A container with $1 \mathrm{kg}$ of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is $700 \mathrm{W} {\mathrm{m}}^{-2}$ and it is absorbed by the water over an effective area of $0.05 {\mathrm{m}}^2.$ Assuming that the heat loss from the water to the surroundings is governed by Newton's law of cooling, the difference (in ${}^{\circ }\mathrm{C}$) in the temperature of water and the surroundings after a long time will be_________. (Ignore effect of the container, and take constant for Newton's law of cooling $=0.001 {\mathrm{s}}^{-1},$ Heat capacity of water $=4200 \mathrm{J} {\mathrm{kg}}^{-1} {\mathrm{K}}^{-1}$)

A particle moves in the \(X-Y\) plane under the influence of a force such that its linear momentum is \(\mathbf{p}(t)=A[\hat{\mathbf{i}} \cos k t-\hat{\mathbf{j}} \sin k t]\), where \(A\) and \(k\) are constants. The angle between the force and the momentum is

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_e$. Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4\pi {\epsilon }_0}$, where ${\epsilon }_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $\left[B\right]={\left[e\right]}^{\alpha }{\left[m_e\right]}^{\beta }{\left[h\right]}^{\gamma }{\left[k\right]}^{\delta }$. The value of $\alpha +\beta +\gamma +\delta$ 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)

An optical bench has $1.5\mathrm{ }\mathrm{m}$ long scale having four equal divisions in each $\mathrm{c}\mathrm{m}.$ While measuring the focal length of a convex lens, the lens is kept at $75\mathrm{ }\mathrm{c}\mathrm{m}$ mark of the scale and the object pin is kept at $45\mathrm{ }\mathrm{c}\mathrm{m}$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135\mathrm{ }\mathrm{c}\mathrm{m}$ mark. In this experiment, the percentage error in the measurement of the focal length of the lens is_____

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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

Three vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$ are shown in the figure. Let, $S$ be any point on the vector $\overrightarrow{R}$ . The distance between the points $P$ and $S$ is $b\left|\overrightarrow{R}\right|$. The general relation among vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{S}$ are

At \(298 \mathrm{~K}\), the limiting molar conductivity of a weak monobasic acid is \(4 \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), for an aqueous solution of the acid the degree of dissociation is \(\boldsymbol{\alpha}\) and the molar conductivity is \(\mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\). At \(298 \mathrm{~K}\), upon 20 times dilution with water, the molar conductivity of the solution becomes \(3 \mathbf{y} \times 10^{2} \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)
The value of $\mathrm{y}$ is _________ .

Let \(I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\) and \(P=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)\). Let \(Q=\left(\begin{array}{ll}\mathrm{x} & \mathrm{y} \\ \mathrm{z} & 4\end{array}\right)\) for some non-zero real numbers \(x, y\), and \(z\), for which there is \(2 \times 2\) matrix \(R\) with all entries being non-zero real numbers, such that \(Q R=R P\). Then which of the following statements is (are) TRUE?

Consider \(L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0\) where \(p\) is a real number and \(C: x^2+y^2+6 x-10 y+30=0\).
Statement 1 If line \(L_1\) is a chord of circle \(C\), then line \(L_2\) is not always a diameter of circle \(C\).
Statement 2 If line \(L_1\) is a diameter of circle \(C\), then line \(L_2\) is not a chord of circle \(C\).

If $\mathrm{I}=\frac{2}{\mathrm{\pi }}\underset{-\frac{\mathrm{\pi }}{4}}{\overset{\frac{\mathrm{\pi }}{4}}{\int }}\frac{\mathrm{d}\mathrm{x}}{\left(1+{\mathrm{e}}^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}}\right)\left(2-\mathrm{c}\mathrm{o}\mathrm{s}2\mathrm{x}\right)}$ then find $27{\mathrm{I}}^2$ equals____

The mole fraction of a solute in a solution is $0.1$. At $298 \mathrm{K}$, the molarity of this solution is the same as its molality. The density of this solution at $298 \mathrm{K}$ is $2.0\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{c}{\mathrm{m}}^{–3}$. The ratio of the molecular weights of the solute and solvent, $\left(\frac{\mathrm{M}{\mathrm{W}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}}}{\mathrm{M}{\mathrm{W}}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}}}\right)$, is:

A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the figure. The electric field inside the emptied space is