UPSEE Solved Model Paper 7

Given f (x) = $\cos[\pi^2]x + \cos[-\pi^2 x]$ As π = 3.141 ⇒ $\pi^2$ = 9.8 (approx) Using definition of greatest integer function $[\pi^2]$ = [9.8] = 9 and $[-\pi^2]$ = [- 9.8] = - 10 ∴ f (x) = cos 9x + cos (- 10) x = cos 9x + cos 10x [Since cos (- θ) = cos θ] $f\left(\frac{\pi}{4}\right)$ = cos $\frac{9\pi}{4} + \cos\frac{10\pi}{4}$ = cos $\left(2\pi + \frac{\pi}{4}\right)$ + cos $\left(2\pi + \frac{2\pi}{4}\right)$ = cos $\frac{\pi}{4}$ + cos $\frac{\pi}{2}$ = $\frac{1}{\sqrt{2}} + 0$ = $\frac{1}{\sqrt{2}}$ Similarly, f (- π) = cos (- π) + cos (- 10π) = cos 9π + cos 10π = - 1 + 1 = 0 Also f (π) = cos -π + cos 10π = - 1 + 1 = 0 $f\left(\frac{\pi}{2}\right)$ = cos $\frac{9\pi}{2}$ + cos $\frac{10\pi}{2}$ = cos $\left(4\pi + \frac{\pi}{2}\right)$ + cos 5π = cos $\frac{\pi}{2}$ - 1 = 0 - 1 = - 1 Hence, option 'D' is correct.