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Let $y = f (x)$ be a solution of the differential equation $\frac{d y}{d x} = - 4 x (y - 2)$ with $f (0) = 2$ and $g (x) = max . {1 + \frac{x}{2} - [\frac{x}{2}], sin x, | x - 1 |}, h (x) = ln x$ [Note: $[y]$ denotes greatest integer function less than or equal to $y$ ] Area enclosed by the curves $y = f (x), y = g (x)$ and the $y$ -axis, is

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Detailed Solution

$\frac{d y}{d x} = - 4 x (y - 2)$

$\frac{d y}{y - 2} = - 4 x d x$

$ln | y - 2 | = - 2 x^{2} + C$

$| y - 2 | = e^{- 2 x^{2} + C} = λ e^{- 2 x^{2}}$

at $x = 0, y = 2 \Rightarrow λ = 0$

$∴ y - 2 = 0 \Rightarrow y = f (x) = 2$

$g (x) = max . {1 + {\frac{x}{2}}, sin x, | x - 1 |}$

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