If $x, y, z \in R^+$ such that $x + y + z = 4$, then maximum possible value of $xyz^2$ is -

Let $x, y, z$ be three non-negative integers such that $x+y+z=10$. The maximum possible value of $x y z+x y+y z+z x$ is

If $a_1, a_2...,a_n$ an are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_1 + a_2 +.... + a_{n - 1} + 2a_n$ is

The common difference of an $A.P.$ whose first term is unity and whose second, tenth and thirty fourth terms are in $G.P.$, is

In the four numbers first three are in $G.P.$ and last three are in $A.P.$ whose common difference is $6$. If the first and last numbers are same, then first will be

Let $m$ be the minimum possible value of $\log _3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$, where $y _1, y _2, y _3$ are real numbers for which $y _1+ y _2+ y _3=9$. Let $M$ be the maximum possible value of $\left(\log _3 x _1+\log _3 x _2+\log _3 x _3\right)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of $\log _2\left(m^3\right)+\log _3\left(M^2\right)$ is. . . . . .