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Assertion :The value of $$\int_{0}^{2 \pi} \cos ^{99} x d x$$ is 0. Reason: $$\int_{0}^{2 a} f(x) d x=2 \int_{0}^{a} f(x) d x,$$ if $$f(2 a-x)=f(x)$$


A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$$\operatorname{Let} I=\int_{0}^{2 \pi} \cos ^{99} x d x\\$$
Then, $$I=2 \int_{0}^{\pi} \cos ^{99} x d x \quad\left[\because \cos ^{99}(2 \pi-x)=\cos ^{99} x\right]\\$$
Now, $$\int_{0}^{\pi} \cos ^{99} x d x=0 \quad\left[\because \cos ^{99}(\pi-x)=-\cos ^{99} x\right]\\$$
$$\Rightarrow I=2 \times 0=0$$

Mathematics

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