Question

Let $\alpha (a)$ and $\beta (a)$ be the roots of the equation $(\sqrt[3]{1+a}-1)x^2+(\sqrt{1+a}-1)x+(\sqrt[6]{1+a}-1)=0$ where $a>-1$. Then ${\lim }_{a\to o^{+}}\alpha (a)$ and ${\lim }_{a\to o^{+}}\beta (a)$ 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

Answer: (B)

Let $1+a=y$
$\Rightarrow \left(y^{1/3}-1\right)x^2+\left(y^{1/2}-1\right)x+y^{1/6}-1=0$
$\Rightarrow \left(\frac{y^{1/3}-1}{y-1}\right)x^2+\left(\frac{y^{1/2}-1}{y-1}\right)x+\frac{y^{1/6}-1}{y-1}=0$
Now taking $\underset{y\to 1}{\lim }$ on both the sides
$\Rightarrow \frac{1}{3}{x}^2+\frac{1}{2}x+\frac{1}{6}=0$
$\Rightarrow 2x^2+3x+1=0$
$x=-1,-\frac{1}{2}$