One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin $(0,0)$. A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\overrightarrow{p}$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $\mathrm{\Delta }l\ll l$ from its equilibrium position and released, it is found to oscillate at frequency $\frac{1}{\delta }\sqrt{\frac{k}{m}},$ The value of $\delta$ is _______.

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)

A block of weight $100\mathrm{}\mathrm{N}$ is suspended by copper and steel wires of same cross-sectional area $0.5\mathrm{}\mathrm{c}{\mathrm{m}}^2$ and, length $\sqrt{3}\mathrm{}\mathrm{m}$ and $1\mathrm{}\mathrm{m},$ respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}},$ respectively. If elongation in copper wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}\right)$ and elongation in steel wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}\right),$ then the ratio $\frac{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}}{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}}$ is _____.
[Young's modulus for copper and steel are $1\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ and $2\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ respectively]

Statement I The total translational kinetic energy of all the molecules of a given mass of an ideal gas is \(1.5\) times the product of its pressure and its volume.
Statement II The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

Four harmonic waves of equal frequencies and equal intensities ${\mathrm{I}}_0$ have phase angles $0,\frac{\mathrm{\pi }}{3},\frac{2\mathrm{\pi }}{3}$ and $\mathrm{\pi }$ . When they are superposed, the intensity of the resulting wave is $\mathrm{n}{\mathrm{I}}_0$ . The value of $\mathrm{n}$ is

A train $\mathrm{S}1,$ moving with a uniform velocity of $108\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h},$ approaches another train $\mathrm{S}2$ standing on a platform. An observer $\mathrm{O}$ moves with a uniform velocity of $36\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h}$ towards $\mathrm{S}2,$ as shown in figure. Both the trains are blowing whistles of same frequency $120\mathrm{}\mathrm{H}\mathrm{z}.$ When $\mathrm{O}$ is $600\mathrm{}\mathrm{m}$ away from $\mathrm{S}2$ and distance between $\mathrm{S}1$ and $\mathrm{S}2$ is $800\mathrm{}\mathrm{m},$ the number of beats heard by $\mathrm{O}$ is____
[Speed of the sound $=330\mathrm{}\mathrm{m}/\mathrm{s}]$

A rod of length $2 \mathrm{cm}$ makes an angle $\frac{2\pi }{3}\mathrm{rad}$ with the principal axis of a thin convex lens. The lens has a focal length of $10 \mathrm{cm}$ and is placed at a distance of $\frac{40}{3} \mathrm{cm}$ from the object as shown in the figure. The height of the image is $\frac{30\sqrt{3}}{13} \mathrm{cm}$ and the angle made by it with respect to the principal axis is \(\alpha\) rad. The value of $\alpha$ is $\frac{\pi }{n}\mathrm{rad}$, where $n$ is

A particle of mass \(m\) is moving in a circular orbit under the influence of the central force \(F(r)=-k r\), corresponding to the potential energy \(V(r)=k r^2 / 2\), where \(k\) is a positive force constant and \(r\) is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by \(L=n \hbar\), where \(\hbar=h /(2 \pi), h\) is the Planck's constant, and \(n\) a positive integer. If \(v\) and \(E\) are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

The order of the oxidation state of the phosphorus atom in $H_{3}P{O}_2, H_{3}P{O}_4, H_{3}P{O}_3$ and $H_4{P}_2{O}_6$ is

The value of \(\log _{10} \mathrm{~K}\) for a reaction \(A \rightleftharpoons B\) is (Given: \(\Delta_{\mathrm{r}} H_{298 \mathrm{~K}}^{\circ}=-54.07 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{\mathrm{r}} S_{298 \mathrm{~K}}^{\circ}\)\(=10 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) and \(R=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} ; 2.303 \times 8.314 \times 298\)\(=5705\) )

An audio transmitter \((T)\) and a receiver \((R)\) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the \(X\) axis. They are pulled from the equilibrium position in opposite directions along the \(X\) axis by a small angular amplitude \(\theta_0=\cos ^{-1}(0.9)\) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^2\) ) is \(\qquad\)

Among the complex ions,
\(\left[\mathrm{Co}\left(\mathrm{NH}_{2}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NH}_{2}\right)_{2} \mathrm{Cl}_{2}\right]^{+},\)\(\left[\mathrm{CrCl}_{2}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{2}\right]^{3-},\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{4}(\mathrm{OH})_{2}\right]^{+}\), \(\left[\mathrm{Fe}\left(\mathrm{NH}_{3}\right)_{2}(\mathrm{CN})_{4}\right]^{-},\)\(\left[\mathrm{Co}(\mathrm{NH}_{2}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{NH}_{2})_{2}\left(\mathrm{NH}_{3}\right) \mathrm{Cl}\right]^{2+}\) and \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{H}_{2} \mathrm{O}\right) \mathrm{Cl}\right]^{2+}\), the number of complex ion(s) that show(s) cis-trans isomerism is

An organic compound \(\mathbf{P}\) having molecular formula \(\mathrm{C}_6 \mathrm{H}_6 \mathrm{O}_3\) gives ferric chloride test and does not have intramolecular hydrogen bond. The compound \(\mathbf{P}\) reacts with 3 equivalents of \(\mathrm{NH}_2 \mathrm{OH}\) to produce oxime \(\mathbf{Q}\). Treatment of \(\mathbf{P}\) with excess methyl iodide in the presence of \(\mathrm{KOH}\) produces compound \(\mathbf{R}\) as the major product. Reaction of \(\mathbf{R}\) with excess iso-butylmagnesium bromide followed by treatment with \(\mathrm{H}_3 \mathrm{O}^{+}\)gives compound \(\mathrm{S}\) as the major product.
The total number of methyl \(\left(-\mathrm{CH}_3\right)\) group(s) in compound \(\mathbf{S}\) is ________. (consider maximum value if there are two values)