JEE MAIN - Mathematics (2024 - 29th January Evening Shift - No. 24)
Let the slope of the line $$45 x+5 y+3=0$$ be $$27 r_1+\frac{9 r_2}{2}$$ for some $$r_1, r_2 \in \mathbb{R}$$. Then $$\lim _\limits{x \rightarrow 3}\left(\int_3^x \frac{8 t^2}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} d t\right)$$ is equal to _________.
Answer
12
Explanation
According to the question,
$$\begin{aligned} & 27 r_1+\frac{9 r_2}{2}=-9 \\ & \lim _\limits{x \rightarrow 3} \frac{\int_\limits3^x 8 t^2 d t}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} \\ & =\lim _\limits{x \rightarrow 3} \frac{8 x^2}{\frac{3 r_2^2}{2}-2 r_2 x-3 r_1 x^2-3} \text { (using LH' Rule) } \\ & =\frac{72}{\frac{3 r_2}{2}-6 r_2-27 r_1-3} \\ & =\frac{72}{-\frac{9 r_2}{2}-27 r_1-3} \\ & =\frac{72}{9-3}=12 \end{aligned}$$
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