Q60 / 90 · JEE Main 2024 (09 Apr Shift 2) · Mathematics
For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, suppose $f^{\prime}(x)=3 f(x)+\alpha$, where $\alpha \in \mathbb{R}$, $f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$. Then $9 f\left(-\log _{\mathrm{e}} 3\right)$ is equal to_________
Correct answer: 61
Solution
$\begin{aligned} & \frac{d y}{d x}-3 y=\alpha \\ & \text { If }=e^{\int-3 d x}=e^{-3 x} \\ & \therefore \quad y-e^{-3 x}=\int e^{-3 x} \cdot \alpha d x \\ & y e^{-3 x}=\frac{\alpha e^{-3 x}}{-3}+c \\ & \left(* e^{3 x}\right) \\ & y=\frac{\alpha}{-3}+C \cdot e^{3 x}\end{aligned}$
on substituting $\mathrm{x}=0, \mathrm{y}=1$
$x \rightarrow-\infty, y=7$
we get $y=7-6 e^{3 x}$
$9 f\left(-\log _e 3\right)=61$
Hence, the correct answer is 61.