NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

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

Choose the correct answer:

Explanation

1. Identify the form:

The expression is in the form $a \cos x + b \sin x$ , where $a = \sqrt{3}$ and $b = 1$ .

2. Find the range of the expression:

The maximum and minimum values of $a \cos x + b \sin x$ are given by $\pm\sqrt{a^2 + b^2}$ .

\text{Maximum value} = \sqrt{(\sqrt{3})^2 + (1)^2}

\text{Maximum value} = \sqrt{3 + 1}

\text{Maximum value} = \sqrt{4} = 2

3. Compare with the given equation:

The given equation is:

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

However, as calculated above, the maximum value that $\sqrt{3} \cos x + \sin x$ can ever reach is 2.

4. Logical Conclusion:

Since $2 < 3$ , the expression $\sqrt{3} \cos x + \sin x$ can never equal $3$ for any real value of $x$ .

Conclusion:

There is no real solution for this equation.

The correct option is (c).