JEE Advanced 2018 — Paper 1 — Question 44

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in

any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first

power of the error. For example, consider the relation $z=x / y$. If the errors in $x, y$ and $z$ are $\Delta x, \Delta y$ and $\Delta z$,

respectively, then

The series expansion for $\left(1 \pm \frac{\Delta y}{y}\right)^{-1}$, to first power in $\Delta y / y$, is $1 \mp(\Delta \mathrm{y} / y)$. The relative errors in

independent variables are always added. So the error in $z$ will be

The above derivation makes the assumption that $\Delta x / x \ll 1, \Delta y / y \ll 1$. Therefore, the higher powers of

these quantities are neglected.

In an experiment the initial number of radioactive nuclei is 3000. It is found that $1000 \pm 40$ nuclei decayed in the first 1.0s. For $|x| \ll 1$, $\ln(1+x) \approx x$ up to first power in $x$. The error $\Delta\lambda$, in the determination of the decay constant $\lambda$, in s$^{-1}$, is