Paragraph Two vectors a and b having unit modulus and angle between them is . Now ( )= _ -( a . b )^2 ^ ( a b )^2 f^2( x ) dx and f satisfies the condition f( y )+f( x )= x + y xy for all x , y R - 0 and h ( )=- ( )+ | a b ₁ |^2 | a b ₁ |^2

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Paragraph Two vectors a and b having unit modulus and angle between them is . Now ( )= _ -( a . b )^2 ^ ( a b )^2 f^2( x ) dx and f satisfies the condition f( y )+f( x )= x + y xy for all x , y R - 0 and h ( )=- ( )+ | a b ₁ |^2 | a b ₁ |^2 , where |b₁ |=2|b| . On the basis of above information, answer the following questions : Question The volume of the parallelopiped formed by a , b and a b , where angle between a

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Correct answer: B. 1 2

( )= _ - ^2 ^ ^2 (f( x ))^2 dx Also f(x)+f(y)= x+y x y Put y=x aligned & 2 f( x )= 2 x f( x )= 1 x & ( )= _ - ^2 ^ ^2 1 x ^2 dx = [- 1 x ]_ - ^2 ^ ^2 aligned Now h( )=- ( )+ | a b ₁ |^2 | a b ₁ |^2=4 cosec ^2 2 +16 ^2 ^2 =4 ( cosec ^2 2 + 1 cosec ^2 2 ) It will be minimum when cosec ^2 2 =1 = 4 So volume of the parallelopiped will be [ a b ( a b )]=( a b )^2= 1 2

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