$6.023 \times 10^{22}$ molecules are present in $10 \ g$ of a substance $x$. The molarity of a solution containing $5 \ g$ of substance $x$ in $2 \ L$ of solution is $\dots \dots \dots \times 10^{-3} \ M$.

If $X$ is a Poisson variate such that $P(X=1)=2 P(X=2)$, then $P(X=3)$ is equal to

If $4$ moles of an ideal monoatomic gas at temperature $400 ~K$ is mixed with $2$ moles of another ideal monoatomic gas at temperature $700 ~K$,the temperature of the mixture is

If $\tan A$ and $\tan B$ are the roots of the quadratic equation $x^2-px+q=0$,then $\sin^2(A+B)$ is equal to

Two identical capacitors have the same capacitance $C$. One of them is charged to a potential $V_1$ and the other to $V_2$. The negative ends of the capacitors are connected together. When the positive ends are also connected,the decrease in energy of the combined system is

$A$ person running horizontally observes that rain is falling on his head vertically with a speed of $10\,m/s$. When he stops,he observes that the rain is falling at an angle of $30^{\circ}$ with the vertical. Find the speed of the rain with respect to the ground.