QJEEM22Mab83b: JEE Main 2022 Mathematics - Class 12 JEEM22Mab83b Solution: Suppose be the solution curve to the differential equation such …

IF $= {e^{-x}}$

$y\,.\,{e^{-x}} = - 2{e^{ - x}} + {{{e^{ - 2x}}} \over 2} + C$

$\Rightarrow y = - 2 + {e^{ - x}} + C{e^x}$

$\mathop {\lim }\limits_{x \to \infty } \,y(x)$ is finite so $C = 0$

$y = - 2 + {e^{ - x}}$

$\Rightarrow {\left. {{{dy} \over {dx}} = - {e^{ - x}} \Rightarrow {{dy} \over {dx}}} \right|_{x = 0}} = - 1$

Equation of tangent

$y + 1 = - 1(x - 0)$

or $y + x = - 1$

So $a = - 1,\,b = - 1$

$\Rightarrow a - 4b = 3$