List of top Questions asked in JEE Advanced

At time $t=0$, a disk of radius $1 \,m$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha=\frac{2}{3}\, rad\, s ^{-2}$ A small stone is stuck to the disk At $t=0$, it is at the contact point of the disk and the plane Later, at time $t=\sqrt{\pi} s$, the stone detaches itself and flies off tangentially from the disk The maximum height (in $m$ ) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$ The value of $x$ is ___[ [Take $g=10 \,m s ^{-2}$]

At time $t=0$, a disk of radius $1 \,m$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha=\frac{2}{3}\, rad\, s ^{-2}$ A small stone is stuck to the disk At $t=0$, it is at the contact point of the disk and the plane Later, at time $t=\sqrt{\pi} s$, the stone detaches itself and flies off tangentially from the disk The maximum height (in $m$ ) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$ The value of $x$ is ___[ [Take $g=10 \,m s ^{-2}$]

A solution is prepared by mixing $0.01\, mol$ each of $H _2 CO _3, NaHCO _3, Na _2 CO _3$, and $NaOH$ in $100 \, mL$ of water $pH$ of the resulting solution is ___ [Given : $p K _{ a 1}$ and $p K _{ a 2}$ of $H _2 CO _3$ are $637$ and $1032$, respectively $; \log 2=030$ ]

Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis Let $P$ be a point on the hyperbola, in the first quadrant Let $\angle \operatorname{SPS}_1=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S _1 P$ at $P _1$ Let $\delta$ be the distance of $P$ from the straight line $SP _1$, and $\beta= S _1 P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is _____.

Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes Let $ \overrightarrow{ a }=3 \hat{ i }+\hat{ j }-\hat{ k }, $ $\overrightarrow{ b }=\hat{ i }+ b _2 \hat{ j }+ b _3 \hat{ k }, b _2, b _3 \in R , $ $ \overrightarrow{ c }= c _1 \hat{ i }+ c _2 \hat{ j }+ c _3 \hat{ k }, c _1, c _2, c _3 \in R$ be three vectors such that $b_2 b_3>0, \vec{a} \cdot \vec{b}=0$ and $\begin{pmatrix}0 & -c_3 & c_2 \\c_3 & 0 & -c_1 \\-c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix} 1 \\b_2 \\b_3\end{pmatrix}=\begin{pmatrix}3-c_1 \\1-c_2 \\-1-c_3\end{pmatrix} $ Then, which of the following is/are TRUE?

The reaction of $HClO _3$ with $HCl$ gives a paramagnetic gas, which upon reaction with $O _3$ produces

The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$ is _____.

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$-axis is $V(z)=\frac{\sigma}{2 \epsilon_0}\left(\sqrt{R^2+z^2}-z\right)$.
 A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z=z_0$ and $z_0>0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec{F}=-c \hat{k}$ with $c>0$ Let $\beta=\frac{2 c \in_0}{q \sigma}$. Which of the following statement(s) is(are) correct?

The greatest integer less than or equal to $\int\limits_1^2 \log _2\left(x^3+1\right) d x+\int\limits_1^{\log _2 9}\left(2^x-1\right)^{\frac{1}{3}} dx$ is _____.

The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range $900-1500 K$ is(are)

Six charges are placed around a regular hexagon of side length a as shown in the figure Five of them have charge $q$, and the remaining one has charge $x$ The perpendicular from each charge to the nearest hexagon side passes through the center $O$ of the hexagon and is bisected by the side Which of the following statement(s) is(are) correct in SI units?

Let \(\beta\) be a real number. Consider the matrix
\(A=\begin{pmatrix}\beta & 0 & 1 \\2 & 1 & -2 \\3 & 1 & -2\end{pmatrix}\)
If \(A^7-(\beta-1) A^6-\beta A^5\) is a singular matrix, then the value of \(9 \beta\) is __.

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$. If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then the greatest integer less than or equal to $\left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}{\sin \beta}+\frac{\sin \beta}{\cos \alpha}\right)^2$ is ____.

Let $\alpha=\displaystyle\sum_{ k =1}^{\infty} \sin ^{2 k }\left(\frac{\pi}{6}\right)$ Let $g:[0,1] \rightarrow R$ be the function defined by $g(x)=2^{\alpha x}+2^{\alpha(1-x)}$. Then, which of the following statements is/are TRUE?

Let $A B C$ be the triangle with $A B=1, A C=3$ and $\angle B A C=\frac{\pi}{2}$ If a circle of radius $r>0$ touches the sides $A B, A C$ and also touches internally the circumcircle of the triangle $A B C$, then the value of $r$ ___

Atom X occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in %) of the resultant solid is closest to

2 mol of Hg(g) is combusted in a fixed volume bomb calorimeter with excess of O₂ at 298 K and 1 atm into HgO(s). During the reaction, temperature increases from 298.0 K to 312.8 K. If heat capacity of the bomb calorimeter and enthalpy of formation of Hg(g) are 20.00 kJ K^–1 and 61.32 kJ mol^–1 at 298 K, respectively, the calculated standard molar enthalpy of formation of HgO(s) at 298 K is X kJ mol^–1. The value of |X| is _______.
[Given: Gas constant R = 8.3 J K^–1 mol^–1]

Consider the strong electrolytes $Z_m X_n, U_m Y_p$ and $V_m X_n$. Limiting molar conductivity $\left(\Lambda^0\right)$ of $U _{ m } Y _{ p }$ and $V _{ m } X _{ n }$ are $250$ and $440 \,S \,cm ^2 mol ^{-1}$, respectively. The value of $( m + n + p )$ is ______ Given:

Ion	Zn+	Up+	Vn+	Xm-	Ym-
λ0 (S cm2 mol-1)	50.0	25.0	100.0	80.0	100.0

 $\lambda^0$ is the limiting molar conductivity of ions. The plot of molar conductivity (A) of $Z_{ m } X _{ n } v s c ^{1 / 2}$ is given below

Two resistances R₁ = X Ω and R₂ = 1 Ω are connected to a wire AB of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0.2 mm at A to 1 mm at B. A galvanometer (G) connected to the center of the wire, 50 cm from each end along its axis, shows zero deflection when A and B are connected to a battery. The value of X is _______.

 

Thermal decomposition of $AgNO _3$ produces two paramagnetic gases. The total number of electrons present in the antibonding molecular orbitals of the gas that has the higher number of unpaired electrons is _____

For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap to two $2 p_z$ orbitals is(are)

Consider the following reaction 

On estimation of bromine in $100 \,g$ of $R$ using Carius method, the amount of $AgBr$ formed (in $g$ ) is __ [Given : Atomic mass of $H =1, C =12, O =16, P =31, Br =80, Ag =108$ ]

Let $PQRS$ be a quadrilateral in a plane, where $QR =1, \angle PQR =\angle QRS =70^{\circ}, \angle PQS =15^{\circ}$ and $\angle PRS =40^{\circ}$. If $\angle RPS =\theta^{\circ}, PQ =\alpha$ and $PS =\beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are