Question

The equation \(3\cos x + 4\sin x = 6\) has ... solution.

• A. finite
• B. infinite
• C. one
• D. no

Hint:

\(a\cos x + b\sin x\) lies between \(-\sqrt{a^2 + b^2}\) and \(\sqrt{a^2 + b^2}\).

Answer 1

The Correct Option is D

To determine the number of solutions for the equation \(3\cos x + 4\sin x = 6\), let's analyze it step-by-step:

1. Start by understanding the general form of the equation: \(a\cos x + b\sin x = c\). The given equation is \(3\cos x + 4\sin x = 6\).
2. The maximum value of \(a\cos x + b\sin x\) is found using the expression: \(\sqrt{a^2 + b^2}\). Here, \(a = 3\) and \(b = 4\).
3. Calculate \(\sqrt{a^2 + b^2}\):
   • Compute \(3^2 + 4^2\): \(9 + 16 = 25\)
   • Find the square root: \(\sqrt{25} = 5\)
4. The maximum value of the expression \(3\cos x + 4\sin x\) is \(5\). Any equation of this form cannot have a solution when \(c\) exceeds this maximum value.
5. Since \(6 \gt 5\), the equation \(3\cos x + 4\sin x = 6\) cannot be satisfied by any real value of \(x\).

Therefore, the given equation has no solution.