If $G.M. = 18$ and $A.M. = 27$, then $H.M.$ is

Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$, terms of a non-constant $A.P.$ be respectively the $1^{\text {st }}, 2^{\text {nd }}$ and $3^{\text {rd }}$ terms of $G.P.$ If the first term of $A.P.$ is $1$ then the sum of first $20$ terms is equal to-

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 $a, b$ and $c$ be the $7^{th},\,11^{th}$ and $13^{th}$ terms respectively of a non -constant $A.P.$ If these are also the three consecutive terms of a $G.P.$ then $\frac {a}{c}$ is equal to

If $a,\,b,\;c$ are in $A.P.$ and ${a^2},\;{b^2},\;{c^2}$ are in $H.P.$, then

Consider two positive numbers $a$ and $b$ . If arithmetic mean of $a$ and $b$ exceeds their geometric mean by $\frac{3}{2}$ and geometric mean of $a$ and $b$ exceeds their harmonic mean by $\frac{6}{5}$ , then the absolute value of $(a^2 -b^2)$ is equal to