QJEEM22Ma6776: JEE Main 2022 Mathematics - Class 12 JEEM22Ma6776 Solution: If , where , then which of the following is NOT correct?

$\mathop {\lim }\limits_{x \to 0} {{\alpha {e^x} + \beta {e^{ - x}} + \gamma \sin x} \over {x{{\sin }^2}x}} = {2 \over 3}$

$\Rightarrow \alpha + \beta = 0$ (to make indeterminant form) ...... (i)

Now,

$\mathop {\lim }\limits_{x \to 0} {{\alpha {e^x} - \beta {e^{ - x}} + \gamma \cos x} \over {3{x^2}}} = {2 \over 3}$ (Using L-H Rule)

$\Rightarrow \alpha - \beta + \gamma = 0$ (to make indeterminant form) ...... (ii)

Now,

$\mathop {\lim }\limits_{x \to 0} {{\alpha {e^x} + \beta {e^{ - x}} - \gamma \sin x} \over {6x}} = {2 \over 3}$ (Using L-H Rule)

$\Rightarrow {{\alpha - \beta + \gamma } \over 6} = {2 \over 3}$

$\Rightarrow \alpha - \beta + \gamma = 4$ ...... (iii)

$\Rightarrow \gamma = - 2$

and (i) + (ii)

$2\alpha = - \gamma$

$\Rightarrow \alpha = 1$ and $\beta = - 1$

and $\alpha {\beta ^2} + \beta {\gamma ^2} + \gamma {\alpha ^2} + 3 = 1 - 4 - 2 + 3 = - 2$