$\frac{\tan 80^{\circ}-\tan 10^{\circ}}{\tan 70^{\circ}}$ is equal to

In the conversion of limestone to lime,$CaCO_{3(s)} \rightarrow CaO_{(s)} + CO_{2(g)}$,the values of $\Delta H^o$ and $\Delta S^o$ are $+179.1 \ kJ \ mol^{-1}$ and $160.2 \ J \ K^{-1} \ mol^{-1}$ respectively at $298 \ K$ and $1 \ bar$. Assuming that $\Delta H^o$ and $\Delta S^o$ do not change with temperature,the temperature above which the conversion of limestone to lime will be spontaneous is ............. $K$.

Three rods of same dimensions have thermal conductivities $3K$,$2K$,and $K$. They are arranged as shown in the figure. In the steady state,the temperature of the junction $P$ is

$A$ player tosses $2$ fair coins. He wins Rs. $5$ if $2$ heads appear,Rs. $2$ if $1$ head appears,and Rs. $1$ if no head appears. The variance of his winning amount is:

$A$ sound source is moving towards a stationary observer with $(1/10)$ of the speed of sound. The ratio of apparent to real frequency is .........

Which stage is indicated by the given figure?