InterviewSolution
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InterviewSolution
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If `alpha,beta,gamma,delta` are the four solutions of the equation `tan(theta+pi/4)=3 tan 3theta.` No two of which have equal tangents, then the value of `tan alpha+tan beta+tan gamma+tan delta=`A. `-1//3`B. `-2`C. 0D. none of these |
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Answer» Correct Answer - A 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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