The length of the common chord of the circles of radii $15$ and $20$,whose centers are $25$ units of distance apart,is

$A$ transverse wave is passing through a stretched string with a speed of $20 \ m/s$. The tension in the string is $20 \ N$. At a certain point $P$ on the string,it is observed that energy is being transferred at a rate of $40 \ mW$ at a given instant. Find the speed of point $P$.

$138 \ g$ of ethyl alcohol is mixed with $72 \ g$ of water. The ratio of mole fraction of alcohol to water is

There is some change in length when a $33000 ~N$ tensile force is applied on a steel rod of area of cross-section $10^{-3} ~m^2$. The change of temperature required to produce the same elongation,if the steel rod is heated,is (The modulus of elasticity is $3 \times 10^{11} ~N/m^2$ and the coefficient of linear expansion of steel is $1.1 \times 10^{-5} /{ }^{\circ} C$). (in $^{\circ} C$)

Consider a reaction $A+R \rightarrow$ Product. The rate of this reaction is measured to be $k[A][R]$. At the start of the reaction, the concentration of $R,[R]_0$, is $10-$times the concentration of $A,[A]_0$. The reaction can be considered to be a pseudo first order reaction with assumption that $k[R]=k^{\prime}$ is constant. Due to this assumption, the relative error $(in \%)$ in the rate when this reaction is $40 \%$ complete, is $....$
$[ k$ and $k^{\prime}$ represent corresponding rate constants$]$

The number of electrons in the valence shell of the central atom of a molecule is $8$. The molecule is