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

Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A $\to$ A such that f(1) + f(2) = 3 $-$ f(3) is equal to

Let A = {2, 3, 4, 5, ....., 30} and '$\simeq$' be an equivalence relation on A $\times$ A, defined by (a, b) $\simeq$ (c, d), if and only…

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f: A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m,…

Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A $\to$ B | 2 $\in$ f(A) and f is not one-one} is…

Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $\mathrm{R}=\{(\mathrm{A}, \mathrm{B}):…