SATHEE: Functions 3 Question 9

Functions 3 Question 9

9. Let $F (x)$ be an indefinite integral of $\sin^{2} x$ .

Statement I The function $F (x)$ satisfies

$F (x + π) = F (x)$ for all real $x$ .

Because

Statement II $\sin^{2} (x + π) = \sin^{2} x$ , for all real $x$ .

$(2007, 3 M)$

Analytical & Descriptive Question

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Answer:

Correct Answer: 9. (d)

Solution:

Let $φ (x) = f (x) - g (x) = \begin{array}{ll} x, & x \in Q - x, & x \notin Q \end{array}$

Now, to check one-one.

Take any straight line parallel to $X$ -axis which will intersect $φ (x)$ only at one point.

$\Rightarrow φ (x)$ is one-one.

To check onto

As $f (x) = \begin{array}{cc} x, & x \in Q - x, & x \notin Q \end{array}$ , which shows

$y = x$ and $y = - x$ for rational and irrational values

$\Rightarrow y \in$ real numbers.

$∴ Range = Codomain \Rightarrow onto$

Thus, $f - g$ is one-one and onto.

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Functions 3 Question 9

9. Let F(x) be an indefinite integral of sin2⁡x.

Analytical & Descriptive Question

Answer:

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9. Let $F (x)$ be an indefinite integral of $\sin^{2} x$ .