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.

A hot air balloon is carrying some passengers, and a few sandbags of mass $1 \mathrm{kg}$ each so that its total mass is $480 \mathrm{kg}$. Its effective volume giving the balloon its buoyancy is $V$. The balloon is floating at an equilibrium height of $100 \mathrm{m}$. When $N$ number of sandbags are thrown out, the balloon rises to a new equilibrium height close to $150 \mathrm{m}$ with its volume $V$ remain unchanged. If the variation of the density of air with height $h$ from the ground is $\rho \left(h\right)={\rho }_0{e}^{-\frac{h}{h_0}},$ where ${\mathrm{\rho }}_0=1.25 \mathrm{kg} {\mathrm{m}}^{-3}$ and $h_0=6000 \mathrm{m},$ the value of $N$ is _________.

Paragraph:
A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
The buoyancy force acting on the gas bubble is (Assume \(R\) is the universal gas constant)

Statement I If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.
Statement II The linear momentum of an isolated system remains constant.

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists :

List - i	List - II
A. Alpha decay	P. \({ }_8^{15} O \rightarrow{ }_7^{15} N+\ldots \ldots \ldots\).
B. \(\beta^{+}\)decay	Q. \({ }_{92}^{238} U \rightarrow{ }_{90}^{234} Th +\ldots \ldots \ldots\).
C. Fission	R. \({ }_{83}^{185} Bi \rightarrow{ }_{82}^{184} Pb+\ldots \ldots \ldots\).
D. Proton emission	S. \({ }_{94}^{239} Pu \rightarrow{ }_{57}^{140} La +\ldots \ldots \ldots\).

A pendulum consists of a bob of mass \(m=0.1 \mathrm{~kg}\) and a massless inextensible string of length \(L=1.0 \mathrm{~m}\). It is suspended from a fixed point at height \(H=0.9 \mathrm{~m}\) above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse \(P=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}\) is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is \(J \mathrm{~kg}-\mathrm{m}^{2} / \mathrm{s}\). The kinetic energy of the pendulum just after the liftoff is \(K\) Joules.
The value of $K$ is ____.

One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $\left(\mathrm{V}-\mathrm{T}\right)$ diagram. The correct statement(s) is/are:
[ $\mathrm{R}$ is the gas constant]

Consider an electron in the \(n=3\) orbit of a hydrogen-like atom with atomic number \(Z\). At absolute temperature \(T\), a neutron having thermal energy \(k_{\mathrm{B}} T\) has the same de Broglie wavelength as that of this electron. If this temperature is given by \(T=\frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B}\), (where \(h\) is the Planck's constant, \(k_B\) is the Boltzmann constant, \(m_{\mathrm{N}}\) is the mass of the neutron and \(a_0\) is the first Bohr radius of hydrogen atom) then the value of \(\alpha\) is ________

Two solid cylinders \(P\) and \(Q\) of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder \(P\) has most of its mass concentrated near its surface, while \(Q\) has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

The shape of \(\mathrm{XeO}_{2} \mathrm{~F}_{2}\) molecule 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 $\left(\mathrm{h}\mathrm{c}=1242\mathrm{ }\mathrm{e}\mathrm{V}\mathrm{ }\mathrm{n}\mathrm{m}\right)$

A point mass is subjected to two simultaneous sinusoidal displacements in \(x\)-direction, \(x_1(t)=A \sin \omega t\) and \(x_2(t)=A \sin \left(\omega t+\frac{2 \pi}{3}\right)\). Adding a third sinusoidal displacement \(x_3(t)=B \sin (\omega t+\phi)\) bring the mass to a complete rest. The values of \(B\) and \(\phi\) are

If the distance between the plane \(A x-2 y+z=d\) and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \quad\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\), then \(|d|\) is