The potential energy of a particle of mass m at a distance r from a fixed point O is given by $V\left(r\right)=k{r}^2/2$, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

A particle of mass M and positive charge Q, moving with a constant velocity ${\overrightarrow{\text{u}}}_1=4\hat{\text{i}}{\text{ms}}^{-1}$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity ${\overrightarrow{\text{u}}}_2=2\left(\sqrt{3}\hat{\text{i}}+\hat{\text{j}}\right){\text{ms}}^{-1}$. The correct statement (s) is (are)

Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=\pi /(2 \omega)\), then the value of \(4 b^2 / a^2\) will be ________ .

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin O. In this distribution, a spherical cavity of radius $R_2$ , centred at $P$ with distance $OP=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\overrightarrow{r}$ is $\overrightarrow{E}\left(\overrightarrow{r}\right)$ , then the correct statement(s) is (are)

A small mass \(m\) is attached to a massless string whose other end is fixed at \(P\) as shown in the figure. The mass is undergoing circular motion in the \(x-y\) plane with centre at \(O\) and constant angular speed \(\omega\). If the angular momentum of the system, calculated about \(O\) and \(P\) are denoted by \(\vec{L}_{O}\) and \(\vec{L}_{P}\) respectively, then

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position $x_0$. Consider two cases : (i) when the block is at $x_0\text{;}$ and (ii) when the block is at $x=x_0+\text{A}\text{.}$ In both the cases, a particle with mass m $\left(<\text{M}\right)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?

A black body of temperature \(T\) is inside chamber of temperature \(T_0\). Now, the closed chamber is slightly opened to sun such that temperature of black body \((T)\) and chamber \(\left(T_0\right)\) remains constant.

Let the straight line \(\mathrm{y}=2 \mathrm{x}\) touch a circle with center \((0, \alpha), \alpha>0\), and radius \(\mathrm{r}\) at a point \(\mathrm{A}_1\). Let \(\mathrm{B}_1\) be the point on the circle such that the line segment \(A_1 B_1\) is a diameter of the circle. Let \(\alpha+r=5+\sqrt{5}\).
Match each entry in List-I to the correct entry in List-II.

The correct option is

The sum of the number of lone pairs of electrons on each central atom in the following species is \(\left[ TeBr _6\right]^{2-},\left[ Br F _2\right]^{+}, S N F_3\) and \(\left[ XeF _3\right]^{-}\)[Atomic number : \(N=7, F=9, S =16, B r=35,\) \(T e=52, X e=54]\)

Let \(\alpha\) and \(\beta\) be the real numbers such that \(\lim _{x \rightarrow 0} \frac{1}{x^3}\left(\frac{\alpha}{2} \int_0^x \frac{1}{1-t^2} d t+\beta x \cos x\right)=2\). Then the value of \(\alpha+\beta\) is _______

The value of \(\int \frac{\left(x^2-1\right) d x}{x^3 \sqrt{2 x^4-2 x^2+1}}\) is

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${}_7{}^{16}\mathrm{N}+{}_2{}^4\mathrm{He}\to {}_1{}^1\mathrm{H}+{}_8{}^{19}\mathrm{O}$ in a laboratory frame is $n$ (in $\mathrm{MeV}$). Assume that ${}_7{}^{16}\mathrm{N}$ is at rest in the laboratory frame. The masses of ${}_7{}^{16}\mathrm{N}, {}_2{}^4\mathrm{He}, {}_1{}^1\mathrm{H}$ and ${}_8{}^{19}\mathrm{O}$ can be taken to be $16.006 \mathrm{u}, 4.003 \mathrm{u}$, $1.008 \mathrm{u}$ and $19.003 \mathrm{u}$, respectively, where $1 \mathrm{u}=930 \mathrm{MeV} {\mathrm{c}}^{-2}$. The value of $n$is
If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

Paragraph :
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 :
What is the speed of the sound?