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If F(X) = Cos (Loge X), Then F ( 1 X ) F ( 1 Y ) − 1 2 { F ( X Y ) + F ( X Y ) } is Equal To(A) Cos (X − Y) (B) Log (Cos (X − Y)) (C) 1 (D) Cos (X + Y) - Mathematics

MCQ

If f(x) = cos (loge x), then \[f\left( \frac{1}{x} \right)f\left( \frac{1}{y} \right) - \frac{1}{2}\left\{ f\left( xy \right) + f\left( \frac{x}{y} \right) \right\}\] is equal to

 

Options

  • (a) cos (x − y)

  • (b) log (cos (x − y))

  • (c) 1

  • (d) cos (x + y)

     
  • (e) 0

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Solution

Given:

\[f\left( x \right) = \cos\left( \log_e x \right)\]
\[\Rightarrow f\left( \frac{1}{x} \right) = \cos\left( \log_e \left( \frac{1}{x} \right) \right)\]
\[ \Rightarrow f\left( \frac{1}{x} \right) = \cos\left( - \log_e \left( x \right) \right)\]
\[ \Rightarrow f\left( \frac{1}{x} \right) = \cos\left( \log_e \left( x \right) \right)\]
Similarly,
\[f\left( \frac{1}{y} \right) = \cos\left( \log_e y \right)\]
Now,
\[f\left( xy \right) = \cos\left( \log_e xy \right) = \cos\left( \log_e x + \log_e y \right)\]
  and
 
\[f\left( \frac{x}{y} \right) = \cos\left( \log_e \frac{x}{y} \right) = \cos\left( \log_e x - \log_e y \right)\]
\[\Rightarrow f\left( \frac{x}{y} \right) + f\left( xy \right) = \cos\left( \log_e x - \log_e y \right) + \cos\left( \log_e x + \log_e y \right)\]
\[ \Rightarrow f\left( \frac{x}{y} \right) + f\left( xy \right) = 2\cos\left( \log_e x \right)\cos\left( \log_e y \right)\]
\[ \Rightarrow \frac{1}{2}\left[ f\left( \frac{x}{y} \right) + f\left( xy \right) \right] = \cos\left( \log_e x \right)\cos\left( \log_e y \right)\]
\[\Rightarrow f\left( \frac{1}{x} \right)f\left( \frac{1}{y} \right) - \frac{1}{2}\left\{ f\left( xy \right) + f\left( \frac{x}{y} \right) \right\} = \cos\left( \log_e x \right)\cos\left( \log_e y \right) - \cos\left( \log_e x \right)\cos\left( \log_e y \right) = 0\]

Notes

Disclaimer: The question in the book has some error, so none of the options are matching with the solution. The solution is created according to the question given in the book.

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 3 Functions
Q 18 | Page 43
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