$A$ block of mass $m$ is placed on a surface with a vertical cross-section given by $y = \frac{x^3}{6}$. If the coefficient of friction is $0.5$,the maximum height above the ground at which the block can be placed without slipping is

The rate of steady volume flow of water through a capillary tube of length $l$ and radius $r$, under a pressure difference of $p$ is $V$. This tube is connected with another tube of the same length but half the radius, in series. Then, the rate of steady volume flow through them is (The pressure difference across the combination is $p$.)

For positive numbers $x, y, \text{ and } z$,the numerical value of the determinant $\left| \begin{array}{ccc} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{array} \right|$ is

$A$ ball moving with velocity $2 \ m/s$ collides head-on with another stationary ball of double the mass. If the coefficient of restitution is $0.5$,then their velocities (in $m/s$) after collision will be

Which of the following orders are correct?
$I$. Thermal stability : $BeSO_4 < MgSO_4 < CaSO_4 < SrSO_4 < BaSO_4$
$II$. Basic nature : $ZnO > BeO > MgO > CaO$
$III$. Solubility in water : $LiOH > NaOH > KOH > RbOH > CsOH$
$IV$. Melting point : $NaCl > KCl > RbCl > CsCl > LiCl$

An ideal gas heat engine operates in a Carnot cycle between $227\,^{\circ}C$ and $127\,^{\circ}C$. It absorbs $6\, kcal$ at the higher temperature. The amount of heat (in $kcal$) converted into work is equal to