Functions 3 Question 9
9. Let $F(x)$ be an indefinite integral of $\sin ^{2} x$.
Statement I The function $F(x)$ satisfies
$F(x+\pi)=F(x)$ for all real $x$.
Because
Statement II $\sin ^{2}(x+\pi)=\sin ^{2} x$, for all real $x$.
$(2007,3 M)$
Analytical & Descriptive Question
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Answer:
Correct Answer: 9. (d)
Solution:
- Let $\varphi(x)=f(x)-g(x)=\begin{array}{ll}x, & x \in Q \ -x, & x \notin Q\end{array}$
Now, to check one-one.
Take any straight line parallel to $X$-axis which will intersect $\varphi(x)$ only at one point.
$\Rightarrow \varphi(x)$ is one-one.
To check onto
As $\quad f(x)=\begin{array}{cc}x, & x \in Q \ -x, & x \notin Q\end{array}$, which shows
$y=x$ and $y=-x$ for rational and irrational values
$\Rightarrow y \in$ real numbers.
$$ \therefore \quad \text { Range }=\text { Codomain } \Rightarrow \text { onto } $$
Thus, $f-g$ is one-one and onto.
