The fossil bone has a ${}^{14}C:{}^{12}C$ ratio,which is $\left( \frac{1}{16} \right)$ of that in a living animal bone. If the half-life of ${}^{14}C$ is $5730 \text{ years}$,then the age of the fossil bone is ......... $\text{years}$.

If $x=9$ is a chord of contact of the hyperbola $x^2-y^2=9$,then the equation of the tangent at one of the points of contact is

Two blocks of masses $10\,kg$ and $4\,kg$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of $14\,m/s$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is ......... $m/s$.

Wires $A$ and $B$ have resistivities $\rho_A$ and $\rho_B$,where $\rho_B = 2 \rho_A$,and have lengths $l_A$ and $l_B$. If the diameter of wire $B$ is twice that of $A$ and the two wires have the same resistance,then the ratio $\frac{l_B}{l_A}$ is:

$A$ particle is released from height $S$ from the surface of the Earth. At a certain height its kinetic energy is three times its potential energy. The height from the surface of the Earth and the speed of the particle at that instant are respectively:

$A$ large tank filled with water to a height $h$ is to be emptied through a small hole at the bottom. The ratio of the time taken for the level of water to fall from $h$ to $\frac{h}{2}$ and from $\frac{h}{2}$ to zero is: