Functions Ques 37

If $f(x)=\cos (\log x)$, then $f(x) \cdot f(y)-\frac{1}{2}[ f (\frac{x}{y})+f(x y)]$ has the value

(1983, 1M)

(a) -1

(b) $\frac{1}{2}$

(c) -2

(d) None of these

Show Answer

Answer:

Correct Answer: 37.(d)

Solution:

Formula:

Operations on functions:

  1. Given, $f(x)=\cos (\log x)$

$\therefore f(x) \cdot f(y)-\frac{1}{2} [f \frac{x}{y}+f(x y)]$

$=\cos (\log x) \cdot \cos (\log y)-\frac{1}{2}[\cos (\log x-\log y)$ $+\cos (\log x+\log y)]$

$=\cos (\log x) \cdot \cos (\log y)-\frac{1}{2}[(2 \cos (\log x) \cdot \cos (\log y)]$

$=\cos (\log x) \cdot \cos (\log y)-\cos (\log x) \cdot \cos (\log y)=0$



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