A balloon is made of a material of surface tension $S$ and its inflation outlet (from where gas is filled in it) has small area $A$. It is filled with a gas of density $\rho$ and takes a spherical shape of radius $R$. When the gas is allowed to flow freely out of it, its radius $r$ changes from $R$ to 0 (zero) in time $T$. If the speed $v(r)$ of gas coming out of the balloon depends on $r$ as $r^\alpha$ and $T \propto S^\alpha A^\beta \rho^\gamma R^\delta$ then

A physical quantity $P$ is related to four observations $a, b, c$ and $d$ as follows:

$$P=a^3 b^2 / c \sqrt{d}$$

The percentage errors of measurement in $a, b, c$ and $d$ are $1 \%, 3 \%, 2 \%$, and $4 \%$ respectively. The percentage error in the quantity $P$ is

Consider the diameter of a spherical object being measured with the help of a Vernier callipers. Suppose its 10 Vernier Scale Divisions (V.S.D.) are equal to its 9 Main Scale Divisions (M.S.D.). The least division in the M.S. is 0.1 cm and the zero of V.S. is at $x=0.1 \mathrm{~cm}$ when the jaws of Vernier callipers are closed. If the main scale reading for the diameter is $M=5 \mathrm{~cm}$ and the number of coinciding vernier division is 8 , the measured diameter after zero error correction, is

In an electrical circuit, the voltage is measured as $$V=(200 \pm 4)$$ volt and the current is measured as $$I=(20 \pm 0.2)$$ A. The value of the resistance is: