If the force is given by $F = at + bt^2$ where $t$ is time, the dimensions of $a$ and $b$ are:

The potential energy of a particle changes with distance $x$ from a fixed origin as $V = \frac{A\sqrt{x}}{x + B}$, where $A$ and $B$ are constants with appropriate dimensions. The dimensions of $AB$ are . . . . . . .

Force $F$ is given in terms of time $t$ and distance $x$ by $F = a \sin(ct) + b \cos(dx)$. Then the dimension of $a/b$ is:

Which of the following relations can be derived using dimensional analysis?

An expression for a dimensionless quantity $P$ is given by $P = \frac{\alpha}{\beta} \log_{e} \left( \frac{kt}{\beta x} \right)$,where $\alpha$ and $\beta$ are constants,$x$ is distance,$k$ is the Boltzmann constant,and $t$ is the temperature. Then the dimensions of $\alpha$ will be:

In the $CGS$ system, the magnitude of the force is $100 \ dynes$. In another system where the fundamental physical quantities are $kilogram$, $meter$, and $minute$, the magnitude of the force is: