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1.Relation and Function
hard
Let $f, g: N -\{1\} \rightarrow N$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^{\alpha}$ divides $a$, and $g(a)=a+1$, for all $a \in N -\{1\}$. Then, the function $f+ g$ is.
A
one-one but not onto
B
onto but not one-one
C
both one-one and onto
D
neither one-one nor onto
(JEE MAIN-2022)
Solution
$f: N-\{1\} \rightarrow N \quad f(a)=\alpha$
Where $\alpha$ is max of powers of prime $P$ such that $p ^{\alpha}$ divides a. Also $g ( a )= a +1$
$f(2)=1$ $g(2)=3$
$f(3)=1$ $g(3)=4$
$f(4)=2$ $g(4)=5$
$f(5)=1$ $g(5)=6$
$f(2)+g(2)=4$
$(f(3)+g(3))=5$
$f(4)+g(4)=7$
$f(5)+g(5)=7$
$\therefore$ Many one $f(x)+g(x)$ does not cotain 1
into function
Ans.$(D)$ [neither one-one nor onto]
Standard 12
Mathematics
