$A$ particle moves with a velocity $\vec{v} = 5\hat{i} - 3\hat{j} + 6\hat{k} \text{ m/s}$ under the influence of a constant force $\vec{F} = 10\hat{i} + 10\hat{j} + 20\hat{k} \text{ N}$. The instantaneous power will be ............... $\text{J/s}$.

If $u=f(r)$,where $r^2=x^2+y^2$,then $\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)$ is equal to

Benzene and toluene form an ideal solution over the entire range of composition. The vapour pressure of pure benzene and toluene at $T \ K$ are $50 \ mm \ Hg$ and $40 \ mm \ Hg$ respectively. What is the mole fraction of toluene in vapour phase when $117 \ g$ of benzene is mixed with $46 \ g$ of toluene? (molar mass of benzene and toluene are $78 \ g \ mol^{-1}$ and $92 \ g \ mol^{-1}$ respectively)

The figure shows three long straight wires $P, Q$ and $R$ carrying currents normal to the plane of the paper. All three currents have the same magnitude. Which arrow best shows the direction of the resultant force on the wire $P$?

Two tuning forks $A$ and $B$ produce $8 \, Hz$ beats when sounded together. $A$ gas column $37.5 \, cm$ long in a pipe closed at one end resonates in its fundamental mode with fork $A$,whereas a column of length $38.5 \, cm$ of the same gas in a similar pipe is required for resonance with fork $B$. The frequencies of these two tuning forks are:

$A$ hoop of radius $r$ and mass $m$ rotating with an angular velocity $\omega_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?