For $p, q \in R$,consider the real-valued function $f(x) = (x - p)^2 - q$,where $x \in R$ and $q > 0$. Let $a_1, a_2, a_3, a_4$ be in an arithmetic progression with mean $p$ and a positive common difference $d$. If $|f(a_i)| = 500$ for all $i = 1, 2, 3, 4$,then the absolute difference between the roots of $f(x) = 0$ is:

Consider two sets $A$ and $B$,each containing three numbers in $A.P.$ Let the sum and the product of the elements of $A$ be $36$ and $p$ respectively,and the sum and the product of the elements of $B$ be $36$ and $q$ respectively. Let $d$ and $D$ be the common differences of the $A.P.s$ in $A$ and $B$ respectively such that $D = d + 3$ and $d > 0$. If $\frac{p + q}{p - q} = \frac{19}{5}$,then $p - q$ is equal to:

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is the square of an integer. The least possible value of the number of digits of $c$ is

If the sum of the first $2n$ terms of $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of $57, 59, 61, \dots$,then $n$ is equal to

If the arithmetic mean between the $p^{th}$ term and $q^{th}$ term of an arithmetic progression is equal to the arithmetic mean between its $r^{th}$ and $s^{th}$ term,then $p + q = ......$

Let $S = \{(a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N} : a+b+c=21, a \leq b \leq c\}$ and $T = \{(a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N} : a, b, c \text{ are in } AP\}$,where $\mathbb{N}$ is the set of all natural numbers. Then,the number of elements in the set $S \cap T$ is: