The frequency $$\left( f \right)$$ of a wire oscillating with a length $$\ell ,$$ in $$p$$ loops, under a tension $$T$$ is given by $$f = \frac{p}{{2\ell }}\sqrt {\frac{T}{\mu }} $$ where $$\mu = $$ linear density of the wire. If the error made in determining length, tension and linear density be $$1\% , - 2\% $$ and $$4\% ,$$ then find the percentage error in the calculated frequency.
A quantity $$X$$ is given by $${\varepsilon _0}L\frac{{\Delta V}}{{\Delta t}}$$ where $${ \in _0}$$ is the permittivity of the free space, $$L$$ is a length, $$\Delta V$$ is a potential difference and $$\Delta t$$ is a time interval. The dimensional formula for $$X$$ is the same as that of-
Pressure depends on distance as, $$P = \frac{\alpha }{\beta }exp\left( { - \frac{{\alpha z}}{{k\theta }}} \right),$$ where $$\alpha ,$$ $$\beta $$ are constants, $$z$$ is distance, $$k$$ is Boltzman’s constant and $$\theta $$ is temperature. The dimension of $$\beta $$ are-