The vectors $\vec{AB} = 3\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{BC} = \hat{i} - 2\hat{k}$ are the adjacent sides of a parallelogram. The angle between its diagonals is

$A$ 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

The enzyme enterokinase helps in the conversion of $............$.

Every planet revolves around the sun in an elliptical orbit. Consider the following statements:
$A.$ The gravitational force acting on a planet is inversely proportional to the square of the distance from the sun.
$B.$ The gravitational force acting on a planet is inversely proportional to the product of the masses of the planet and the sun.
$C.$ The centripetal force acting on the planet is directed away from the sun.
$D.$ The square of the time period of revolution of a planet around the sun is directly proportional to the cube of the semi-major axis of the elliptical orbit.
Choose the correct statements from the options given above:

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 ......... $\text{years}$.

Two cylinders $A$ and $B$ fitted with pistons contain equal number of moles of an ideal monoatomic gas at $400 ~K$. The piston of $A$ is free to move while that of $B$ is held fixed. Same amount of heat energy is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $42 ~K$,the rise in temperature of the gas in $B$ is (in $~K$)