List of top Questions asked in JEE Advanced

In a hotel, four rooms are available Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ______

A beaker of radius r is filled with water (refractive index \(\left(\frac{4}{3}\right)\)) up to a height H as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of water level at the center and at the circumference of the beaker is h ( \(h < < H , h < < r\) ), as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature $R$. Which of the following is/are correct? ($g$ is the acceleration due to gravity)

Two large circular discs separated by a distance of $0.01\, m$ are connected to a battery via a switch as shown in the figure Charged oil drops of density $900\, kg \,m ^{-3}$ are released through a tiny hole at the center of the top disc Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of $200\, V$ across the discs As a result, an oil drop of radius $8 \times 10^{-7} m$ stops moving vertically and floats between the discs The number of electrons present in this oil drop is (neglect the buoyancy force, take acceleration due to gravity $=10 \,ms ^{-2}$ and charge on an electron $(e) =1.6 \times 10^{-19} C$ )

Newman projections P, Q, R and S are shown below:

 Which one of the following options represents identical molecules?

Choose the correct statement(s) among the following:

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 $10^{3} \,kg\, m ^{-3} 0.07 \,Nm ^{-1}$ and $10 \,ms ^{-2}$, respective, the value of $h($ in $mm )$ is ________

Let $a, b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^{2}=4 \lambda x$, and suppose the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ 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

A uniform electric field, $\vec{ E }=-400 \sqrt{3} \hat{ y } NC ^{-1}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of $2 \sqrt{10} \times 10^{6} \,ms ^{-1}$. This particle is aimed to hit a target $T$, which is $5 m$ away from its entry point into the field as shown schematically in the figure Take $\frac{ q }{ m }=10^{10} \,C\,kg ^{-1}$. Then

Let the function $f: R \rightarrow R$ be defined by $f(x)=x^{3}-x^{2}+(x-1) \sin x$ and let $g: R \rightarrow R$ be an arbitrary function. Let $f g: R \rightarrow R$ be the product function defined by $(f, g)(x)=f(x) g(x)$. Then which of the following statements is/are TRUE?

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20 x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2}-20 x+2020$. Then the value of $a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)$ is

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 $x_{R}$ 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 $x_{R}$ (in $cm$ ) is____.

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 \(\theta\) from the horizontal as shown in the figure below. The maximum value of \(\theta\) so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

Let \(m\) be the minimum possible value of \(\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)\), where \(y_{1}, y_{2}, y_{3}\) are real numbers for which \(y_{1}+y_{2}+y_{3}=9\) .Let \(M\) be the maximum possible value of \(\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)\), where \(x_{1}, x_{2}, x_{3}\) are positive real numbers for which \(x_{1}+x_{2}+x_{3}=9\) .Then the value of \(\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)\) is_______

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

In thermodynamics, the $P-V$ work done is given by
$w=-\int d V P_{ ext }$
For a system undergoing a particular process, the work done is,
$w=-\int d V\left(\frac{R T}{V-b}-\frac{a}{V^{2}}\right)$
This equation is applicable to a

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$ ) of potassium permanganate $\left( KMnO _{4}\right)$ of different concentrations $n _{1}$ and $n _{2}\left( n _{1} > n _{2}\right)$ molecules per unit volume with $\Delta n =\left( n _{1}- n _{2}\right) < < n _{1}$. When they are connected by a tube of small length $\ell$ and cross-sectional 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 gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta n )^{2}$, which of the following is/are correct? ( $k_{B}$ is the Boltzmann constant)

Sometimes it is convenient 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 different quantities are given in terms of a quantity $X$ as follows: $[$ position $]=\left[X^{\alpha}\right] ;[$ Speed $]=\left[X^{\beta}\right] ;[$ acceleration $]=\left[X^{p}\right] ;[$ Linear momentum $]=\left[X^{q}\right] ;$ [force] $=\left[X^{ r }\right]$. Then

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 the roller has moved \(50 \,cm\), the position of the scale will look like (figures are schematic and not drawn to scale)

Shown in the figure is a semicircular metallic strip that has thickness $t$ and resistivity $\rho$. Its inner radius is $R_{1}$ and outer radius is $R_{2}$. If a voltage $V_{0}$ is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage $\Delta V$ 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)

The filament of a light bulb has surface area \(64\, mm ^{2}\) The filament can be considered as a black body at temperature \(2500\, K\) emitting radiation like a point source 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 =\(567 \times 10^{-8}\), \(Wm ^{-2} K ^{-4}\), Wien's displacement constant =\(290 \times 10^{-3} \),\(m - K\), Planck's constant =\(663 \times 10^{-34} \,Js\), speed of light in vacuum =\(300 \times 108\, ms ^{-1} )\)

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 $n ^{\text {th }}$ orbit (here $h$ is the Planck's constant)

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 \(B_o\) 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)

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)

An open-ended U-tube of uniform cross-sectional area contains water 
\((density\) \(10^{3} \,kg\, m ^{-3} )\). 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 \(\left(\frac{ h _{1}}{ h _{2}}\right)\) of the heights of the liquid in the two arms is

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^{2}+z+1\right|=1$. Then which of the following statements is/are TRUE?