Question

Let$f\left(x\right)={\log }_e\left(\sin x\right),(0<x<\pi )$ and $g\left(x\right)={\sin }^{-1}\left(e^x\right),(x\ge 0).$If $\alpha$ is a positive real number such that $a=\left(fog{)}^{\prime }\right(\alpha )$ and $b=\left(fog\right)\left(\alpha \right),$ then :

• A. $a{\alpha }^2-b\alpha -a=0$
• B. $a{\alpha }^2+b\alpha +a=0$
• C. $a{\alpha }^2+b\alpha -a=-2{\alpha }^2$
• D. $a{\alpha }^2-b\alpha -a=1$

Answer 1

The correct option is D $a{\alpha }^2-b\alpha -a=1$

$f\left(x\right)={\log }_e\left(\sin x\right),(0<x<\pi )$ &
$g\left(x\right)={\sin }^{-1}\left(e^{-x}\right),(x\ge 0)$
$f\left(g\right(x\left)\right)=-x\Rightarrow \left(f\right(g\left(x\right)){)}^{\prime }=-1$
$f\left(g\right(\alpha )=-\alpha =b\Rightarrow (f\left(g\right(\alpha \left)\right){)}^{\prime }=-1=a$
$b=-\alpha$
$a=-1$
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