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1.Relation and Function
hard
Let, $f(x)=\left\{\begin{array}{l} x \sin \left(\frac{1}{x}\right) \text { when } x \neq 0 \\ 1 \text { when } x=0 \end{array}\right\}$ and $A=\{x \in R: f(x)=1\} .$ Then, $A$ has
A
exactly one element
B
exactly two elements
C
exactly three elements
D
infinitely many elements
(KVPY-2019)
Solution
(a)
Given function
$f(x)=\left[\begin{array}{cl} x \sin \left(\frac{1}{x}\right) & , \text { when } x \neq 0 \\ 1 & , \text { when } x=0 \end{array}\right.$
Now, for $x=0, f(x)=1$ and for $x \neq 0, f(x)=1 \Rightarrow \sin \frac{1}{x}=\frac{1}{x}$ has no solution.
$\therefore$ The $\operatorname{set} A=\{x \in R: f(x)=1\}$ has exactly one element.
Standard 12
Mathematics
