For a differentiable function f : I R rightarrow I R , suppose f prime (x)=3 f(x)+α , where α in IR , f(0)=1 and lim x rightarrow-∞ f(x)=7 . Then 9 f (-log e 3 ) is equal to .

Mathematics · Differential Equations

JEE Main 2024 — 9 April, Shift 2 — Question 23

For a differentiable function $\mathrm{f}: I R \rightarrow I R$ , suppose $f^{\prime}(x)=3 f(x)+\alpha$ , where $\alpha \in \operatorname{IR}, f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$ . Then $9 \mathrm{f}\left(-\log _{\mathrm{e}} 3\right)$ is equal to $\qquad$ .

Answer: 61

Numerical answer — enter this value.

Step-by-step solution

$\frac{d y}{d x}-3 y=\alpha$

If $=\mathrm{e}^{\int-3 \mathrm{dx}}=\mathrm{e}^{-3 \mathrm{x}}$

$\therefore y-e^{-3 x}=\int e^{-3 x} \cdot \alpha \mathrm{dx}$

$y e^{-3 x}=\frac{\alpha e^{-3 x}}{-3}+c$

$\left(* \mathrm{e}^{3 \mathrm{x}}\right)$

$y=\frac{\alpha}{-3}+C \cdot e^{3 x}$

on substituting $\mathrm{x}=0, \mathrm{y}=1$

$x \rightarrow-\infty, y=7$ we get $y=7-6 e^{3 x}$

$9 f\left(-\log _{e} 3\right)=61$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 9 April, Shift 2
Subject: Mathematics
Chapter: Differential Equations
Topic: Methods of solving a First Order,First Degree Differential

If log e y=3 sin ^-1 x , then (1-x)^2 y^prime prime-x y^prime at x=1/2 is equal to :

Answer: 61

Step-by-step solution

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