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QJEEM22Ma8d6f: JEE Main 2022 Mathematics - Class 11 JEEM22Ma8d6f Solution: The number of functions , from the set to the set such that ,…

The number of functions $f$ , from the set $\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}$ to the set $\mathrm{B}=\left\{\mathrm{n}^{2}: \mathrm{n} \in \mathbf{N}\right\}$ such that $f(x) \leq(x-3)^{2}+1$ , for every $x \in \mathrm{A}$ , is ___________.

Step-by-Step Verified Solution

$A = {\begin{matrix} x \in N, & x^{2} - 10 x + 9 \leq 0 \end{matrix}}$

$= \{ 1,2,3,\,....,\,9\}$

$B = \{ 1,4,9,16,\,.....\}$

$f(x) \le {(x - 3)^2} + 1$

$f(1) \le 5,\,f(2) \le 2,\,\,..........\,f(9) \le 37$

$x = 1$ has 2 choices

$x = 2$ has 1 choice

$x = 3$ has 1 choice

$x = 4$ has 1 choice

$x = 5$ has 2 choices

$x = 6$ has 3 choices

$x = 7$ has 4 choices

$x = 8$ has 5 choices

$x = 9$ has 6 choices

$\therefore$ Total functions = $2\times1\times1\times1\times2\times3\times4\times5\times6=1440$

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