InterviewSolution
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InterviewSolution
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If `alpha, beta,gamma` are the solutions of the equation `tan (theta+(pi)/(4))=3tan 3theta`, no two of which have equal tangents. The value of `(1)/(tan alpha)+(1)/(tan beta)+(1)/(tan gamma)+(1)/(tan delta)` isA. `-8`B. `8`C. `2//3`D. `1//3` |
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Answer» Correct Answer - B We have `tan( theta + (pi)/(4))= 3tan 3 theta ` or `(1+ tan theta)/(1-tan theta) = 3 xx (3tan theta - tan^(3) theta)/(1-3tan^(2) theta)` `rArr (1+t)/(1-t) = 3 ((3t-t^(3))/(1-3t^(2)))" "` (putting `t = tan theta`) `or 3t^(4) - 6t^(2) + 8t- 1 =0` Hence, `S_1 `= sum of roots `=t_1 + t_2 + t_3 + t_4 =0` `S_2 ` = sum of product of roots taken two at a time = -2 `S_3` = sum of product of roots taken three at time `= -8//3` `S_4` = product of all roots `=-1//3` `(1)/(t_1) + (1)/(t_2) + (1)/(t_3) + (1)/(t_4) = (sum t_1 t_2 t_3)/(t_2t_2t_3t_4) = (-8//3)/(-1//3) =8` |
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