Question

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

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

Answer 1

Answer: (B)

We know that,
$-\sqrt{a^{2} +b^{2}} \leq a \cos \theta +b \sin \theta \leq \sqrt{a^{2}+b^{2}}$
$\therefore -\sqrt{3+1} \leq \sqrt{3} \sin x+\cos x \leq \sqrt{3+1}$
$\Rightarrow -2 \leq \sqrt{3} \sin x+\cos x \leq \sqrt{2}$
But $\sqrt{3} \sin x+\cos x=4$
Hence, given equation has no solution.