Question

The values of $x$ satisfying $\frac{d}{dx}\left(\cos \left(x\right)+a\right)=\sin \left(2x\right)$ are

• A. $\left(\frac{3\mathrm{\pi }n}{2}\right)$
• B. $\left(\frac{3\mathrm{\pi }n}{4}\right)$
• C. $\left(\frac{2\mathrm{\pi }n}{3}\right)$
• D. None of these

Answer 1

The correct option is C

$\left(\frac{2\mathrm{\pi }n}{3}\right)$

Explanation for correct option:

Given,

$$\begin{gathered} \frac{d\left(\cos \left(x\right)+a\right)}{dx}=\sin \left(2x\right) \\ \Rightarrow -\sin \left(x\right)=\sin \left(2x\right) \\ \Rightarrow 2\sin \left(\frac{3x}{2}\right)\cos \left(\frac{x}{2}\right)=0 \\ \therefore \sin \left(\frac{3x}{2}\right)=0\mathrm{or}\cos \left(\frac{\mathrm{x}}{2}\right)=0 \\ \frac{3\mathrm{x}}{2}=\mathrm{n\pi }\mathrm{or}\frac{\mathrm{x}}{2}=\left(2\mathrm{n}+1\right)\frac{\mathrm{\pi }}{2} \\ \therefore \mathrm{x}=\frac{2\mathrm{n\pi }}{3}\mathrm{or}\mathrm{x}=\left(2\mathrm{n}+1\right)\mathrm{\pi } \end{gathered}$$

Hence, option C is correct.