$A$ hollow sphere of radius $R$ is filled completely with an ideal liquid of density $\rho$. The sphere is moving horizontally with an acceleration $2g$,where $g$ is the acceleration due to gravity. If the minimum pressure of the liquid is $P_0$,then find the pressure at the centre of the sphere.

In the conversion of limestone to lime,$CaCO_{3(s)} \rightarrow CaO_{(s)} + CO_{2(g)}$,the values of $\Delta H^o$ and $\Delta S^o$ are $+179.1 \ kJ \ mol^{-1}$ and $160.2 \ J \ K^{-1} \ mol^{-1}$ respectively at $298 \ K$ and $1 \ bar$. Assuming that $\Delta H^o$ and $\Delta S^o$ do not change with temperature,the temperature above which the conversion of limestone to lime will be spontaneous is ............. $K$.

$I$. Two non-zero,non-collinear vectors are linearly independent.
$II$. Any three coplanar vectors are linearly dependent.
Which of the above statements is/are true?

Current $I$ flows through a long conducting wire bent at right angles as shown in the figure. The magnetic field at a point $P$ on the right bisector of the angle $XOY$ at a distance $r$ from $O$ is

The probability of a student qualifying for $IITJEE$ and $EAMCET$ are $\frac{1}{5}$ and $\frac{3}{5}$ respectively. The probability that the student qualifies for at least one of these tests is:

$A$ magnetic needle lying parallel to a magnetic field is turned through $60^{\circ}$. The work done on it is $W$. The torque required to maintain the magnetic needle in the position mentioned above is