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If sin⁻¹(x-1)+cos⁻¹(x-3)+tan⁻¹ ((x)/(2-x²))=cos⁻¹k+π, then what is the value of k?

1. 1
2. −1/√2
3. 1/√2
4. None of these

Correct answer: 1/√2

Solution

sin^-1(x-1) needs x in [0,2]; cos^-1(x-3) needs x in [2,4]; their intersection is x=2. At x=2: pi/2 + pi + tan^-1(2/(2-4)) = pi/2 + pi - pi/4 = 5pi/4. Setting = cos^-1(k) + pi gives cos^-1(k) = pi/4, so k = 1/sqrt(2).

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