Q144 / 200 · AP EAMCET 2001 · Mathematics

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

  • A
    only one solution
  • B
    two solution
  • C
    infinitely many solution
  • Correct answer
    D
    no solution

Solution

We have, $\begin{array}{cc} & \sqrt{3} \sin x+\cos x=4 \\ \Rightarrow & \frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x=2 \\ \Rightarrow & \sin x \cos \frac{\pi}{6}+\cos x \sin \frac{\pi}{6}=2 \\ \Rightarrow & \sin \left(x+\frac{\pi}{6}\right)=2, \text { which is not possible. }\end{array}$ Therefore, this equation has no solution.

Hence, the correct answer is option: 'D'.

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