JEE 2024 — Mathematics PYQ

\begin{aligned}
45x+5y+3 & =0 \\
\mathrm{Slope} & =-\frac{45}{5} \\
 & =27r_{i}+\frac{9r_{2}}{2} \\
\mathrm{-9} & =27r_{1}+\frac{9r_{2}}{2} \\
\mathrm{-2} & =6r+r_{\circ} \\
6r_{1}+r_{2}+2 & =0^{1\cdots2} \\
\operatorname*{lim}_{x\to3}\left(\int_{3}^{x}\frac{3t^{2}}{\frac{3r_{2}x}{2}-r_{2}x^{2}-}\right. & \frac{}{r_{1}x^{3}-3x}dt \\
 & & \lim_{x\to3}\frac{\frac{8}{3}(x^{3}-27)}{\frac{3r_{2}x}{2}-r_{2}x^{2}-r_{1}x^{3}-3x} \\
 & & \mathrm{L}^{1}\text{Hospital rule} \\
\lim_{x\to3}\frac{\frac{8}{3}.3x^{2}}{\frac{3r_{2}}{2}-2r_{2}x-3r_{1}x^{2}-3} \\
 & \frac{8(9)}{\frac{3r_{2}}{2}-6r_{2}-27r_{1}-3}=\frac{8\times9\times2}{3r_{2}-12r_{2}-54r_{1}-6} \\
 & =\frac{144}{-9(r_{2}+6r_{1})-6} \\
 & =\frac{144}{12} \\
 & =12
\end{aligned}