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-

Let $\frac{1}{16}, a$ and $b$ be in $G.P.$ and $\frac{1}{ a }, \frac{1}{ b }, 6$ be in $A.P.,$ where $a , b >0$. Then $72( a + b )$ is equal to ...... .

Let $a, b, c$ be the sides of a triangle. If $t$ denotes the expression $\frac{\left(a^2+b^2+c^2\right)}{(a b+b c+c a)}$, the set of all possible values of $t$ is

In a $G.P.$ the sum of three numbers is $14$, if $1 $ is added to first two numbers and subtracted from third number, the series becomes $A.P.$, then the greatest number is

If the first and ${(2n - 1)^{th}}$ terms of an $A.P., G.P.$ and $H.P.$ are equal and their ${n^{th}}$ terms are respectively $a,\;b$ and $c$, then

Let three real numbers $a, b, c$ be in arithmetic progression and $\mathrm{a}+1, \mathrm{~b}, \mathrm{c}+3$ be in geometric progression. If $\mathrm{a}>10$ and the arithmetic mean of $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ is $8$ , then the cube of the geometric mean of $a, b$ and $c$ is