Consider an electric field \(\mathbf{E}=E_0 \hat{\mathbf{x}}\) where \(\bar{E}_0\) is a constant. The flux through the shaded area ( as shown in the figure) due to this field is

A drop of liquid of radius $R={10}^{-2}m$ having surface tension $S=\frac{0.1}{4\pi }N{m}^{-1}$ divides itself into K identical drops. In the process the total change in the surface energy $\mathrm{\Delta }U={10}^{-3}J$ . If $K={10}^{\alpha }$ then the value of $\alpha$ is

A massless rod \(B D\) is suspended by two identical massless strings \(A B\) and \(C D\) of equal lengths. A block of mass ' \(m\) ' is suspended point \(P\) such that \(B P\) is equal to ' \(x\) ', if the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of \(x\) is

A small object of uniform density rolls up a curved surface with an initial velocity \(v\). It reaches up to a maximum height of \(\frac{3 v^2}{4 g}\) with respect to the initial\(
\text { position. The object is }
\)

A train is moving along a straight line with a constant acceleration a. A boy standing in the train throws a ball forward with a speed of \(10 \mathrm{~m} / \mathrm{s}\), at an angle of \(60^{\circ}\) to the horizontal. The boy has to move forward by \(1.15 \mathrm{~m}\) inside the train to catch the ball back at the initial height. The acceleration of the train in \(\mathrm{m} / \mathrm{s}^2\), is

A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas $\left(\gamma =\frac{5}{3}\right)$ and one mole of an ideal diatomic gas $\left(\gamma =\frac{7}{5}\right)$. Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of $66$ Joule when heated at constant pressure. The change in its internal energy is _____ Joule.

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Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question:

The average current in the steady state registered by the ammeter in the circuit will be

Consider all possible isomeric ketones, including stereoisomers of MW = 100. All these isomers are independently reacted with $\mathrm{N}\mathrm{a}\mathrm{B}{\mathrm{H}}_4$ (NOTE: stereoisomers are also reacted separately). The total number of ketones that give a racemic product(s) is/are

A number is chosen at random from the set $\left\{1,2,3,\dots ,2000\right\}$. Let $p$ be the probability that the chosen number is a multiple of 3 or $7$. Then the value of $500p$ is ____.

The treatment of an aqueous solution of 3.74 g of \(\text{Cu} \left(\text{NO} _3\right)_2\) with excess KI results in a brown solution along with the formation of a precipitate. Passing \(\text{H} _2 \text{S}\) through this brown solution gives another precipitate X . The amount of X (in g ) is _____________ [Given: Atomic mass of \(\text{H} =1, \text{N}=14, \text{O} =16, \text{S}=32, \text{K}=39,\) \(\text{Cu} =63, \text{I} =127]\)

A vertical line passing through the point $\left(\text{h},\text{0}\right)$ intersects the ellipse $\frac{{\text{x}}^2}{4}+\frac{{\text{y}}^2}{3}=1$ at the points $\text{P}$ and $\text{Q}$. Let the tangents to the ellipse at $\text{P}$ and $\text{Q}$ meet at the point $\text{R}$. If $\Delta \left(\mathrm{h}\right)=$ area of triangle $\mathrm{PQR},{\Delta }_1=\underset{\frac{1}{2}\le \mathrm{h}\le 1}{\max }\Delta \left(\mathrm{h}\right)$ and ${\mathrm{\Delta }}_2=\underset{\frac{1}{2}\le \mathrm{h}\le 1}{\min }\Delta \left(\mathrm{h}\right)$, then $\frac{8}{\sqrt{5}}{\Delta }_1-8{\Delta }_2=$

The plot of ${\mathrm{logk}}_{\mathrm{f}}$ versus $\frac{1}{\mathrm{T}}$ for a reversible reaction $\mathrm{A}\left(\mathrm{g}\right)\rightleftharpoons \mathrm{P}\left(\mathrm{g}\right)$ is shown.

Pre-exponential factors for the forward and backward reactions are ${10}^{15} {\mathrm{s}}^{-1}$ and ${10}^{11} {\mathrm{s}}^{-1}$, respectively. If the value of $\mathrm{logK}$ for the reaction at $500 \mathrm{K}$ is $6$, the value of $\left|{\mathrm{logk}}_{\mathrm{b}}\right|$ at $250 \mathrm{K}$ is
$\left[\mathrm{K}=\right.$equilibrium constant of the reaction, ${\mathrm{k}}_{\mathrm{f}}=$rate constant of forward reaction, ${\mathrm{K}}_{\mathrm{b}}=$rate constant of backward reaction]

Statement I Alkali metals dissolve in liquid ammonia to give blue solutions. Statement II Alkali metals in liquid ammonia give solvated species of the type \(\left(M\left(\mathrm{NH}_3\right)_n\right]^{+}(M=\) alkali metals \()\).