Let $x, y>0$. If $x^{3} y^{2}=2^{15}$, then the least value of $3 x +2 y$ is

The number of different possible values for the sum $x+y+z$, where $x, y, z$ are real number such that $x^4+4 y^4+16 z^4+64=32 x y z$ 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

The ratio of the $A.M.$ and $G.M.$ of two positive numbers $a$ and $b,$ is $m: n .$ Show that $a: b=(m+\sqrt{m^{2}-n^{2}}):(m-\sqrt{m^{2}-n^{2}})$

If all the terms of an $A.P.$ are squared, then new series will be in

The sum of three consecutive terms in a geometric progression is $14$. If $1$ is added to the first and the second terms and $1$ is subtracted from the third, the resulting new terms are in arithmetic progression. Then the lowest of the original term is