Question

The maximum value of $3\cos \left(x\right)+4\sin \left(x\right)+5$ is

• A. $5$
• B. $6$
• C. $7$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

We know that the range of $a\cos \left(\theta \right)+b\sin \left(\theta \right)+c$ is $\left(c-\sqrt{(a^2+b^2)},c+\sqrt{(a^2+b^2)}\right)$

Thus, the maximum value of $a\cos \left(\theta \right)+b\sin \left(\theta \right)+c$ is $c+\sqrt{(a^2+b^2)}$

Substituting, $a=3,b=4,c=5$ in $c+\sqrt{(a^2+b^2)}$, we get,

$$\begin{gathered} =5+\sqrt{(9+16)} \\ =5+\sqrt{25} \\ =5+5 \\ =10 \end{gathered}$$

Thus, the maximum value of the given expression $3\cos \left(x\right)+4\sin \left(x\right)+5$ is $10$.

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