InterviewSolution
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InterviewSolution
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If `log_(cosx) sinx>=2` and `x in [0,3pi]` then `sinx` lies in the intervalA. `[(sqrt(5)-1)/(2), 1]`B. `(0, (sqrt(5)-1)/(2)]`C. `[0, 1//2]`D. none of these |
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Answer» Correct Answer - B We observe that `"log"_("cos"x)sin x` is defined for ` x in (0, pi//2) cup (2pi, 5pi//2)`. Now, `"log_("cos"x) "sin"x ge 2` `rArr "sin" x le ("cos" x)^(2) " "[because 0 lt "cos" x lt 1]` `rArr "sin"^(2) x + "sin"x -1 lt 0` `rArr ("sin" x + (1)/(2))^(2) - (5)/(4) le 0` `rArr ("sin" x + (1)/(2) - (sqrt(5))/(2))("sin" x + (1)/(2) + (sqrt(5))/(2)) le 0` `rArr "sin" x + (1)/(2) -(sqrt(5))/(2) le 0 " " [because "sin" x + (1)/(2) + (sqrt(5))/(2) gt 0]` `rArr "sin" x le (sqrt(5)-1)/(2) rArr "sin" x in (0, (sqrt(5)-1)/(2)]` |
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