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JEE Main 20249 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
9 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Functions

JEE Main 2024 · 9 April · Evening Shift · Mathematics · Functions

For a differentiable function f: ℝ arrow ℝ , suppose f^ (x)=3...

For a differentiable function f:R→Rf: \mathbb{R} \rightarrow \mathbb{R}, suppose f′(x)=3f(x)+αf^{\prime}(x)=3 f(x)+\alpha, where α∈R\alpha \in \mathbb{R}, f(0)=1andf(0)=1 and\lim _{x \rightarrow-\infty} f(x)=7.Then. Then9 f(−log⁡e3)\left(-\log _{\mathrm{e}} 3\right) is equal to_________

Correct answer

0

Step-by-step solution

dydx−3y=α If =e∫−3dx=e−3x∴y−e−3x=∫e−3x⋅αdxye−3x=αe−3x−3+c(∗e3x)y=α−3+C⋅e3x\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 x=0,y=1x→−∞,y=7\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
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