Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^3\) with correct significant figures:

A ray \(O P\) of monochromatic light is incident on the face \(A B\) of prism \(A B C D\) near vertex \(B\) at an incident angle of \(60^{\circ}\) (see figure). If the refractive index of the material of the prism is \(\sqrt{3}\), which of the following is (are) correct?

Consider an elliptically shaped rail PQ in the vertical plane with $\mathrm{O}\mathrm{P}\mathrm{}=\mathrm{}3\mathrm{}\mathrm{m}$ and $\mathrm{O}\mathrm{Q}\mathrm{}=\mathrm{}4\mathrm{}\mathrm{m}$ . A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is $(\mathrm{n}\times 10)$ Joules. The value of n is (take acceleration due to gravity = $10\mathrm{}\mathrm{m}{\mathrm{s}}^{–2}$ )

Two spherical stars $A$ and $B$ have densities ${\rho }_A$ and ${\rho }_B$, respectively. $A$ and $B$ have the same radius, and their masses $M_A$ and $M_B$ are related by $M_B=2M_A$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains ${\rho }_A$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being ${\rho }_A$. If $v_A$ and $v_B$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_B}{v_A}=\sqrt{\frac{10n}{{15}^{{\displaystyle \frac{1}{3}}}}}$. The value of $n$ is

A particle of mass \(m\) is under the influence of the gravitational field of a body of mass \(M(\gg m)\). The particle is moving in a circular orbit of radius \(r_0\) with time period \(T_0\) around the mass \(M\). Then, the particle is subjected to an additional central force, corresponding to the potential energy \(V_{\mathrm{c}}(r)=m \alpha / r^3\), where \(\alpha\) is a positive constant of suitable dimensions and \(r\) is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius \(r_0\) in the combined gravitational potential due to \(M\) and \(V_{\mathrm{c}}(r)\), but with a new time period \(T_1\), then \(\left(T_1^2-T_0^2\right) / T_1^2\) is given by
[ \(G\) is the gravitational constant.]

A thermodynamic system is taken from an initial state $\mathrm{i}$ with internal energy ${\mathrm{U}}_{\mathrm{i}}\mathrm{}=100\mathrm{}\mathrm{J}$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the paths af, ib and bf are ${\mathrm{w}}_{\mathrm{a}\mathrm{f}}=\mathrm{}200\mathrm{}\mathrm{J},\mathrm{}{\mathrm{W}}_{\mathrm{i}\mathrm{b}}\mathrm{}=\mathrm{}50\mathrm{}\mathrm{J}$ and ${\mathrm{W}}_{\mathrm{b}\mathrm{f}}\mathrm{}=\mathrm{}100\mathrm{}\mathrm{J}$ respectively. The heat supplied to the system along the path iaf, ib and bf are iaf ib ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}},\mathrm{}{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}$ and ${\mathrm{Q}}_{\mathrm{b}\mathrm{f}}$ respectively. If the internal energy of the system in the state b is ${\mathrm{U}}_{\mathrm{b}}=200\mathrm{}\mathrm{J}$ and ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}}=500\mathrm{}\mathrm{J}$ , the ratio $\frac{{\mathrm{Q}}_{\mathrm{b}\mathrm{f}}}{{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}}\mathrm{}$ is

At time $t=0$, a disk of radius $1 \mathrm{m}$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha =\frac{2}{3} \mathrm{rad} {\mathrm{s}}^{-2}$. A small stone is stuck to the disk. At $t=0$, it is at the contact point of the disk and the plane. Later, at time $t=\sqrt{\pi } \mathrm{s}$, the stone detaches itself and flies off tangentially from the disk. The maximum height (in $\mathrm{m}$) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$. The value of $x$ is [Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.] If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

The total number of stereoisomers that can exist for M is

The concentration of \(R\) in the reaction \(R \longrightarrow P\) was measured as a function of time and the following data is obtained :
\(\begin{array}{lllll}{[R] \text { (molar) }} & 1.0 & 0.75 & 0.40 & 0.10 \\ t \text { (min) } & 0.0 & 0.05 & 0.12 & 0.18\end{array}\)
The order of the reaction is

Under the same reaction conditions, initial concentration of \(1.386 \mathrm{~mol} \mathrm{dm}^{-3}\) of a substance becomes half in 40 s and 20 s through first-order and zero-order kinetics, respectively. Ratio \(\left(\frac{k_1}{k_0}\right)\) of the rate constants for first order \(\left(k_1\right)\) and zero \(\operatorname{order}\left(k_0\right)\) of the reaction is

Consider an $\mathrm{LC}$ circuit, with inductance $L=0.1 \mathrm{H}$ and capacitance $C={10}^{-3} \mathrm{F}$, kept on a plane. The area of the circuit is $1 {\mathrm{m}}^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_0+\beta \mathrm{t}$ with $\beta =0.04 \mathrm{T} {\mathrm{s}}^{-1}$. The maximum magnitude of the current in the circuit is____$\mathrm{m} \mathrm{A}$.

We have taken a saturated solution of \(\mathrm{AgBr}, K_{\mathrm{sp}}\) is \(12 \times 10^{-14}\). If \(10^{-7}\) mole of \(\mathrm{AgNO}_3\) are added to 1 litre of this solution, find conductivity (specific conductance) of this solution in terms of \(10^{-7} \mathrm{Sm}^{-1}\) units.
\( \text {Given, } \lambda_{\left(\mathrm{Ag}^{+}\right)}^{\circ} =6 \times 10^{-3} \mathrm{Sm}^2 \mathrm{~mol}^{-1},\) \(\lambda_{\left(\mathrm{Br}^{-}\right)}^{\circ}=8 \times 10^{-3} \mathrm{Sm}^2 \mathrm{~mol}^{-1}, \)
\( \lambda_{\left(\mathrm{NO}_3^{-}\right)} =7 \times 10^{-3} \mathrm{Sm}^2 \mathrm{~mol}^{-1},\)

The correct statement $\left(\mathrm{s}\right)$ related to the metal extraction processes is(are)