Question

The equation $\sqrt{3}\, \sin \,x+\cos\,x = 4$ has

• A. only one solution
• B. two solutions
• C. infinitely many solutions
• D. no solution

Answer 1

Answer: (D)

no solution

Answer 2

Given, $\sqrt{3}\, \sin \,x+\cos\,x=4$
$\Rightarrow 2\left[\frac{\sqrt{3}}{2}\sin\,x+\frac{1}{2}\cos\,x\right]=4$
$\Rightarrow \sin \left(x+\frac{\pi}{6}\right)=2 \ldots\left(i\right)$
But $\sin \left(x+\frac{\pi}{6}\right) \le\,1$
$\therefore$ No soltuion exist
Alternative
From E $\left(i\right)$, $\sin \left(x+\frac{\pi}{6}\right)=2$

Let $y=\sin \left(x+\frac{\pi}{6}\right)=2$
It is clear from the graph two curves does not intersect