Let $A = \{ {x_1},\,{x_2},\,............,{x_7}\} $ and $B = \{ {y_1},\,{y_2},\,{y_3}\} $ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f : A \to B$ that are onto, if there exist exactly three elements $x$ in $A$ such that $f(x)\, = y_2$, is equal to

Let $f(x)=2 x^{2}-x-1$ and $S =\{n \in Z :|f(n)| \leq 800\}$ . Then value of $\sum_{n \in S} f(n)$ is . . . .  .

Let $f(x)$ be a non-constant polynomial with real coefficients such that $f\left(\frac{1}{2}\right)=100$ and $f(x) \leq 100$ for all real $x$. Which of the following statements is NOT necessarily true?

If the domain of the function $f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$ is $[\alpha, \beta) \cup(\gamma, \delta]$, then $|3 \alpha+10(\beta+\gamma)+21 \delta|$ is equal to $.......$.

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P ( x )=f\left( x ^{3}\right)+ xg \left( x ^{3}\right)$ is divisible by $x^{2}+x+1,$ then $P(1)$ is equal to ....... .