A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3\times {10}^5$ times heavier than the Earth and is at a distance $2.5\times {10}^4$ times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is $v_e=11.2 km s^{-1}$ . The minimum initial velocity $\left(v_s\right)$ required for the rocket to be able to leave the Sun-Earth system is closest to
(Ignore the rotation and revolution of the Earth and the presence of any other planet)

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of \(\sigma_0\). The separation between any two consecutive sheets is \(1 \mu \mathrm{~m}\). The various regions between the sheets are denoted as \(1,2,3,4\) and 5. If \(\sigma_0=9 \mu \mathrm{C} / \mathrm{m}^2\), then which of the following statements is/are correct: (Take permittivity of free space \(\epsilon_0=9 \times 10^{-12} \mathrm{~F} / \mathrm{m}\)):

The smallest division on the main scale of a Vernier calipers is $0.1 \mathrm{cm}$. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is

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In a mixture of \(\mathrm{H}-\mathrm{H}^{+}\)gas \(\left(\mathrm{He}^{+}\right.\)is singly ionised \(\mathrm{He}\) atom), \(\mathrm{H}\) atoms and \(\mathrm{He}^{+}\)ions are excited to their respective first excited states. Subsequently, \(\mathrm{H}\) atoms transfer their total excitation energy to \(\mathrm{He}^{+}\)ions (by collisions). Assume that the Bohr model of atom is exactly valid.
Question:
The quantum number \(n\) of the state finally populated in \(\mathrm{He}^{+}\)ions is

A planet of mass M, has two natural satellites with masses $m_1\mathrm{a}\mathrm{n}\mathrm{d}{m}_2$ . The radii of their circular orbits are $R_1 \mathrm{a}\mathrm{n}\mathrm{d} R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1,L_1,K_1 \mathrm{a}\mathrm{n}\mathrm{d} T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite $1; \mathrm{a}\mathrm{n}\mathrm{d} v_2,L_2,K_2\mathrm{a}\mathrm{n}\mathrm{d} T_2$ and to be the corresponding quantities of satellite 2. Given $m_1/m_2=2 \mathrm{a}\mathrm{n}\mathrm{d} R_1/R_2=\frac{1}{4}$ , match the ratios in List-I to the numbers in List-II.

	LIST -I		LIST-II
A)	$\frac{v_1}{v_2}$	P)	$\frac{1}{8}$
B)	$\frac{L_1}{L_2}$	Q)	1
C)	$\frac{K_1}{K_2}$	R)	2
D)	$\frac{T_1}{T_2}$	S)	8

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4V_1$. After this, the gas expands adiabatically and its volume becomes $32V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\mathrm{iso}}$ and $W_{\text{adia}},$ $\frac{W_{\mathrm{iso}}}{W_{\mathrm{adia}}}=f\ln \left(2\right)$, then $f$ is _________ .

Consider a hydrogen atom with its electron in the ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is $\left(\mathrm{h}\mathrm{c}=1242\mathrm{ }\mathrm{e}\mathrm{V}\mathrm{ }\mathrm{n}\mathrm{m}\right)$

Considering the following reaction sequence, the correct statement(s) is(are)

Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.
Question:
\(25 \mathrm{~mL}\) of household solution was mixed with \(30 \mathrm{~mL}\) of \(0.50\) M KI and \(10 \mathrm{~mL}\) of \(4 \mathrm{~N}\) acetic acid. In the titration of the liberated iodine, \(48 \mathrm{~mL}\) of \(0.25 \mathrm{~N~} \mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}\) was used to reach the end point. The molarity of the household bleach solution is

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Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

If two structures of same cross-sectional area, but different numerical apertures \(N A_{1}\) and \(N A_{2}\left(N A_{2} < N A_{1}\right)\) are joined longitudinally, the numerical aperture of the combined structure is

At room temperature, disproportionation of an aqueous solution of in situ generated nitrous acid \(\left(\mathrm{HNO}_2\right)\) gives the species

In terms of potential difference $V$ , electic current $I$ , permittivity ${\epsilon }_0$ , permeability ${\mu }_0$ and speed of light $c$ , the dimensionally correct equation(s) is (are)

The number of hexagonal faces that are present in a truncated octahedron is