The equation of state for some gases is given by $(P + \frac{a}{V^2})(V - b) = RT$,where $P$ is pressure,$V$ is volume,$T$ is absolute temperature,and $a, b, R$ are constants. What are the dimensions of '$a$'?

If $\text{force} = \frac{\alpha}{\text{density} + \beta^3}$,then the dimensional formulae of $\alpha$ and $\beta$ are respectively:

The equation of the stationary wave is $y = 2A \sin \left( \frac{2\pi ct}{\lambda} \right) \cos \left( \frac{2\pi x}{\lambda} \right)$. Which statement is not true?

In an experiment,four quantities $a, b, c$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%$ and $4\%$ respectively. Quantity $w$ is calculated as follows: $w = \frac{a^4 b^3}{c^2 \sqrt{d}}$. The percentage error in the measurement of $w$ is .......... $\%$.

$1 \, J$ of energy is to be converted into a new system of units in which length is measured in $10 \, m$,mass in $10 \, kg$,and time in $1 \, minute$. The numerical value of $1 \, J$ in the new system is:

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 -
$(A)$ $\alpha + p = 2\beta$
$(B)$ $p + q - r = \beta$
$(C)$ $p - q + r = \alpha$
$(D)$ $p + q + r = \beta$