$A$ manometer connected to a closed tap reads $4.5 \times 10^5 \, Pa$. When the tap is opened,the reading of the manometer falls to $4 \times 10^5 \, Pa$. The velocity of the flow of water is ........ $m/s$.

The area (in square units) of the region bounded by the curve $y=|\sin 2x|$ and the $X$-axis in the interval $[0, 2\pi]$ is:

If $\theta$ is the angle between the curves $xy=2$ and $x^2+4y=0$, then $\tan \theta$ is equal to:

The internal energy change in a system that has absorbed $2 \, kcal$ of heat and done $500 \, J$ of work is ........... $J$.

Let $\vec{V} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{W} = \hat{i} + 3\hat{k}$. If $\vec{U}$ is a unit vector,then the maximum value of the scalar triple product $[\vec{U} \, \vec{V} \, \vec{W}]$ is:

At a given instant of time,two particles have position vectors $4 \hat{i} + 4 \hat{j} + 57 \hat{k} \text{ m}$ and $2 \hat{i} + 2 \hat{j} + 5 \hat{k} \text{ m}$ respectively. If the velocity of the first particle is $0.4 \hat{i} \text{ ms}^{-1}$,what is the velocity of the second particle in $\text{ms}^{-1}$ if they collide after $10 \text{ s}$?