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1.Relation and Function
hard
જો $S=\{1,2,3,4,5,6,7\} $ આપેલ છે. વિધેય $f:S \rightarrow S$ કેટલા શક્ય બને કે જેથી દરેક $m, n \in S$ માટે $f(m \cdot n)=f(m) \cdot f(n)$ અને $m . n \in S$ થાય.
A
$500$
B
$600$
C
$570$
D
$490$
(JEE MAIN-2021)
Solution
$f(m n)=f(m) \cdot f(n)$
Put $m=1 f(n)=f(1) \cdot f(n) \Rightarrow f(1)=1$
Put $m=n=2$
$f(4)=f(2) \cdot f(2)$
$f(2)=1 \Rightarrow f(4)=1 \text { or } f(2)=2 \Rightarrow f(4)=4$
Put $m=2, n=3$
$f(6)=f(2) \cdot f(3)$
$\text { when } f(2)=1 \Rightarrow f(3)=1 \text { to } 7$
$f(2)=2 \Rightarrow f(3)=1 \text { or } 2 \text { or } 3$
$f(5), f(7)$ can take any value
Total $=(1 \times 1 \times 7 \times 1 \times 7 \times 1 \times 7)$
$+(1 \times 1 \times 3 \times 1 \times 7 \times 1 \times 7)$
$=490$
Standard 12
Mathematics
