The general solution of \sqrt{3} \cos x+\sin x=3 is

Correct Answer is : no solution

NIMCET 2011 Question Paper with solutions

Question : 4 of 120

Marks: +1, -0

The general solution of

\sqrt{3} \cos x+\sin x=3

Solution:

Given,

√{3} \cos x+\sin x=3

{LHS}=√{3+1} ≤ √{3} \cos x+\sin x ≤ √{3+1}

⇒ -2 ≤ √{3} \cos x+\sin x ≤ 2

Here, maximum value is 2 , but right hand side is 3 , hence no solution exists.
Alternate method.

√{3} \cos x+\sin x=3

{√{3}}/{2} \cos x+ {1}/{2} \sin x= {3}/{2}

\sin π / 3 ⋅ \cos x ⋅ \cos π / 3 ⋅ \sin x=3 / 2

\sin (π / 3+x)=3 / 2

But

\sin ( {π}/{3}+x) ≠ {3}/{2}

∵ -1 ≤ \sin x ≤ 1

So, no solution exist.

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