Let ${a_1},\;{a_2},.........{a_{10}}$ be in $A.P.$ and ${h_1},\;{h_2},........{h_{10}}$ be in $H.P.$ If ${a_1} = {h_1} = 2$ and ${a_{10}} = {h_{10}} = 3$, then  ${a_4}{h_7}$ is

Let $f: R \rightarrow R$ be such that for all $\mathrm{x} \in \mathrm{R}\left(2^{1+\mathrm{x}}+2^{1-\mathrm{x}}\right), f(\mathrm{x})$ and $\left(3 ^\mathrm{x}+3^{-\mathrm{x}}\right)$ are in $A.P.$, then the minimum value of $f(x)$ is

If $p,q,r$ are in $G.P$ and ${\tan ^{ - 1}}p$, ${\tan ^{ - 1}}q,{\tan ^{ - 1}}r$ are in $A.P.$ then $p, q, r$ are satisfies the relation

Let $a, b$ and $c$ be in $G.P$ with common ratio $r,$ where $a \ne 0$ and $0\, < \,r\, \le \,\frac{1}{2}$.  If $3a, 7b$ and $15c$ are the first three terms of an $A.P.,$ then the $4^{th}$ term of this $A.P$ is

If $x\in (0,\frac{\pi}{4})$ then the expression $ \frac{cos x}{sin^2 x(cos x-sin x)}$ can not take the value

If the $A.M., G.M.$ and $H.M.$ between two positive numbers $a$ and $b$ are equal, then