𝑒−1 𝑥𝑎 is equal to a nonzero real number, is _____

Numerical answer type