If f(x)=cos[π^2]x+cos[-π^2]x, where [.] stands for greatest integer function,...

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Question Id : 11668 | Context :NIMCET 2024

Question

If f(x)=cos[

$\pi$ ^2]x+cos[-

$\pi$ ^2]x, where [.] stands for greatest integer function, then

$f(\pi/2)$ =

🎥 Video solution / Text Solution of this question is given below:

? Function with Greatest Integer and Cosine

Given:

$f(x) = \cos\left([\pi^2]x\right) + \cos\left([-\pi^2]x\right)$

Find: $f\left(\frac{\pi}{2}\right)$

Step 1: Estimate Floor Values

$\pi^2 \approx 9.8696 \Rightarrow [\pi^2] = 9,\quad [-\pi^2] = -10$

Step 2: Plug into the Function

$f\left(\frac{\pi}{2}\right) = \cos\left(9 \cdot \frac{\pi}{2}\right) + \cos\left(-10 \cdot \frac{\pi}{2}\right) = \cos\left(\frac{9\pi}{2}\right) + \cos(-5\pi)$

Step 3: Simplify

$\cos\left(\frac{9\pi}{2}\right) = 0,\quad \cos(-5\pi) = -1$

✅ Final Answer:

$\boxed{-1}$

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