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Consider the equation `sectheta+ cottheta = 31/12` On the basis of above, answer the following If `0 lt theta lt (pi)/(2)`, then minimum value of `[tan theta]` is equal to (where [ ] is G.I.F)A. `0`B. `1`C. `2`D. `3` |
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Answer» Correct Answer - A Let `t =tan,(theta)/(2)` `implies (1+t^(2))/(1-t^(2))+(1-t^(2))/(2t)=(31)/(12)` `implies6t^(4)+43t^(3)-12t^(2)-19t+6=0` `implies(3t-1)(2t^(3)+15t^(2)+t-6)=0` `implies t=(1)/(3),alpha,beta,gamma` where `alpha in(-8,-7)implies(theta)/(2)in((pi)/(2),(3pi)/(4))` `beta in (-1,(-1)/(2))implies(theta)/(2) in ((3pi)/(4),(7pi)/(8))` `gamma in((1)/(2),1)implies(theta)/(2)in(0,(pi)/(4))` (i) if `0lttheta lt (pi)/(2)` then `t=(1)/(3)` or `t in ((1)/(2),1)` `implies tan theta =(3)/(4) "or if" tan. (theta)/(2)in((1)/(2),1)implies "tan" theta in ((4)/(3),oo)` `implies"Min value of" [tan theta] =0` (ii)No. of values of `theta in ((pi)/(2),(3pi)/(2))` will be 1 |
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