Which one of the following options represents the magnetic field $\overrightarrow{B}$at $O$ due to the current flowing in the given wire segments lying on the $xy$plane?

A spherical body of radius $\mathrm{R}$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $\mathrm{P}\left(\mathrm{r}\right)$ is the pressure at $\mathrm{r}(\mathrm{r}<\mathrm{R})$ , then the correct option(s) is (are)

A block of mass $2\mathrm{M}$ is attached to a massless spring with spring-constant $\mathrm{k}.$ This block is connected to two other blocks of masses $\mathrm{M}$ and $2\mathrm{M}$ using two massless pulleys and strings. The accelerations of the blocks are ${\mathrm{a}}_1,{\mathrm{a}}_2$ and ${\mathrm{a}}_3$ as shown in figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is ${\mathrm{x}}_0.$ Which of the following option(s) is/are correct?
[ $\mathrm{g}$ is the acceleration due to gravity. Neglect friction]

The dimensions of a cone are measured using a scale with a least count of \(2 \mathrm{~mm}\). The diameter of the base and the height are both measured to be \(20.0 \mathrm{~cm}\). The maximum percentage error in the determination of the volume is _______ .

A parallel plate capacitor having plates of area $\mathrm{S}$ and plate separation $\mathrm{d}$ , has capacitance ${\mathrm{C}}_1$ in air. When two dielectrics of different relative permittivities ( ${\mathrm{\epsilon }}_1=2\mathrm{}$ and ${\mathrm{\epsilon }}_2=4$ ) are introduced between the two plates as shown in the figure, the capacitance becomes ${\mathrm{C}}_2$ . The ratio $\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1}$ is

A steady current $I$ flows along an infinitely long hollow cylindrical conductor of radius $R$. This cylinder is placed co-axially inside an infinite solenoid of radius $2R$. The solenoid has $n$ turns per unit length and carries a steady current $I$. Consider a point $P$ at a distance $r$ from the common axis. The correct statement(s) is(are):

Two spheres $P$ and $Q$ of equal radii have densities ${\rho }_1$ and ${\rho }_2$ , respectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities ${\sigma }_1$ and ${\mathrm{\sigma }}_2$ and viscosities ${\eta }_1$ and ${\eta }_2$ , respectively. They float in equilibrium with the sphere $P$ in $L_1$ and sphere $Q$ in $L_2$ and the string being taut (see figure). If sphere $P$ alone in $L_2$ has terminal velocity ${\overrightarrow{V}}_P$ and $Q$ alone in $L_1$ has terminal velocity ${\overrightarrow{V}}_Q$ , then

Two identical plates P and Q , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures \(\mathrm{T}_{\mathrm{P}}\) and \(\mathrm{T}_{\mathrm{Q}}\), respectively, with \(\mathrm{T}_{\mathrm{Q}}<\mathrm{T}_{\mathrm{P}}\), as shown in Fig. 1. The radiated power transferred per unit area from P to Q is \(W_0\). Subsequently, two more plates, identical to P and Q, are introduced between P and Q, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from P to Q (Fig. 2) in the steady state is \(W_S\), then the ratio \(\frac{W_0}{W_S}\) is ________.

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.

Consider the following volume-temperature \((\mathrm{V}-\mathrm{T})\) diagram for the expansion of 5 moles of an ideal monoatomic gas.

Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence \(\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Z}\) is ________.
[Use the given data: Molar heat capacity of the gas for the given temperature range, \(\mathrm{C}_{\mathrm{V}, \mathrm{m}}=12 \mathrm{~J} \mathrm{~K}^{-1}\) \(\mathrm{mol}^{-1}\) and gas constant, \(\left.\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)

\(\mathrm{NiCl}_{2}\left\{\mathrm{P}\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)\right\}_{2}\) exhibits temperature depend-ent magnetic behaviour (paramagnetic/diamagnetic). The coordination geometries of \(\mathrm{Ni}^{2+}\) in the paramagnetic and diamagnetic states are respectively

Three lines
$L_1:\overrightarrow{r}=\lambda \widehat{i}, \lambda \in R,$
$L_2:\overrightarrow{r}=\widehat{k}+\mu \widehat{j}, \mu \in R$ and
$L_3: \overrightarrow{r}=\hat{i}+\hat{j}+\nu \hat{k}, \nu \in R$
are given. For which point(s) $Q$ and $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P, Q$ and $R$ are collinear?

The value of \(\int_{\sqrt{\log 2}}^{\sqrt{\log 3}} \frac{x \sin x^2}{\sin x^2+\sin \left(\log 6-x^2\right)} d x\) is