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The minimum integral value of `alpha` for which the quadratic equation `(cot^(-1)alpha)x^(2)-(tan^(-1)alpha)^(3//2)x+2(cot^(-1)alpha)^(2)=0` has both positive rootsA. 1B. 2C. 3D. 4 |
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Answer» Correct Answer - B `(cot^(-1)alpha)x^(2)-(tan^(-1)alpha)^(3//2)x + 2(cot^(-1)alpha)^(2)=0` Equation has real roots `therefore D ge 0 rArr (tan^(-1)alpha)^(3)-8(cot^(-1)alpha)^(2)=0` `therefore tan^(-1)alpha ge 2 cot^(-1)alpha = pi-2 tan^(-1)alpha` `rArr tan^(-1)alpha ge (pi)/(3)rArr alpha ge sqrt(3)` Sum of roots gt 0 `rArr ((tan^(-1)alpha)^(3//2))/(2cot^(-1)alpha)gt 0` Product of roots gt 0 `rArr 2cot^(-1)alpha gt 0 rArr alpha in R` `rArr alpha ge sqrt(3)` |
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