InterviewSolution
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InterviewSolution
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For `x, y, z, t in R, sin^(-1) x + cos^(-1) y + sec^(-1) z ge t^(2) - sqrt(2pi t) + 3pi` The value of `x + y + z` is equal toA. 1B. 0C. 2D. `-1` |
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Answer» Correct Answer - D `sin^(-1) x in [-(pi)/(2), (pi)/(2)]` `cos^(-1) y in [0, pi]` `sec^(-1) z in [0, (pi)/(2)) uu ((pi)/(2), pi]` `rArr sin^(-1) x + cos^(-1) y + sec^(-1) z le (pi)/(2) + pi + pi = (5pi)/(2)` Also `t^(2) - sqrt(2pi) t + 3pi = t^(2) -2 sqrt((pi)/(2)) t + (pi)/(2) - (pi)/(2) + 3pi` `= (t -sqrt((pi)/(2)))^(2) + (5pi)/(2) ge (5pi)/(2)` The given inequation exists if equality holds, i.e., L.H.S. = R.H.S. `= (5pi)/(2)` `rArr x = 1, y = -1, z = -1 and t = sqrt((pi)/(2))` `rArr cos^(-1) (cos 5t^(2)) = cos^(-1) (cos 5 t^(2)) = cos^(-1) (cos ((5pi)/(2))) = (pi)/(2)` `cos^(-1) ("min") {x, y, z}) = cos^(-1) (-1) = pi` |
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