The value of gravitational acceleration in the $C.G.S.$ system is $980 \; cm/s^2$. Find the value of $g$ in the $M.K.S.$ system.

The dimensions of the product $\mu_{0} \varepsilon_{0}$ are related to those of velocity as

When a wave traverses a medium, the displacement of a particle located at $x$ at a time $t$ is given by $y = a \sin (bt - cx)$, where $a, b,$ and $c$ are constants of the wave. Which of the following is a quantity with dimensions?

If the formula is $X = 3 Y Z^{2}$,where $X$ and $Z$ have dimensions of capacitance and magnetic induction respectively,then the dimensions of $Y$ in the $MKSQ$ system 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 . . . . . . .

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: $[position] = [X^\alpha]$; $[speed] = [X^\beta]$; $[acceleration] = [X^p]$; $[linear momentum] = [X^q]$; $[force] = [X^r]$. Then -
$(A)$ $\alpha + p = 2\beta$
$(B)$ $p + q - r = \beta$
$(C)$ $p - q + r = \alpha$
$(D)$ $p + q + r = \beta$