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Question

# Assertion :The area bounded by f(x)=1x2−2x+2 and x-axis is π square units. Reason: The function f(x)=1x2−2x+2 is continuous ∀xϵR and attains maximum value at x=1, which is also the line of symmetry for f(x). Therefore the area is given by 2∫∞1f(x)dx

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
Both Assertion and Reason are incorrect
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Solution

## The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertionf(x)=1x2−2x+2=1(x−1)2+1 which is define for all values of xϵR So f(x) is continuous ∀xϵR and f(x) attains its maximum value for (x−1)2=0 i. e. x=1Again f(x)=1x2−2x+2 is symmetrical about the line x=1 as it satisfies f(1+x)=f(1−x)∴ Required area =2∫∞1f(x)dx=2∫∞11(x−1)2+12=2[tan−1(x−1)]∞1=2tan−1∞−tan−10=2×π2=π

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