Question

The number of solution of $2\sin \left(x\right)+\cos \left(x\right)=3$ is

• A. $1$
• B. $2$
• C. Infinite
• D. No solution

Answer 1

The correct option is D

No solution

Explanation for correct option

Given, $2\sin \left(x\right)+\cos \left(x\right)=3$

The range of the equation $a\cos \left(x\right)+b\sin \left(x\right)$ is

$-\sqrt{a^2+b^2}\le a\cos \left(x\right)+b\sin \left(x\right)\le \sqrt{a^2+b^2}$

Now, let $a=1,b=2$

$$\begin{gathered} \Rightarrow -\sqrt{1^2+2^2}\le 2\sin \left(x\right)+\cos \left(x\right)\le \sqrt{1^2+2^2} \\ \Rightarrow -\sqrt{5}\le 2\sin \left(x\right)+\cos \left(x\right)\le \sqrt{5} \end{gathered}$$

Thus $\sqrt{5}<3$

So, there exists no possible solution for the equation.

Hence, the correct option is $\left(\mathrm{D}\right)$.