The common difference of an $A.P.$ whose first term is unity and whose second, tenth and thirty fourth terms are in $G.P.$, is

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2$, then the value of $\frac{a^2+a-14}{a+1}$ is

${2^{\sin \theta }} + {2^{\cos \theta }}$ is greater than

Three non-zero real numbers form an $A.P.$ and the squares of these numbers taken in the same order form a $G.P.$ Then the number of all possible common ratios of the $G.P.$ is

If  $f(x) = \sqrt {{x^2} + x}  + \frac{{{{\tan }^2}\alpha }}{{\sqrt {{x^2} + x} }},\alpha  \in (0,\pi /2),x > 0$ then value of $f(x)$ is greater than or equal to-

If $a,\;b,\;c$ are in $G.P.$ and $\log a - \log 2b,\;\log 2b - \log 3c$ and $\log 3c - \log a$ are in $A.P.$, then $a,\;b,\;c$ are the length of the sides of a triangle which is