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If F ( X ) = 2 X + 2 − X 2 , Then F(X + Y) F(X − Y) is Equal to (A) 1 2 [ F ( 2 X ) + F ( 2 Y ) ](B) 1 2 [ F ( 2 X ) − F ( 2 Y ) ](C) 1 4 [ F ( 2 X ) + F ( 2 Y ) ] - CBSE (Science) Class 11 - Mathematics

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Question

If  \[f\left( x \right) = \frac{2^x + 2^{- x}}{2}\] , then f(x + yf(x − y) is equal to

 

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

     

  • (b)  \[\frac{1}{2}\left[ f\left( 2x \right) - f\left( 2y \right) \right]\]

     

  • (c)  \[\frac{1}{4}\left[ f\left( 2x \right) + f\left( 2y \right) \right]\]

     

  • (d) \[\frac{1}{4}\left[ f\left( 2x \right) - f\left( 2y \right) \right]\]

     

Solution

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

Given: \[f\left( x \right) = \frac{2^x + 2^{- x}}{2}\] Now,
f(x + yf(x − y) = \[\left( \frac{2^{x + y} + 2^{- x - y}}{2} \right)\left( \frac{2^{x - y} + 2^{- x + y}}{2} \right)\]

⇒ f(x + yf(x − y) = \[\frac{1}{4}\left( 2^{2x} + 2^{- 2y} + 2^{2y} + 2^{- 2x} \right)\] ⇒ f(x + yf(x − y) = \[\frac{1}{2}\left( \frac{2^{2x} + 2^{- 2x}}{2} + \frac{2^{2y} + 2^{- 2y}}{2} \right)\]

⇒ f(x + yf(x − y) = \[\frac{1}{2}\left[ f\left( 2x \right) + f\left( 2y \right) \right]\]
 
 
 
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Solution If F ( X ) = 2 X + 2 − X 2 , Then F(X + Y) F(X − Y) is Equal to (A) 1 2 [ F ( 2 X ) + F ( 2 Y ) ](B) 1 2 [ F ( 2 X ) − F ( 2 Y ) ](C) 1 4 [ F ( 2 X ) + F ( 2 Y ) ] Concept: Functions.
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