Q229 / 270 · NDA 2023 (Phase 1) · Mathematics

Paragraph: Let $f(x)=\sin \left[\pi^2\right] x+\cos \left[-\pi^2\right] x$ where [.] is a greatest integer function.
Question: What is $f\left(\frac{\pi}{2}\right)$ equal to?

  • A
    $-1$
  • Correct answer
    B
    $0$
  • C
    $1$
  • D
    $2$

Solution

$f(x)=\sin \left[\pi^2\right] x+\cos \left[-\pi^2\right] x$ $\because \quad \pi^2=(3 \cdot 14)^2$ Hence $\left[\pi^2\right]=9$ and $\left[-\pi^2\right]=-10$ $f(x)=\sin 9 x+\cos (-10 x)=\sin 9 x+\cos 10 x$ $f\left(\frac{\pi}{2}\right)=\sin \frac{9 \pi}{2}+\cos \frac{10 \cdot \pi}{2}$ $=\sin \left(4 \pi+\frac{\pi}{2}\right)+\cos 5 \pi=0$

Hence, the correct answer is option: 'B'.

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