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If F(X) = Cos (Log X), Then the Value of F(X2) F(Y2) −1 2 { F ( X 2 Y 2 ) + F ( X 2 Y 2 ) } Is(A) −2 (B) −1 (C) 1/2 (D) None of These - CBSE (Science) Class 11 - Mathematics

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Question

If f(x) = cos (log x), then the value of f(x2f(y2) −

\[\frac{1}{2}\left\{ f\left( \frac{x^2}{y^2} \right) + f\left( x^2 y^2 \right) \right\}\] is
 
  • (a) −2

  • (b) −1

  • (c) 1/2

  • (d) None of these

     

Solution

(d) None of these

Given: \[f\left( x \right) = \cos\left( \log x \right)\]

\[\Rightarrow f\left( x^2 \right) = \cos\left( \log\left( x^2 \right) \right)\]
\[ \Rightarrow f\left( x^2 \right) = \cos\left( 2\log\left( x \right) \right)\]

Similarly,

\[f\left( y^2 \right) = \cos\left( 2\log\left( y \right) \right)\]
Now,
 
\[f\left( \frac{x^2}{y^2} \right) = \cos\left( \log\left( \frac{x^2}{y^2} \right) \right) = \cos\left( \log x^2 - \log y^2 \right)\]and
\[f\left( x^2 y^2 \right) = \cos\left( \log x^2 y^2 \right) = \cos\left( \log x^2 + \log y^2 \right)\]
\[\Rightarrow f\left( \frac{x^2}{y^2} \right) + f\left( x^2 y^2 \right) = \cos\left( \left( 2\log x - 2\log y \right) \right) + \cos\left( \left( 2\log x + 2\log y \right) \right)\]
\[ \Rightarrow f\left( \frac{x^2}{y^2} \right) + f\left( x^2 y^2 \right) = 2\cos\left( 2\log x \right)\cos\left( 2\log y \right)\]
\[ \Rightarrow \frac{1}{2}\left[ f\left( \frac{x^2}{y^2} \right) + f\left( x^2 y^2 \right) \right] = \cos\left( 2\log x \right)\cos\left( 2\log y \right)\]
\[\Rightarrow f\left( x^2 \right)f\left( y^2 \right) - \frac{1}{2}\left\{ f\left( x^2 y^2 \right) + f\left( \frac{x^2}{y^2} \right) \right\} = \cos\left( 2\log x \right)\cos\left( 2\log y \right) - \cos\left( 2\log x \right)\cos\left( 2\log y \right) = 0\]
 
 
 
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Solution If F(X) = Cos (Log X), Then the Value of F(X2) F(Y2) −1 2 { F ( X 2 Y 2 ) + F ( X 2 Y 2 ) } Is(A) −2 (B) −1 (C) 1/2 (D) None of These Concept: Functions.
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