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

$A$ 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.

An expression of energy density is given by $u = \frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)$,where $\alpha, \beta$ are constants,$x$ is displacement,$k$ is Boltzmann constant,and $t$ is the temperature. The dimensions of $\beta$ will be.

$A, B, C$ and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. We know that the equation $AD = C \ln(BD)$ holds true. Which of the following combinations is not a meaningful quantity?

Write the dimensions of $a/b$ in the relation $P = \frac{a - t^2}{bx}$,where $P$ is pressure,$x$ is distance,and $t$ is time.

$A$ neutron star with magnetic moment of magnitude $m$ is spinning with angular velocity $\omega$ about its magnetic axis. The electromagnetic power $P$ radiated by it is given by $\mu_{0}^{x} m^{y} \omega^{z} c^{u}$,where $\mu_{0}$ and $c$ are the permeability and speed of light in free space,respectively. Then,