Let `alpha(a)` and `beta(a)` be the roots of the equation `((1+a)^(1/3)-1)x^2 +((1+a)^(1/2)-1)x+((1+a)^(1/6)-1)=0` where `agt-1` then `lim_(a->0^+)alpha(a)` and `lim_(a->0^+)beta(a)`

Options

A. (a) `(-5/2)` and `1`
B. (b) `(-1/2)` and `-1`
C. (c) `(-7/2)` and `2`
D. (d) `(-9/2)` and `3`

Answer

D

Explanation

Let `a+1=h^(6)`
`:.(h^(2)-1)x^(2)+(h^(3)-1)x+(h-1)=0`
`implies((h^(2)-1)/(h-1))x^(2)+((h^(3)-1)/(h-1))x+1=0`
As `ato0` then `hto1`
`lim_(hto1)((h^(2)-1)/(h-1))x^(2)+lim_(hto1)((h^(3)-1)/(h-1))x+1=0`
`implies2x^(2)+3x+1=0`
`implies2x^(2)+2x+x+1=0`
`implies(2x+1)(x+1)=0`
`:.x=-1` and `x=-1/2`