NIMCET 2027 Function Previous Year Questions (PYQs) – Page 2 of 3

Given:
$$f\left(\frac{1 - x}{1 + x}\right) = x + 2$$

To Find: \( f(1) \)

Let \( \frac{1 - x}{1 + x} = 1 \Rightarrow x = 0 \)

Then, \( f(1) = f\left(\frac{1 - 0}{1 + 0}\right) = 0 + 2 = 2 \)

Answer: $$\boxed{2}$$

? 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:

\[ \boxed{-1} \]