IIT-JEE 2012 · Paper 2 · Mathematics Q45

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MathematicsSingle correct+3 / -1Question 45 of 60

Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation
$\left(\sqrt[3]{1+a}-1\right)x^2+\left(\sqrt{1+a}-1\right)x+\left(\sqrt[6]{1+a}-1\right)=0$
where $a>-1$ . Then $\displaystyle\lim_{a\to0^+}\alpha(a)$ and $\displaystyle\lim_{a\to0^+}\beta(a)$ are

-\dfrac52

and 1

-\dfrac12

and −1

-\dfrac72

and 2

-\dfrac92

and 3

Answer and worked solution

Official answer: B

Small $a$ . $\sqrt[3]{1+a}-1\approx\frac a3$ , $\sqrt{1+a}-1\approx\frac a2$ , $\sqrt[6]{1+a}-1\approx\frac a6$ .

Limiting equation. Divide by $a$ and let $a\to0^+$ :
$\frac{x^2}3+\frac x2+\frac16=0\implies2x^2+3x+1=0\implies x=-\frac12,\ -1.$
The roots depend continuously on the coefficients, so these are the limits.

(B)

Topics:Limits Roots and coefficients

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IIT-JEE 2012 · Paper 2, Mathematics question 45