Let α ( a) and β (a) be the roots of the equation (√ [3] 1 + a - 1) x2 - (√ 1 + a - 1) x + (√ [ 6] 1 + a - 1) = 0, where a - 1. Then, lima → 0+ , α (a) and lima → 0+ β (a) are

Question

Let α ( a) and β (a) be the roots of the equation (√ [3] 1 + a - 1) x2 - (√ 1 + a - 1) x + (√ [ 6] 1 + a - 1) = 0, where a > - 1. Then, lima → 0+ , α (a) and lima → 0+ β (a) are (A) - ( 5/2) and 1 (B)  - ( 1/2) and -1 (C)  - ( 7/2) and 2 (D) - ( 9/2) and 3

Tardigrade Admin · Accepted Answer

PLAN To make the quadratic into simple form we should
eliminate radical sign.
Description of Situation As for given equation,
when a

\to

0 the equation reduces to identity in x.
i, e.,

a x^{2} + b x + c = 0, \forall x \in R

or a = b = c

\to

0
Thus, first we should make above equation independent
from coefficients as 0 .
Let

a + 1 = t^{6} >

Thus when

a \to 0, t \to 1

.

(t^{2} - 1) x^{2} + (t^{3} - 1) x + (t - 1) = 0

\Rightarrow (t - 1) {(t + 1) x^{2} + (t^{2} + t + 1) x + 1} = 0,

as t

\to 1

2 x^{2} + 3 x + 1 = 0

\Rightarrow

2 x^{2} + 2 x + x + 1 = 0

\Rightarrow

( 2x + 1) (x + 1) = 0
Thus,

x = - 1, - 1 / 2
or

l i m_{a \to 0^{+}} α (a) = - 1/2

and

l i m_{a \to 0^{+}} β (a) = - 1

.

Q. Let $α (a) an d β (a)$ be the roots of the equation
$(3 1 + a - 1) x^{2} - (1 + a - 1) x + (6 1 + a - 1) = 0$ , where
a > - 1. Then, $l i m_{a \to 0^{+}}$ , $α (a) an d l i m_{a \to 0^{+}} β$ (a) are

$- \frac{5}{2}$ and 1

$- \frac{1}{2}$ and -1

$- \frac{7}{2}$ and 2

$- \frac{9}{2}$ and 3

Q. Let α(a)andβ(a) be the roots of the equation (31+a​−1)x2−(1+a​−1)x+(61+a​−1)=0, where a > - 1. Then, lima→0+​, α(a)andlima→0+​β (a) are

−25​ and 1

−21​ and -1

−27​ and 2

−29​ and 3

Q. Let $α (a) an d β (a)$ be the roots of the equation
$(3 1 + a - 1) x^{2} - (1 + a - 1) x + (6 1 + a - 1) = 0$ , where
a > - 1. Then, $l i m_{a \to 0^{+}}$ , $α (a) an d l i m_{a \to 0^{+}} β$ (a) are

$- \frac{5}{2}$ and 1

$- \frac{1}{2}$ and -1

$- \frac{7}{2}$ and 2

$- \frac{9}{2}$ and 3