Consider the graphs of the functions \(f(x) = 2cos(x/2) + 3\) and \(g(x) = 4\). The number of points of intersection of the two graphs in the interval \([0, 4π]\) is

\(2cosx/2 + 3 = 4 cosx/2 = 1/2\) For \(x ∈ [0, 4π]\), \(x/2 ∈ [0, 2π]\). Then \(x/2 = π/3\) or \(5π/3\), so \(x = 2π/3\) or \(10π/3\). That's 2 points of intersection. Answer: (D) 2

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Question 45

Mathematics Trigonometric Equations Hard

Consider the graphs of the functions \(f(x) = 2\cos\left(\frac{x}{2}\right) + 3\) and \(g(x) = 4\). The number of points of intersection of the two graphs in the interval \([0, 4\pi]\) is

(A) 4

(B) 3

(C) 1

(D) 2

✓ Correct Answer: Option D Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

\(2\cos\frac{x}{2} + 3 = 4 \Rightarrow \cos\frac{x}{2} = \frac{1}{2}\)

For \(x \in [0, 4\pi]\), \(\frac{x}{2} \in [0, 2\pi]\). Then \(\frac{x}{2} = \frac{\pi}{3}\) or \(\frac{5\pi}{3}\), so \(x = \frac{2\pi}{3}\) or \(\frac{10\pi}{3}\).

That's 2 points of intersection.

Answer: (D) 2

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