Consider a thin square sheet of side \(L\) and thickness \(t\), made of a material of resistivity \(\rho\). The resistance between two opposite faces, shown by the shaded areas in the figure is

A positive point charge of \(10^{-8} \mathrm{C}\) is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm . The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2\) (where \(\epsilon_0\) is the permittivity of free space), which of the following
statements is/are correct: :

Statement I A block of mass \(m\) starts moving on a rough horizontal surface with a velocity \(v\). It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of \(30^{\circ}\) with the horizontal and the same block is made to go up on the surface with the same initial velocity \(v\). The decrease in the mechanical energy in the second situation is smaller than that in the first situation.
Statement II The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.

Paragraph :
A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The maximum value of \(v_0\) for which the disk will roll without slipping 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 slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3h$ from the ground, as shown in the figure. A spherical ball of mass $m$is released on the slide from rest at a height $h$from the top of the terrace. The ball leaves the slide with a velocity ${\overrightarrow{u}}_0=u_0\stackrel{\wedge }{x}$and falls on the ground at a distance $d$from the building making an angle $\theta$with the horizontal. It bounces off with a velocity $\overrightarrow{v}$and reaches a maximum height $h_1$. The acceleration due to gravity is $g$and the coefficient of restitution of the ground is $\frac{1}{\sqrt{3}}$. Which of the following statement(s) is(are) correct?

Paragraph :
When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.


The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
The acceleration of this particle for \(|x|>X_0\) is

Column II shows five systems in which two objects are labelled as \(X\) and \(Y\). Also in each case a point \(P\) is shown. Column-I gives some statements about \(X\) and/or \(Y\). Match these statements to the appropriate system(s) from Column-II.

The value of
\(\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec (\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec (\frac{7 \pi}{12}\) \(+\frac{(k+1) \pi}{2}))\) in the interval \(\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]\) equals

With reference to aqua regia, choose the correct option(s).

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]

For any positive integer $n$, let $S_n:(0,\infty )\to R$ be defined by
$S_n\left(x\right)=\sum _{k=1}^n{\cot }^{-1}\left(\frac{1+k(k+1)x^2}{x}\right)$
where for any $x\in R,{\cot }^{-1}\left(x\right)\in (0,\pi )$ and ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then which of the following statements is (are) TRUE ?

Among the species given below, the total number of diamagnetic species is _____________.
\(H\) atom, \(N O_2\) monomer, \(O_2^{-}\)(superoxide), dimeric sulphur in vapour phase,
\(Mn_3 O_4, \left(NH_4\right)_2\left[FeCl_4\right], \left(NH_4\right)_2\left[NiCl_4\right],\) \(K_2 MnO_4, K_2 CrO_4\)