Assertion :If α,β are the roots of the equation 18(tan−1x)2−9πtan−1x+π2=0 then α+β=4√3. Reason: sec2(cos−1(14))+cosec2(sin−1(15))=41
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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion Reason: sec2(cos−114)+csc2(sin−115)=sec2(sec−14)+csc2(csc−15)=16+25=41 Assertion: 18(tan−1x)2−9πtan−1x+π2=0
⇒(tan−1x)2−π2tan−1x+π218=0
⇒(tan−1x)2−(π6+π3)tan−1x+(π6×π3)=0
(tan−1(x)−π6)(tan−1(x)−π3)=0
Given α,β are the roots⇒tan−1α=π6,tan−1β=π3
α=tanπ6=√3,β=tanπ3=1√3
∴α+β=tanπ6+tanπ3=1√3+√3=4√3
Hence,both Assertion and Reason are correct but Reason is not the correct explanation for Assertion