A resistance of \(2 \Omega\) is connected across one gap of a meter-bridge (the length of the wire is \(100 \mathrm{~cm}\) ) and an unknown resistance, greater than \(2 \Omega\), is connected across the other gap. When these resistances are interchanged, the balance point shifts by \(20 \mathrm{~cm}\). Neglecting any corrections, the unknown resistance is

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $\frac{a}{4}$ is pivoted on this rod with its centre at a distance $\frac{a}{4}$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary conserver finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left[\left(\frac{M{a}^2\Omega }{48}\right)n\right]$. The value of $n$ is _________ .

During an experiment with a metre bridge, the galvanometer shows a null point when the jockey is pressed at $40.0\mathrm{cm}$using a standard resistance of $90\mathrm{}\mathrm{\Omega }$ ,as shown in the figure. The least count of the scale used in the metre bridge is $1\mathrm{mm}$. The unknown resistance 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.

A large square container with thin transparent vertical walls and filled with water $\left(\text{refractive index}\frac{4}{3}\right)$ is kept on a horizontal table. A student holds a thin straight wire vertically inside the water $12\mathrm{cm}$ from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in $\mathrm{cm}$) between these images is ________.

Parallel rays of light of intensity $\mathrm{I}=912\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}$ are incident on a spherical black body kept in surroundings of temperature 300 K. Take Stefan-Boltzmann constant $\mathrm{\sigma }=5.7\times {10}^{-8}\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}\mathrm{ }{\mathrm{K}}^{-4}$ and assume that the energy exchange with the surroundings is only through radiation. The final steady state temperature of the black body is close to

A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index \(n\) of the first lens is \(1.5\) and that of the second lens is \(1.2\). Both the curved surface are of the same radius of curvature \(R=14\) \(\mathrm{cm}\). For this bi-convex lens, for an object distance of 40 \(\mathrm{cm}\), the image distance will be

The following carbohydrate is

The ionization isomer of \(\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}\right)_4 \mathrm{Cl}\left(\mathrm{NO}_2\right)\right] \mathrm{Cl}\) is

For any integer $k,$ let ${\alpha }_k=\cos \left(\frac{k\pi }{7}\right)+i\sin \left(\frac{k\pi }{7}\right), where i= \sqrt{-1}.$ The value of the expression $\frac{\sum _{k=1}^{12}\left|{\alpha }_{k+1}-{\alpha }_k\right|}{\sum _{k=1}^3\left|{\alpha }_{4k-1}-{\alpha }_{4k-2}\right|}$ is

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

Which is correct statement if \(\mathrm{N}_2\) is added at equilibrium condition?

Let \(S\) be the set of all \((\alpha, \beta) \in \mathbb{R} \times \mathbb{R}\) such that
\(\lim _{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\log _e(1+x)\right)^\beta}=0\).
Then which of the following is (are) correct?