The speed of a water wave $v$ depends on its wavelength $\lambda$,the density of water $\rho$,and the acceleration due to gravity $g$. Determine the relationship between these quantities.

Which of the following relations cannot be deduced using dimensional analysis? [the symbols have their usual meanings]

$A$ quantity $z$, to be estimated, has a dependency on the variables $a, b$ and $c$ as $z = a b^2 c^{-2}$. The percentage errors in the measurement of $a, b$ and $c$ are $2.1 \%$, $1.3 \%$ and $2.2 \%$, respectively. The percentage error in the measurement of $z$ would then be: (in $\%$)

In the equation $(P+\frac{a}{V^2})(V-b)=RT$,where $P$ is pressure,$V$ is volume,$T$ is temperature,$R$ is universal gas constant,and $a$ and $b$ are constants. The dimensions of $a$ are

If electronic charge $e$,electron mass $m$,speed of light in vacuum $c$,and Planck's constant $h$ are taken as fundamental quantities,the permeability of vacuum $\mu_0$ can be expressed in units of

$A$ book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:
$(a) \; y = a \sin \left(\frac{2 \pi t}{T}\right)$
$(b) \; y = a \sin v t$
$(c) \; y = \left(\frac{a}{T}\right) \sin \frac{t}{a}$
$(d) \; y = (a \sqrt{2}) \left(\sin \frac{2 \pi t}{T} + \cos \frac{2 \pi t}{T}\right)$
($a =$ maximum displacement of the particle,$v =$ speed of the particle,$T =$ time-period of motion). Rule out the wrong formulas on dimensional grounds.