If the $A.M.$ and $H.M.$ of two numbers is $27$ and $12$ respectively, then $G.M.$ of the two numbers will be

The sum of three numbers in $G.P.$ is $56.$ If we subtract $1,7,21$ from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

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

Consider an arithmetic series and a geometric series having four initial terms from the set $\{11,8,21,16,26,32,4\}$ If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to .......

Let $a_{1}, a_{2}, \ldots, a_{10}$ be an $AP$ with common difference $-3$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{10}$ be a $GP$ with common ratio $2.$ Let $c_{k}=a_{k}+b_{k}, k=1,2, \ldots, 10 .$ If $c_{2}=12$ and $\mathrm{c}_{3}=13$, then $\sum_{\mathrm{k}=1}^{10} \mathrm{c}_{\mathrm{k}}$ is equal to ..... .

If $a$ be the arithmetic mean of $b$ and $c$ and ${G_1},\;{G_2}$ be the two geometric means between them, then $G_1^3 + G_2^3 = $