The area (in square unit) of the region enclosed by the curves $y=x^2$ and $y=x^3$ is

$A$ gramophone record is revolving with an angular velocity $\omega$. $A$ coin is placed at a distance $R$ from the centre of the record. The static coefficient of friction is $\mu$. The coin will revolve with the record if

The human body does not produce

$a^{m \log_a n} = $

An archaeologist analyses the wood in a prehistoric structure and finds that the ratio of ${C^{14}}$ (half-life $= 5700 \, years$) to ${C^{12}}$ is only one-fourth of that found in the cells of living plants. The age of the wood is about ........ $years$.

$A$ variable circle passes through the fixed point $A(p, q)$ and touches the $x$-axis. The locus of the other end of the diameter through $A$ is