Statement I In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision.
Statement II In an elastic collision, the linear momentum of the system is conserved.

In the $\text{xy - plane}$, the region $y>0$ has a uniform magnetic field $B_1\widehat{k}$ and the region $y<0$ has another uniform magnetic field $B_2\widehat{k}$. A positively charged particle is projected from the origin along the positive $\text{y}-\text{axis}$ with speed $v_0=\pi \mathrm{m}{s}^{-1}$ at $t=0$, as shown in the figure. Neglect gravity in this problem. Let $t=T$ be the time when the particle crosses the $\text{x}-\text{axis}$ from below for the first time. If $B_2=4B_1$, the average speed of the particle, in $\mathrm{m}{\mathrm{s}}^{-1}$, along the $\text{x}-\text{axis}$ in the time interval $T$ is ________.

The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1m at ${10}^{o}C.$ Now the end P is maintained at ${10}^{o}C,$ while the end S is heated and maintained at ${400}^{o}C.$ The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is $1.2\times {10}^{-5} K^{-1},$ the change in length of the wire PQ is

Figure shows three resistor configurations \(R_1, R_2\) and \(R_3\) connected to \(3 \mathrm{~V}\) battery. If the power dissipated by the configuration \(R_1, R_2\) and \(R_3\) is \(P_1, P_2\) and \(P_3\), respectively, then

A uniform magnetic field B exists in the region between $x=0$ and $x=\frac{3R}{2}$ (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge $+Q$ and momentum p directed along $x$ -axis enters region 2 from region 1 at point $P_1\left(y=-R\right).$ Which of the following option(s) is/are correct?

A circular coil of radius $R$ and $N$ turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge $Q$ through a switch. The coil is in a horizontal uniform magnetic field $B_o$ parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)

System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass \(2 m\) and \(m\) just after the string is cut will be

For non-negative integers $n,$ let
$f\left(n\right)=\frac{\sum _{k=0}^n\sin \left(\frac{k+1}{n+2}\pi \right)\sin \left(\frac{k+2}{n+2}\pi \right)}{\sum _{k=0}^n{\sin }^2\left(\frac{k+1}{n+2}\pi \right)}$
Assuming ${\cos }^{-1}x$ takes values in $\left[0, \pi \right],$ which of the following options is/are correct?

A colorless aqueous solution contains nitrates of two metals, $\mathrm{X}$ and $\mathrm{Y}$. When it was added to an aqueous solution of $\mathrm{NaCl}$, a white precipitate was formed. This precipitate was found to be partly soluble in hot water to give a residue $\mathrm{P}$ and a solution $\mathrm{Q}$. The residue $\mathrm{P}$ was soluble in aq. ${\mathrm{NH}}_3$ and also in excess sodium thiosulfate. The hot solution $\mathrm{Q}$ gave a yellow precipitate with $\mathrm{KI}$. The metals $\mathrm{X}$ and $\mathrm{Y}$, respectively, are

A solid sphere of radius R and density ρ is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density 3 ρ The complete arrangement is placed in a liquid of density 2ρ and is allowed to reach equilibrium. The correct statement(s) is (are)

Let \(S_n=\sum_{k=0}^n \frac{n}{n^2+k n+k^2}\) and \(T_n=\sum_{k=0}^{n-1} \frac{n}{n^2+k n+k^2}\), for \(n=1,2,3, \ldots\), then

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}\).
Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is

Match each of the compounds is Column I with its characteristic reaction(s) is Column II.