In an experiment of determine the Young's modulus of wire of a length exactly $1\; m$, the extension in the length of the wire is measured as $0.4\,mm$ with an uncertainty of $\pm 0.02\,mm$ when a load of $1\,kg$ is applied. The diameter of the wire is measured as $0.4\,mm$ with an uncertainty of $\pm 0.01\,mm$. The error in the measurement of Young's modulus $(\Delta Y)$ is found to be $x \times 10^{10}\,Nm ^{-2}$. The value of $x$ is

$\left[\right.$ Take $\left.g =10\,m / s ^{2}\right]$

If $P = \frac{{{A^3}}}{{{B^{5/2}}}}$ and $\Delta A$ is absolute error in $A$ and $\Delta B$ is absolute error in $B$ then absolute error $\Delta P$ in $P$ is

The random error in the arithmetic mean of $100$ observations is $x$; then random error in the arithmetic mean of $400$ observations would be

A physical quantity $P$ is related to four observables $a, b, c$ and $d$ as follows: $P=\frac{a^{2} b^{2}}{(\sqrt{c} d)}$ The percentage errors of measurement in $a, b, c$ and $d$ are $1 \%, 3 \%, 4 \%$ and $2 \%$ respectively. What is the percentage error in the quantity $P$ ? If the value of $P$ calculated using the above relation turns out to be $3.763,$ to what value should you round off the result?

If $Z=\frac{A^{2} B^{3}}{C^{4}}$, then the relative error in $Z$ will be

In a simple pendulum experiment, the maximum percentage error in the measurement of length is $2\%$ and that in acceleration due to gravity $g$ is $4\%$. Then the maximum percentage error in determination of the time-period is