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 .......... $\%$.

If $x$ and $a$ represent distance,then for what value of $n$ is the given equation dimensionally correct? The equation is $\int \frac{dx}{\sqrt{a^2 - x^n}} = \sin^{-1} \frac{x}{a}$.

The dimensional formula for a physical quantity $x$ is $[M^{-1} L^{3} T^{-2}]$. The errors in measuring the quantities $M, L$ and $T$ respectively are $2\%, 3\%$ and $4\%$. The maximum percentage of error that occurs in measuring the quantity $x$ is (in $\%$)

The number of particles crossing a unit area perpendicular to the $X$-axis in unit time is given by $n = -D \frac{n_2 - n_1}{x_2 - x_1}$,where $n_1$ and $n_2$ are the number of particles per unit volume at positions $x_1$ and $x_2$ respectively. Find the dimensions of $D$,which is known as the diffusion constant.

If mass is written as $m=kc^{p} G^{-1 / 2} \,h^{1 / 2}$ then the value of $P$ will be : (Constants have their usual meaning with $k$ a dimensionless constant)

The energy $E$ of a system is a function of time $t$ and is given by $E(t) = \alpha t - \beta t^3$,where $\alpha$ and $\beta$ are constants. The dimensions of $\alpha$ and $\beta$ are