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The exhaustive set of values of a for which `a - cot^(-1) 3x = 2 tan^(-1) 3x + cos^(-1) x sqrt3 + sin^(-1) x sqrt3` may have solution, isA. `[-(pi)/(4), (pi)/(4)]`B. `((pi)/(2), (3pi)/(2))`C. `[(2pi)/(3), (4pi)/(3)]`D. `[-(3pi)/(6), (7pi)/(6)]` |
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Answer» Correct Answer - C `a - cot^(-1) 3x = 2 tan^(-1) 3x + cos^(-1) x sqrt3 + sin^(-1) x sqrt3` Clearly, given equation is meaningful when `-1 le x sqrt3 le 1` Given equation becomes `a= tan^(-1) 3x + (tan^(-1) 3x+ cot^(-1) 3x) + (cos^(-1) x sqrt3 + sin^(-1) x sqrt3)` `= tan^(-1) 3x + pi//2 + pi//2` `pi + tan^(-1) 3x` Now `-1 le x sqrt3 le 1` `rArr -sqrt3 le 3x le sqrt3` `-(pi)/(3) le tan^(-1) 3x le (pi)/(3)` `rArr (2pi)/(3) le pi + tan^(-1) 3x le (4pi)/(3)` `rArr a in [(2pi)/(3), (4pi)/(3)]` |
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