What is the mole ratio of benzene $({P_B}^0 = 150 \ torr)$ and toluene $({P_T}^0 = 50 \ torr)$ in the vapour phase if the given solution has a vapour pressure of $120 \ torr$?

In a potentiometer, a wire of length $10 \, m$ having resistance $50 \, \Omega$ is used. $A$ battery of $5 \, V$ and a resistor of $450 \, \Omega$ are connected in series to the wire. If an unknown battery of emf $E$ balances the potentiometer at $450 \, cm$, then the value of $E$ is (in $V$)

$A$ machine gun fires a bullet of mass $40\, g$ with a velocity $1200\, m/s$. The man holding it can exert a maximum force of $144\, N$ on the gun. How many bullets can he fire per second at the most?

The spring balance $A$ reads $2 \ kg$ with a block $m$ suspended from it. $A$ balance $B$ reads $5 \ kg$ when a beaker with liquid is placed on the pan of the balance. The two balances are now arranged such that the hanging mass is inside the liquid in the beaker as shown in the figure. In this situation:

If $x^y=y^x$, then $x(x-y \log x) \frac{d y}{d x}$ is equal to :

If $z = \frac{y}{x} \left[ \sin \frac{x}{y} + \cos \left( 1 + \frac{y}{x} \right) \right]$,then $x \frac{\partial z}{\partial x}$ is equal to