Question

Assertion :If $e^{xy}+\log \left(xy\right)+\sin \left(xy\right)-2=0$ then $\frac{dy}{dx}=\frac{-y}{x}$ Reason: $\frac{d\left(xy\right)}{dx}=0$ $\Rightarrow$ $\frac{dy}{dx}=\frac{-y}{x}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Assertion:
$e^{xy}+\log \left(xy\right)+\sin \left(xy\right)-2=0$
Differentiate both side w.r.t $x$
$\Rightarrow$ $e^{xy}\frac{d}{dx}\left(xy\right)+\frac{1}{xy}\frac{d}{dx}\left(xy\right)+\cos \left(xy\right)\frac{d}{dx}\left(xy\right)=0$
$\Rightarrow$ $\frac{d}{dx}\left(xy\right)[e^{xy}+\frac{1}{xy}+\cos \left(xy\right)]=0$
$\Rightarrow$ $\frac{d}{dx}\left(xy\right)=0$ or $e^{xy}+\frac{1}{xy}+\cos \left(xy\right)=0$ (rejected)
$\Rightarrow$ $x\frac{dy}{dx}+y.1=0$ $\Rightarrow$ $\frac{dy}{dx}=-\frac{y}{x}$
Thus both statements are correct and Assertion is followed by Reason.