A real gas behaves like an ideal gas if its

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 _________.

A particle of mass $m$ is projected from the ground with an initial speed $u_{\circ }$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_{\circ }$. The angle that the composite system makes with the horizontal immediately after the collision is

Statement I If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic \(X\)-rays do not change.
Statement II When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

In a radioactive decay chain, ${\mathrm{ }}_{90}^{232}\mathrm{Th}$ nucleus decays to ${\mathrm{ }}_{82}^{212}\mathrm{Pb}$ nucleus. Let $N_{\alpha } \text{and} N_{\beta }$ be the number of $\alpha \text{and} {\beta }^{-}$ particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

A metal surface is illuminated by light of two different wavelengths 248 nm and 310 nm. The maximum speeds of the photoelectrons corresponding to these wavelengths are u₁ and u₂, respectively. If the ratio $u_1:u_2=2:1$ , the work function of the metal is nearly

A monochromatic beam of light is incident at ${60}^{\mathrm{o}}$ on one face of an equilateral prism of refractive index $\mathrm{n}$ and emerges from the opposite face making an angle $\mathrm{\theta }\left(\mathrm{n}\right)$ with the normal (see the figure). For $\mathrm{n}=\sqrt{3}$ the value of $\mathrm{\theta }$ is ${60}^{\mathrm{o}}$ and $\frac{\mathrm{d}\mathrm{\theta }}{\mathrm{d}\mathrm{n}}=\mathrm{m}$ . The value of $\mathrm{m}\mathrm{}$ is

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If \(\rho_{c}\) is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

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Two plane harmonic sound waves are expressed by the equations.
\(\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}\)
(All parameters are in MKS)
Question:
At \(x=0\) how many times the amplitude of \(y_1+y_2\) is zero in one second at \(x=0\) ?

The center of a disk of radius \(r\) and mass \(m\) is attached to a spring of spring constant \(k\), inside a ring of radius \(R>r\) as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as \(T=\frac{2 \pi}{\omega}\). The correct expression for \(\omega\) is ( \(g\) is the acceleration due to gravity):

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \mathrm{mm}$. The circular scale has $100$ divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	$0$ division	$4$ divisions
Attempt-$1$: With wire	$4$ divisions	$20$ divisions
Attempt-$2$: With wire	$4$ divisions	$16$ divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

Six point charges are kept at the vertices of a regular hexagon of side \(\mathrm{L}\) and centre \(O\), as shown in the figure.
Given that \(K=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{L^{2}}\), which of the following statement(s) is (are) correct?