JEE Main 2025 Apr 7 Shift 2, Mathematics Q23: Methods to Evaluate Limits

By Vieta's formula, $a+b=\underset{t\to -1^{+}}{\lim }\left(-\frac{(t+2{)}^{1/6}-1}{(t+2{)}^{1/7}-1}\right)$.

Substitute $y=t+2$, so $y\to 1^{+}$ as $t\to -1^{+}$. Both numerator and denominator of $\frac{y^{1/6}-1}{y^{1/7}-1}$ vanish at $y=1$, and dividing each by $(y-1)$ gives standard limits: $\frac{y^{1/6}-1}{y-1}\to \frac{1}{6}$ and $\frac{y^{1/7}-1}{y-1}\to \frac{1}{7}$.

So $\frac{y^{1/6}-1}{y^{1/7}-1}\to \frac{1/6}{1/7}=\frac{7}{6}$, giving $(a+b{)}^2={\left(\frac{7}{6}\right)}^2=\frac{49}{36}$ regardless of the overall sign.

$\therefore 72(a+b{)}^2=72\times \frac{49}{36}=2\times 49=98$.