The number of solutions of 2 sin x + cos x = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The given equation is of the form $a \sin x + b \cos x = c$, where $a = 2$, $b = 1$, and $c = 3$.For the equation $a \sin x + b \cos x = c$ to hav

Vedclass · Accepted Answer

No solution

Vedclass · Answer

(D) The given equation is of the form $a \sin x + b \cos x = c$, where $a = 2$, $b = 1$, and $c = 3$.For the equation $a \sin x + b \cos x = c$ to have a solution, the condition $|c| \leq \sqrt{a^2 + b^2}$ must be satisfied.Here, $\sqrt{a^2 + b^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$.Since $\sqrt{5} \approx 2.236$, we have $c = 3 > \sqrt{5}$.Because the maximum value of $2 \sin x + \cos x$ is $\sqrt{5}$, it can never reach the value $3$.Therefore, the equation has no solution.

The number of solutions of $2 \sin x + \cos x = 3$ is

Similar Questions

If $x \neq (2n+1) \frac{\pi}{4}$,then the general solution of $\cos x + \cos 3x = \sin x + \sin 3x$ is

The principal solutions of the equation $\sec x + \tan x = 2 \cos x$ are

The number of solutions of the equation $\tan \theta + \sec \theta = 2 \cos \theta$ where $\cos \theta \neq 0$ lying in the interval $(0, 2 \pi)$ is

The number of values of $x$ for which $\sin 2x + \cos 4x = 2$ is

The number of values of $x$ in the interval $\left(\frac{\pi}{4}, \frac{7 \pi}{4}\right)$ for which $14 \operatorname{cosec}^{2} x - 2 \sin^{2} x = 21 - 4 \cos^{2} x$ holds,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module