Question

If \( f(x) = \cos([\pi^2] \cdot x) + \cos([-\pi^2] \cdot x) \), where \([ \cdot ]\) denotes the greatest integer function, then \( f\left(\frac{\pi}{2}\right) = \)

• A. \(-1\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

Use the identities \(\cos(n\pi) = (-1)^n\) and periodicity of cosine to simplify.

Answer 1

The Correct Option is A

Step 1: Evaluate the greatest integer values. We have: \[ \pi^2 \approx 9.8696 \Rightarrow [\pi^2] = 9, \quad [-\pi^2] = -10 \] Step 2: Substitute into the function. The given function is: \[ f(x) = \cos(9x) + \cos(-10x) \] Now, substitute \( x = \frac{\pi}{2} \): \[ f\left(\frac{\pi}{2}\right) = \cos\left(9 \cdot \frac{\pi}{2}\right) + \cos\left(-10 \cdot \frac{\pi}{2}\right) \] \[ f\left(\frac{\pi}{2}\right) = \cos\left(\frac{9\pi}{2}\right) + \cos\left(-5\pi\right) \] \[ f\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) + \cos(5\pi) \] We know that: \[ \cos\left(\frac{\pi}{2}\right) = 0, \quad \cos(5\pi) = -1 \] Thus: \[ f\left(\frac{\pi}{2}\right) = 0 + (-1) = -1 \]