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

Young's modulus is determined by the equation $Y = 49000 \frac{M}{\ell} \text{ dyne/cm}^2$,where $M$ is the mass and $\ell$ is the extension of the wire used in the experiment. The error in Young's modulus $(Y)$ is estimated by taking data from the $M-\ell$ plot on graph paper. The smallest scale divisions are $5 \text{ g}$ and $0.02 \text{ cm}$ along the load axis and extension axis,respectively. If the values of $M$ and $\ell$ are $500 \text{ g}$ and $2 \text{ cm}$ respectively,then the percentage error of $Y$ is: (in $\%$)

$A$ student determined Young's Modulus of elasticity using the formula $Y = \frac{M g L^{3}}{4 b d^{3} \delta}$. The value of $g$ is taken to be $9.8 \, m/s^2$, without any significant error. His observations are as follows:

Physical Quantity	Least count and Observed value
Mass $(M)$	$1 \, g$ and $2 \, kg$
Length of bar $(L)$	$1 \, mm$ and $1 \, m$
Breadth of bar $(b)$	$0.1 \, mm$ and $4 \, cm$
Thickness of bar $(d)$	$0.01 \, mm$ and $0.4 \, cm$
Depression $(\delta)$	$0.01 \, mm$ and $5 \, mm$

Then the fractional error in the measurement of $Y$ is:

$A$ set of defective observation weights is used by a student to find the mass of an object using a physical balance. $A$ large number of readings will reduce:

$A$ particle covers a distance of $(13.8 \pm 0.2) \ m$ in $(4 \pm 0.3) \ s$. Its velocity under error limits will be

The relative density of the material of a body is found by weighing it first in air and then in water. If the weight in air is $(5.00 \pm 0.05) \ N$ and the weight in water is $(4.00 \pm 0.05) \ N$,then the relative density along with the maximum permissible percentage error is: