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 roots

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

`(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)`