The component of vector $\vec{A} = 2\hat{i} + 3\hat{j}$ along the vector $\vec{B} = \hat{i} + \hat{j}$ is:

The path difference between the two waves $y_1 = a_1 \sin \left( \omega t - \frac{2\pi x}{\lambda} \right)$ and $y_2 = a_2 \cos \left( \omega t - \frac{2\pi x}{\lambda} + \phi \right)$ is

An ideal gas is taken through the cycle $A \to B \to C \to A$,as shown in the figure. If the net heat supplied to the gas in the cycle is $5 \ J$,the work done by the gas in the process $C \to A$ is .... $J$.

The densities of a liquid at $0^{\circ} C$ and $100^{\circ} C$ are respectively $1.0127$ and $1$. $A$ specific gravity bottle is filled with $300 \ g$ of the liquid at $0^{\circ} C$ up to the brim and it is heated to $100^{\circ} C$. Then,the mass of the liquid expelled in grams is: (Coefficient of linear expansion of glass $= 9 \times 10^{-6} /^{\circ} C$)

$A$ ball $A$ of mass $m$ moving along the positive $x$-direction with kinetic energy $K$ and momentum $p$ undergoes an elastic head-on collision with a stationary ball $B$ of mass $M$. After the collision,ball $A$ moves along the negative $x$-direction with kinetic energy $K/9$. The final momentum of ball $B$ is: