An experiment measures quantities $a, b,$ and $c$ and then $x$ is calculated from $x = \frac{a^{1/2} b^2}{c^3}.$ If the percentage errors in $a, b,$ and $c$ are $\pm 1\%, \pm 3\%,$ and $\pm 2\%$ respectively,then the percentage error in $x$ is:

Consider an expression $Q V = k P T L^\alpha$ where $V, P, T, L$ are volume, pressure, time, and length respectively. The quantity $[Q]$ has dimension $M L^{-1} T^{-1}$. $k$ is a dimensionless constant. The value of the integer $\alpha$ is:

The speed of ripples $(v)$ on a water surface depends on surface tension $(\sigma)$,density $(\rho)$,and wavelength $(\lambda)$. Then the square of speed $(v^2)$ is proportional to

Velocities $(V)$ and accelerations $(a)$ in two systems of units $1$ and $2$ are related as $V_2 = \frac{n}{m^2} V_1$ and $a_2 = \frac{a_1}{mn}$ respectively. Here $m$ and $n$ are constants. Dimensionally, the relations between distances ($S_1$ and $S_2$) and times ($t_1$ and $t_2$) in the two systems are respectively:

In a new system of units,the unit of mass is $a \ kg$,and the units of length and time are $b \ m$ and $c \ s$,respectively. What is the magnitude of $6 \ W$ of power in this system?

$A$ small steel ball of radius $r$ is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta$. After some time,the velocity of the ball attains a constant value known as terminal velocity $v_T$. The terminal velocity depends on $(i)$ the mass of the ball $m$,$(ii)$ $\eta$,$(iii)$ $r$,and $(iv)$ acceleration due to gravity $g$. Which of the following relations is dimensionally correct?