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Let alpha (a) beta (a) be the roots of t...

Let `alpha (a) beta (a)` be the roots of the equations : `(root(3)(1+a) -1)x^(2) + (sqrt(1 + a) - 1)x + (root(6)(1 + a) - 1) = 0`, where `a gt - 1`. Then `lim_(a rarr 0^(+)) alpha (a) and lim_(a rarr 0^(+)) beta(a)` are :

A

(a) `(-5/2)` and `1`

B

(b) `(-1/2)` and `-1`

C

(c) `(-7/2)` and `2`

D

(d) `(-9/2)` and `3`

Text Solution

Verified by Experts

The correct Answer is:
D
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`
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