InterviewSolution
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InterviewSolution
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Let `S={xepsilon(-pi,pi):x!=0,+pi/2}`The sum of all distinct solutions of the equation `sqrt3secx+cosecx+2(tan x-cot x)=0` in the set S is equal toA. `-(7pi)/(9)`B. `-(2pi)/(9)`C. 0D. `(5pi)/(9)` |
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Answer» Correct Answer - C Give `(sqrt(3)) sec x + coses x + 2 (tan x - cot c) = 0` `(-pi lt x lt pi) - {0,pm pi//2}` `rArr sqrt(3) x cos x + 2 (sin^(2) x - cos^(2)x) = 0` `rArr sqrt(3) sin x + cos x - 2 cos 2 x =0` Multiplying and dividing by `sqrt(a^(2) + b^(2)) i.e., sqrt(3+1) = 2`. We get `2((sqrt(3))/(2) sin x + (1)/(2)cosx)- 2 cps 2x = 0` `rArr (cos x cos .(pi)(3) + sin x.sin.(pi)/(2))-cos 2 x cos = 0 cos (x-(pi)/(3)) = cos2x` `therefore " " 2x = 2npi+x -(pi)/(3)" " [underset(rArr theta = 2npipmalpha)"Since",cos theta =cos alpha] ` `rArr 2x = 2npi + x- (pi)/(3)` `or " " 3x = 2xpi-x + (pi)/(3)` `rArr" "x = 2npi - (pi)/(3)` `or " " 3x = 2npi + (pi)/(3)` `rArr x = 2npi - (pi)/(3)` `or " " x= (2npi)/(3) + (pi)/(9)` `therefore" " x= (-pi)/(3)` ` or" "x=(pi)/(9),(-5pi)/(9) ,(7pi)/(9)` Now, sum of all distinct solutions `= (-pi)/(3) + (pi)/(9)-(5pi)/(9) + (7pi)/(9) = 0` |
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