If the sum of two extreme numbers of an $A.P.$ with four terms is $8$ and product of remaining two middle term is $15$, then greatest number of the series will be

If $2x,\;x + 8,\;3x + 1$ are in $A.P.$, then the value of $x$ will be

For $p, q \in R$, consider the real valued function $f ( x )=( x - p )^{2}- q , x \in R$ and $q >0$. Let $a _{1}, a _{2}, a _{3}$ and $a _{4}$ be in an arithmetic progression with mean $P$ and positive common difference. If $\left| f \left( a _{ i }\right)\right|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f ( x )=0$ is.

If $a,b,c$ are in $A.P.$, then $\frac{1}{{\sqrt a + \sqrt b }},\,\frac{1}{{\sqrt a + \sqrt c }},$ $\frac{1}{{\sqrt b + \sqrt c }}$ are in

Three numbers are in $A.P.$ whose sum is $33$ and product is $792$, then the smallest number from these numbers is