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Differential Equations question

2025 · 22 Jan · Shift 1 · Q43

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JEE Main
Mathematics
Differential Equations
2025 · 22 Jan · Shift 1 · Q43

Differential Equations question

2025 · 22 Jan · Shift 1 · Q43

JEE MainMathematicsDifferential EquationsMCQ+4 / −1

Let

f: \mathbf{R} \rightarrow \mathbf{R}

be a twice differentiable function such that

f(x+y)=f(x) f(y)

for all

x, y \in \mathbf{R}

. If

f^{\prime}(0)=4 \mathrm{a}

and

f

satisfies

f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0, \mathrm{a}\gt 0

, then the area of the region

\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(a x), 0 \leq x \leq 2\}

is :

A
$\mathrm{e}^2-1$
B
$e^4+1$
C
$\mathrm{e}^2+1$
D
$e^4-1$

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