If the curves $x^2 - 6x + y^2 + 8 = 0$ and $x^2 - 8y + y^2 + 16 - k = 0$ $(k > 0)$ touch each other at a point,then the largest value of $k$ is

The floral formula (image) is that of

$A$ conveyor belt is moving at a constant speed of $2\, m/s$. $A$ box is gently dropped on it. The coefficient of friction between them is $\mu = 0.5$. The distance that the box will move relative to the belt before coming to rest on it,taking $g = 10\, m/s^2$,is ........ $m$.

In an experiment with $15$ observations of $x$,the results $\sum x^2 = 2830$ and $\sum x = 170$ are obtained. If one observation $20$ is found to be incorrect and is replaced by the correct observation $30$,what is the correct variance?

The number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{5})^{256}$ is

Three moles of an ideal monoatomic gas undergo a cyclic process as shown in the figure. The temperatures of the gas in different states marked as $1, 2, 3$ and $4$ are $400 \ K, 700 \ K, 2500 \ K$ and $1100 \ K$ respectively. The work done by the gas during the process $1-2-3-4-1$ is (universal gas constant is $R$): (in $R$)