Question

Let $\alpha$ (a) and $\beta$ (b) be the roots of the equation of
$\left(\sqrt[3]{31+a}\right)-1)x^2+(\sqrt{1+a})-1)x+\left(\sqrt[6]{61+a}\right)-1)=0,a>-1$
Then $\underset{a\to 0}{lim}$ $\alpha$(a) and $\underset{a\to 0}{lim}$ $\beta$(a) are

• A. $\frac{-5}{2}$ and $1$
• B. $\frac{-1}{2}$ and $-1$
• C. $\frac{-7}{2}$ and $2$
• D. $\frac{-9}{2}$ and $3$

Answer 1

The correct option is B $\frac{-1}{2}$ and $-1$

[(1 + a)${}^{\frac{1}{3}}$ -1)x${}^2$ + [(a+1)${}^{\frac{1}{2}}$ -1] x + [(a+1)${}^{\frac{1}{6}}$-1] =0
Let 1 + a = p${}^6$.
Then (p${}^2$ -1) x${}^2$ + (p${}^3$ -1) x + (p -1) = 0
$\Rightarrow$ (p + 1)x${}^2$ +(p${}^2$ +p + 1)x+1 = 0
As a$\to$ 0, p$\to$ 1
$\therefore$ 2x${}^2$+3x + 1 = 0 $\to$ x = -1 and x = $\frac{-1}{2}$