If $a,\,b,\,c$ are three unequal numbers such that $a,\,b,\,c$ are in $A.P.$ and $b -a, c -b, a$ are in $G.P.$, then $a : b : c$ is

If the product of three terms of $G.P.$ is $512$. If $8$ added to first and $6$ added to second term, so that number may be in $A.P.$, then the numbers are

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, the new numbers are in $A.P.$ then the common ratio of the $G.P.$ is:

If ${A_1},\;{A_2};{G_1},\;{G_2}$ and ${H_1},\;{H_2}$ be $AM's,\;GM's$ and $HM's$ between two quantities, then the value of $\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}}$ 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

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