$A$ force $F$ is given by $F = at + bt^2$,where $t$ is time. What are the dimensions of $a$ and $b$?

The velocity of a particle depends upon the time $t$ according to the equation $v = \sqrt{ab} + bt + \frac{c}{d + t}$. The physical quantities which are represented by $a, b, c$ and $d$,are in the following order:

$A$ liquid drop placed on a horizontal plane has a near spherical shape (slightly flattened due to gravity). Let $R$ be the radius of its largest horizontal section. $A$ small disturbance causes the drop to vibrate with frequency $v$ about its equilibrium shape. By dimensional analysis,the ratio $\frac{v}{\sqrt{\sigma / \rho R^3}}$ can be (Here,$\sigma$ is surface tension,$\rho$ is density,$g$ is acceleration due to gravity and $k$ is an arbitrary dimensionless constant)

The equation of state of some gases can be expressed as $\left( {P + \frac{a}{{{V^2}}}} \right)\,(V - b) = RT$. Here $P$ is the pressure,$V$ is the volume,$T$ is the absolute temperature and $a, b, R$ are constants. The dimensions of $a$ are

Heat produced in a current-carrying conducting wire depends on current $I$,resistance $R$ of the wire,and time $t$ for which the current is passed. Using these facts,obtain the formula for heat energy.

In a particular system of units,a physical quantity can be expressed in terms of the electric charge $e$,electron mass $m_e$,Planck's constant $h$,and Coulomb's constant $k = \frac{1}{4 \pi \epsilon_0}$,where $\epsilon_0$ is the permittivity of vacuum. In terms of these physical constants,the dimension of the magnetic field is $[B] = [e]^\alpha [m_e]^\beta [h]^\gamma [k]^\delta$. The value of $\alpha + \beta + \gamma + \delta$ is . . . . .