CBSE (Science) Class 11CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

If F(X) = Cos (Log X), Then the Value of F(X) F(Y) − 1 2 { F ( X Y ) + F ( X Y ) } Is(A) −1 (B) 1/2 (C) −2 (D) None of These - CBSE (Science) Class 11 - Mathematics

Login
Create free account


      Forgot password?

Question

If f(x) = cos (log x), then the value of f(xf(y) −\[\frac{1}{2}\left\{ f\left( \frac{x}{y} \right) + f\left( xy \right) \right\}\] is

 
  • (a) −1

  • (b) 1/2

  • (c) −2

  • (d) None of these

     

Solution

(d) None of these

Given:

\[f\left( x \right) = \cos\left( \log x \right)\]
∴ \[f\left( y \right) = \cos\left( \log y \right)\]
 
Now,
\[f\left( \frac{x}{y} \right) = \cos\left( \log\left( \frac{x}{y} \right) \right) = \cos\left( \log x - \log y \right)\] and
\[f\left( xy \right) = \cos\left( \log xy \right) = \cos\left( \log x + \log y \right)\]  \[\Rightarrow f\left( \frac{x}{y} \right) + f\left( xy \right) = \cos\left( \log x - \log y \right) + \cos\left( \log x + \log y \right)\]
\[ \Rightarrow f\left( \frac{x}{y} \right) + f\left( xy \right) = 2\cos\left( \log x \right)\cos\left( \log y \right)\]
\[ \Rightarrow \frac{1}{2}\left[ f\left( \frac{x}{y} \right) + f\left( xy \right) \right] = \cos\left( \log x \right)\cos\left( \log y \right)\] \[\Rightarrow f\left( x \right)f\left( y \right) - \frac{1}{2}\left\{ f\left( xy \right) + f\left( \frac{x}{y} \right) \right\} = \cos\left( \log x \right)\cos\left( \log y \right) - \cos\left( \log x \right)\cos\left( \log y \right) = 0\]
\[\Rightarrow f\left( x \right)f\left( y \right) - \frac{1}{2}\left\{ f\left( xy \right) + f\left( \frac{x}{y} \right) \right\} = \cos\left( \log x \right)\cos\left( \log y \right) - \cos\left( \log x \right)\cos\left( \log y \right) = 0\]
  
 
  Is there an error in this question or solution?

APPEARS IN

Solution If F(X) = Cos (Log X), Then the Value of F(X) F(Y) − 1 2 { F ( X Y ) + F ( X Y ) } Is(A) −1 (B) 1/2 (C) −2 (D) None of These Concept: Functions.
S
View in app×