JEE Main 2024 — 29 January, Shift 2 — Question 30

According to the question , $\begin{aligned}& 27 r_{1}+\frac{9 r_{2}}{2}=-9 \end{aligned}$

$\lim _{x \rightarrow 3} \frac{\int_{3}^{x} 8 t^{2} d t}{\frac{3 r_{2} x}{2}-r_{2} x^{2}-r_{1} x^{3}-3 x}$

$=\lim _{x \rightarrow 3} \frac{8 x^{2}}{\frac{3 r_{2}^{2}}{2}-2 r_{2} x-3 r_{1} x^{2}-3}$ (using LH' Rule)

$=\frac{72}{\frac{3 r_{2}}{2}-6 r_{2}-27 r_{1}-3}$

$=\frac{72}{-\frac{9 r_{2}}{2}-27 r_{1}-3}$

$=\frac{72}{9-3}=12$