MHT CET · Maths · Functions
If \(f(x)=\cos (\log x)\), then \(f(x) \cdot f(y)-\frac{1}{2}\left(f\left(\frac{x}{y}\right)+f(x y)\right)\) has the value
- A \(-2\)
- B \(-1\)
- C \(0\)
- D \(\frac{1}{2}\)
Answer & Solution
Correct Answer
(C) \(0\)
Step-by-step Solution
Detailed explanation
\(f(x)=\cos (\log x) \)
\( \text { Now, } f(x) \cdot f(y)-\frac{1}{2}\left(f\left(\frac{x}{y}\right)+f(x y)\right) \)
\( =\cos (\log x) \cdot \cos (\log y)-\frac{1}{2}(\cos \log (\frac{x}{y})+\cos\) \(\log (x y)) \)
\( =\cos (\log x) \cdot \cos (\log y)-\frac{1}{2}\{\cos (\log x-\log y)~+\) \(\cos (\log x+\log y)\} \)
\( =\cos (\log x) \cdot \cos (\log y)-\frac{1}{2} \times 2 \cos (\log x) \cdot \cos (\log y) \)
\( =0\)
\( \text { Now, } f(x) \cdot f(y)-\frac{1}{2}\left(f\left(\frac{x}{y}\right)+f(x y)\right) \)
\( =\cos (\log x) \cdot \cos (\log y)-\frac{1}{2}(\cos \log (\frac{x}{y})+\cos\) \(\log (x y)) \)
\( =\cos (\log x) \cdot \cos (\log y)-\frac{1}{2}\{\cos (\log x-\log y)~+\) \(\cos (\log x+\log y)\} \)
\( =\cos (\log x) \cdot \cos (\log y)-\frac{1}{2} \times 2 \cos (\log x) \cdot \cos (\log y) \)
\( =0\)
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