The percentage error in the measurement of a physical quantity $m$ given by $m = \pi \tan \theta$ is minimum when $\theta = \dots \dots \dots \dots \dots ^\circ$ (Assume that the error in $\theta$ remains constant).

The energy of a system as a function of time $t$ is given as $E(t)=A^2 \exp(-\alpha t)$,where $\alpha=0.2 \ s^{-1}$. The measurement of $A$ has an error of $1.25 \%$. If the error in the measurement of time is $1.50 \%$,the percentage error in the value of $E(t)$ at $t=5 \ s$ is

The percentage errors in quantities $P, Q, R$ and $S$ are $0.5\%, 1\%, 3\%$ and $1.5\%$ respectively in the measurement of a physical quantity $A = \frac{P^3 Q^2}{\sqrt{R} S}$. The maximum percentage error in the value of $A$ will be ........... $\%$.

$A$ physical quantity $P$ is related to four observations $a, b, c$ and $d$ as follows: $P = \frac{a^3 b^2}{c \sqrt{d}}$. 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: (in $\%$)

The distance $s$ travelled by a particle in time $t$ is $s = ut - \frac{1}{2} gt^2$. The initial velocity of the particle was measured to be $u = 1.11 \pm 0.01 \, m/s$ and the time interval of the experiment was $t = 1.01 \pm 0.1 \, s$. The acceleration was taken to be $g = 9.8 \pm 0.1 \, m/s^2$. With these measurements,the student estimates the total distance travelled. How should the student report the result?

In an experiment,the values of refractive indices of glass were found to be $1.54, 1.53, 1.44, 1.54, 1.56$ and $1.45$ in successive measurements. The mean absolute error is: