A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated ($P$) by the metal. The sensor has a scale that displays ${\log }_2\left(\frac{P}{P_0}\right)$, where $P_0$ is a constant. When the metal surface is at a temperature of $487 °\mathrm{C}$, the sensor shows a value $1$. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $2767 °\mathrm{C}$?

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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 :
For periodic motion of small amplitude \(A\), the time period \(T\) of this particle is proportional to

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In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).

Question :
Now consider the partition to be free to move without friction so that the pressure of gases in both compartments is the same. Then total work done by the gases till the time they achieve equilibrium will be

The activity of a freshly prepared radioactive sample is \(10^{10}\) disintegrations per second, whose mean life is \(10^{-9} \mathrm{~s}\). The mass of an atom of this radioisotope is \(10^{-25} \mathrm{~kg}\). The mass (in \(\mathrm{mg}\) ) of the radioactive sample is

A lamina is made by removing a small disc of diameter \(2 R\) from a bigger disc of uniform mass density and radius \(2 R\), as shown in the figure. The moment of inertia of this lamina about axes passing though \(O\) and \(P\) is \(I_{O}\) and \(I_{P}\) respectively. Both these axes are perpendicular to the plane of the lamina. The ratio \(I_{P} / I_{O}\) to the nearest integer is

In the given circuit, the \(\mathrm{AC}\) source has \(\omega=100 \mathrm{rad} / \mathrm{s}\). Considering the inductor and capacitor to be ideal, the correct choice(s) is (are)

A thin convex lens is made of two materials with refractive indices ${\mathrm{n}}_1$ and ${\mathrm{n}}_2,$ as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. $\mathrm{f}$ is the focal length of the lens when ${\mathrm{n}}_1={\mathrm{n}}_2=\mathrm{n}.$ The focal length is $\mathrm{f}+\mathrm{\Delta }\mathrm{f}$ when ${\mathrm{n}}_1=\mathrm{n}$ and ${\mathrm{n}}_2=\mathrm{n}+\mathrm{\Delta }\mathrm{n}.$ Assuming $\mathrm{\Delta }\mathrm{n}<<\left(\mathrm{n}-1\right)$ and $1<\mathrm{n}<2,$ the correct statement(s) is/are:

Mixture(s) showing positive deviation from Raoult's law at ${35}^{\mathrm{o}}\mathrm{C}$ is (are)

An optical arrangement consists of two concave mirrors $M_1$ and $M_2$, and a convex lens $L$ with a common principal axis, as shown in the figure. The focal length of $L$ is $10 \mathrm{cm}$. The radii of curvature of $M_1$ and $M_2$ are $20 \mathrm{cm}$ and $24 \mathrm{cm}$, respectively. The distance between $L$ and $M_2$ is $20 \mathrm{cm}$. A point object $S$ is placed at the mid-point between $L$ and $M_2$ on the axis. When the distance between $L$ and $M_1$ is $\frac{n}{7} \mathrm{cm}$, one of the images coincides with $S$. The value of $n$ is _____(Consider the case which gives minimum value).

For a dilute solution containing \(2.5 \mathrm{~g}\) of a non-volatile non-electrolyte solute in \(100 \mathrm{~g}\) of water, the elevation in boiling point at 1 atm pressure is \(2^{\circ} \mathrm{C}\). Assuming concentration of solute is much lower than the concentration of solvent, the vapour pressure ( \(\mathrm{mm}\) of \(\mathrm{Hg}\) ) of the solution is (take \(K_{b}=0.76 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\) )

Let \(f_{1}:(0, \infty) \rightarrow \mathbb{R}\) and \(f_{2}:(0, \infty) \rightarrow \mathbb{R}\) be defined by

\[

f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x > 0

\(

and

\]

f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x > 0

\)

where, for any positive integer \(n\) and real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}\) denotes the product of \(a_{1}, a_{2}, \ldots, a_{n} .\) Let \(m_{i}\) and \(n_{i}\), respectively, denote the number of points of local minima and the number of points of local maxima of function \(f_{i}, i=1,2\), in the interval \((0, \infty)\).

The value of $6m_2+4n_2+8m_2{n}_2$ is ____.

Let $\alpha$ be a positive real number. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\left(\alpha ,\infty \right)\to \mathrm{ℝ}$ be the functions defined by $f\left(x\right)=\sin \left(\frac{\pi x}{12}\right)$ and $g\left(x\right)=\frac{2{\log }_e\left(\sqrt{x}-\sqrt{\alpha }\right)}{{\log }_e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha }}\right)}$.
Then the value of $\underset{x\to {\alpha }^{+}}{\lim }f\left(g\left(x\right)\right)$ is ______.

A particle is moved along a path $\mathrm{A}\mathrm{B}-\mathrm{B}\mathrm{C}-\mathrm{C}\mathrm{D}-\mathrm{D}\mathrm{E}-\mathrm{E}\mathrm{F}-\mathrm{F}\mathrm{A},$ as shown in figure, in presence of a force $\overrightarrow{\mathrm{F}}=\left(\mathrm{\alpha }\mathrm{y}\widehat{\mathrm{i}}+2\mathrm{\alpha }\mathrm{x}\widehat{\mathrm{j}}\right)\mathrm{N},$ where $x$ and $\mathrm{y}$ are in meter and $\mathrm{\alpha }=-1\mathrm{N}/{\mathrm{m}}^{-1}.$ The work done on the particle by this force $\overrightarrow{\mathrm{F}}$ will be ____ Joule.