An electric dipole is formed by two charges $+q$ and $-q$ located in $xy$-plane at $(0, 2) \mathrm{mm}$ and $(0, -2) \mathrm{mm}$, respectively, as shown in the figure. The electric potential at point $P(100, 100) \mathrm{mm}$ due to the dipole is $V_0$. The charges $+q$ and $–q$ are then moved to the points $(-1, 2) \mathrm{mm}$ and $(1, -2) \mathrm{mm}$, respectively. What is the value of electric potential at $$

In order to measure the internal resistance $r_1$ of a cell of emf $\epsilon ,$ a meter bridge of wire resistance $R_0=50 \mathrm{\Omega }$, a resistance $\frac{R_0}{2}$, another cell of emf $\frac{\epsilon }{2}$ (internal resistance $r$) and a galvanometer $G$ are used in a circuit, as shown in the figure. If the null point is found at $l=72 \mathrm{cm},$ then the value of $r_1=\_\_\_\_\mathrm{\Omega }.$

A solid sphere of radius \(R\) has moment of inertia \(J\) about its qeometrical axis. If it is melted into a disc of radius \(r\) and thickness \(t\). If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to \(I\), then the value of \(r\) is equal to

A series LCR circuit is connected to a $45 \sin \left(\omega t\right)$ Volt source. The resonant angular frequency of the circuit is ${10}^5 \mathrm{rad} {\mathrm{s}}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega =8\times {10}^4 \mathrm{rad} {\mathrm{s}}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L=50 \mathrm{mH}$, match each entry in List-$\mathrm{I}$ with an appropriate value from List-$\mathrm{II}$ and choose the correct option.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	$I_0$ in $\mathrm{mA}$	$1$	$44.4$
$\mathrm{Q}$	The quality factor of the circuit	$2$	$18$
$\mathrm{R}$	The bandwidth of the circuit in $\mathrm{rad} {\mathrm{s}}^{-1}$	$3$	$400$
$\mathrm{S}$	The peak power dissipated at resonance in Watt	$4$	$2250$
		$5$	$500$

In an $X$-ray tube, electrons emitted from a filament (cathode) carrying current $I$ hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\frac{I}{2},$ the potential difference $V$ is increased to $2V,$ and the separation distance $d$ is reduced to $\frac{d}{2},$ then

Match the following for the given process

Paragraph :
When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.


The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
The acceleration of this particle for \(|x|>X_0\) is

For $x\in R$, the number of real roots of the equation $3x^2-4\left|x^2-1\right|+x-1=0$ is

A circuit with an electrical load having impedance \(Z\) is connected with an AC source as shown in the diagram. The source voltage varies in time as \(V(t)=300 \sin (400 t) \mathrm{V}\), where \(t\) is time in s. List-I shows various options for the load. The possible currents \(i(t)\) in the circuit as a function of time are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

	List-I		List-II
\((P)\)		\((1)\)
\((Q)\)		\((2)\)
\((R)\)		\((3)\)
\((S)\)		\((4)\)
		\((5)\)

${ }^{131}I is$ an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of $8 days$ . A small amount of a serum labelled with ${ }^{131}I$ is injected into the blood of a person. The activity of the amount of ${ }^{131}I$ injected was $2.4\times {10}^5$ Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person's body, and gives an activity of $115 Bq$ . The total volume of blood in the person's body, in litres is approximately (you may use \(\mathrm{e}^{\mathrm{x}} \approx 1+\mathrm{x}\) for \(|\mathrm{x}|<<1\) and \(\ln 2 \approx 0.7\)).

Match each coordination compound in List-I with an appropriate pair of characteristics from List-II and select the correct answer using the code given below the lists
\(\left\{\right.\)en \(= H _2 NCH _2 CH _2 NH _2 ;\) atomic numbers \(\left.: T _{ i }=22, Cr =24 ; Co =27 ; Pt =78\right\}\)

	List – I		List – II
A.	$\left[\mathrm{C}\mathrm{r}{\left(\mathrm{N}{\mathrm{H}}_3\right)}_4\mathrm{C}{\mathrm{l}}_2\right]\mathrm{C}\mathrm{l}$	P.	Paramagnetic and exhibits ionization isomerism
B.	$\left[\mathrm{T}\mathrm{i}{\left({\mathrm{H}}_2\mathrm{O}\right)}_5\mathrm{ }\mathrm{C}\mathrm{l}\right]{\left(\mathrm{N}{\mathrm{O}}_3\right)}_2$	Q.	Diamagnetic and exhibits cis - trans isomerism
C.	$\left[\mathrm{P}\mathrm{t}\left(\mathrm{e}\mathrm{n}\right)\left(\mathrm{N}{\mathrm{H}}_3\right)\mathrm{C}\mathrm{l}\right]\mathrm{N}{\mathrm{O}}_3$	R.	Paramagnetic and exhibits cis - trans isomerism
D.	$\left[\mathrm{C}\mathrm{o}{\left(\mathrm{N}{\mathrm{H}}_3\right)}_4{\left(\mathrm{N}{\mathrm{O}}_3\right)}_2\right]\mathrm{N}{\mathrm{O}}_3$	S.	Diamagnetic and exhibits ionization isomerism

Equation of the plane containing the straight line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\) and perpendicular to the plane containing the straight lines \(\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\) and \(\frac{x}{4}=\frac{y}{2}=\frac{z}{3}\) is

Two parallel wires in the plane of the paper are distance ${\mathrm{X}}_0$ apart. A point charge is moving with speed u between the wires in the same plane at a distance ${\mathrm{X}}_1$ from one of the wires. When the wires carry current of magnitude $\mathrm{ }\mathrm{I}$  in the same direction, the radius of curvature of the path of the point charge is ${\mathrm{R}}_1$ . In contrast, if the currents $\mathrm{ }\mathrm{I}$ in the two wires have directions opposite to each other, the radius of curvature of the path is ${\mathrm{R}}_2$ . If $\frac{{\mathrm{X}}_0}{{\mathrm{X}}_1}=3$ , value of $\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}$ is