The number of one-one functions $f : \{a, b, c, d\} \rightarrow \{0, 1, 2, \dots, 10\}$ such that $2f(a) - f(b) + 3f(c) + f(d) = 0$ is

The function $f: R \rightarrow R$ defined by $f(x) = e^x + e^{-x}$ is

If the function $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} 2x-3, & \text{if } x < -2 \\ x^2-1, & \text{if } -2 \leq x \leq 2 \\ 3x+2, & \text{if } x > 2 \end{cases}$ then $f$ is

If $f: N \times N \rightarrow N$ defined by $f(m, n) = mn$,then $f$ is . . . . . . .

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

If the set $x$ contains $7$ elements and set $y$ contains $8$ elements,then the number of bijections from $x$ to $y$ is