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

The correct option is D no solution

a $\cos x+b\sin x=c$
Since, we know that for the above equation to have solutions,
the necessary condition is,
$|c|\le \sqrt{a^2+b^2}$
Hence, given equation is
$\sqrt{3}\sin x+\cos x=4$
$\therefore \sqrt{a^2+b^2}=\sqrt{1^2+(\sqrt{3}{)}^2}=\sqrt{4}=2$
but, $|c|=4$
$\Rightarrow |c|>\sqrt{a^2+b^2}$
$\therefore$ the given equation has no solution.