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समुच्चय
$A -\left\{ x \in N : x ^2-10 x +9 \leq 0\right\}$ से समुच्चय
$B =\left\{ n ^2: n \in N \right\}$ में ऐसे फलनों $f$, जिनके लिए
$f ( x ) \leq( x -3)^2+1, x \in A$ है, की संख्या है $........$
$1440$
$1450$
$1460$
$1470$
Solution
$\left( x ^{2}-10 x +9\right) \leq 0 \Rightarrow( x -1)( x -9) \leq 0$
$x \in[1,9] \Rightarrow A =\{1,2,3,4,5,6,7,8,9\}$
$f ( x ) \leq( x -3)^{2}+1$
$x =1: f (1) \leq 5 \Rightarrow 1^{2}, 2^{2}$
$x =2: f (2) \leq 2 \Rightarrow 1^{2}$
$x =3: f (3) \leq 1 \Rightarrow 1^{2}$
$x =4: f (4) \leq 2 \Rightarrow 1^{2}$
$x =5: f (5) \leq 5 \Rightarrow 1^{2}, 2^{2}$
$x =6: f (6) \leq 10 \Rightarrow 1^{2}, 2^{2}, 3^{2}$
$x =7: f (7) \leq 17 \Rightarrow 1^{2}, 2^{2}, 3^{2}, 4^{2}$
$x =8: f (8) \leq 26 \Rightarrow 1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}$
$x =9: f (9) \leq 37 \Rightarrow 1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}$
Total number of such function
$=2(6 !)=2(720)=1440$
