If f(x) = sin
Question:

If f(x) = sin [π2x + sin [−π]2 x, where [x] denotes the greatest integer less than or equal to x, then

(a) f(π/2) = 1

(b) f(π) = 2

(c) f(π/4) = −1

(d) None of these

Solution:

(a) f(π/2) = 1

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

$\Rightarrow f(x)=\sin [9.8] x+\sin [-9.8] x$

 

$\Rightarrow f(x)=\sin 9 x-\sin 10 x$

$f\left(\frac{\pi}{2}\right)=\sin 9 \times \frac{\pi}{2}-\sin 10 \times \frac{\pi}{2}$

$\Rightarrow f\left(\frac{\pi}{2}\right)=1-0=1$

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