Upon heating with $\mathrm{C}{\mathrm{u}}_2\mathrm{S}$ , the reagent(s) that give copper metal is/are

The total number of possible isomers for $\left[\mathrm{Pt}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]{\mathrm{Br}}_2$ is___.

Extraction of metal from the ore cassiterite involves

The correct statement $\left(\mathrm{s}\right)$ related to colloids is(are)

The substituents \(R_{1}\) and \(R_{2}\) for nine peptides are listed in the table given below. How many of these peptides are positively charged at \(\mathrm{pH}=7.0\) ?

\(\begin{array}{|c|c|c|}\hline Peptide & \mathbf{R}_{\mathbf{1}} & \mathbf{R}_{\mathbf{2}} \\\hline I & \mathrm{H} & \mathrm{H} \\\hline II & \mathrm{H} & \mathrm{CH}_{3} \\\hline III & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{H} \\\hline IV & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline V & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VI & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline VII & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VIII & \mathrm{CH}_{2} \mathrm{OH} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline IX & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \mathrm{CH}_{3} \\\hline\end{array}\)

Match the anionic species given in Column I that are present in the ores(s) given in Column II.

Column I	Column II
A. Carbonate	P. Siderite
B. Sulphide	Q. Malachite
C. Hydroxide	R. Bauxite
D. Oxide	S. Calamine
	T. Argentite

A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of $400\mathrm{ }\mathrm{pm}$ . If the density of the substance in the crystal is $8 \mathrm{g}\mathrm{ }{\mathrm{cm}}^{-3},\mathrm{ }$ then the number of atoms present in $256 \mathrm{g}$ of the crystal $\mathrm{is}\mathrm{ }\mathrm{N}\mathrm{ }\times \mathrm{ }{10}^{24}$ . The value of $\mathrm{N}$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

Charges Q, 2Q and 4Q are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of radii R/2, R and 2R respectively, as shown in the figure. If magnitudes of the electric fields at point P at a distance R from the centre of spheres 1, 2 and 3 are ${\mathrm{E}}_1,\mathrm{}\mathrm{}{\mathrm{E}}_2$ and ${\mathrm{E}}_3$ respectively, then

A composite block is made of slabs \(A, B, C, D\) and \(E\) of different thermal conductivities (given in terms of a constant, \(K\) ) and sizes (given in terms of length, \(L\) ) as shown in the figure. All slabs are of same width. Heat \(Q\) flows only from left to right through the blocks. Then, in steady state

Let \(P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}\) and \(Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}\) be two sets. Then,

Let $Q$ be the cube with the set of vertices $\left\{\left(x_1, x_2, x_3\right)\in {\mathrm{ℝ}}^3 : x_1, x_2, x_3\in \left\{0, 1\right\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $\left(0, 0, 0\right)$ and $\left(1, 1, 1\right)$ is in $S$. For lines ${\ell }_1$ and ${\ell }_2$, let $d\left({\ell }_1, {\ell }_2\right)$ denote the shortest distance between them. Then the maximum value of $d\left({\ell }_1, {\ell }_2\right)$, as ${\ell }_1$ varies over $F$ and ${\ell }_2$ varies over $S$, is

Paragraph:

One twirls a circular ring (of mass \(M\) and radius \(R\) ) near the tip of one's finger as shown in Figure 1 . In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is \(r\). The finger rotates with an angular velocity \(\omega_{0}\). The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is \(\mu\) and the acceleration due to gravity is \(g\).








Question:

The minimum value of \(\omega_{0}\) below which the ring will drop down is