If $f(x + y,x - y) = xy\,,$ then the arithmetic mean of $f(x,y)$ and $f(y,x)$ is

If the sum of $n$ terms of an $A.P.$ is $nA + {n^2}B$, where $A,B$ are constants, then its common difference will be

Let ${\left( {1 - 2x + 3{x^2}} \right)^{10x}}  = {a_0} + {a_1}x + {a_2}{x^2} + .....+{a_n}{x^n},{a_n} \ne 0$, then the arithmetic mean of $a_0,a_1,a_2,...a_n$ is

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 the sum of the series $2 + 5 + 8 + 11............$ is $60100$, then the number of terms are

Let the sum of the first $n$ terms of a non-constant $A.P., a_1, a_2, a_3, ……$ be $50\,n\, + \,\frac{{n\,(n\, - 7)}}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.,$ then the ordered pair $(d,a_{50})$ is equal to