Relations and F...

If $$f\left( x+\cfrac { 1 }{ x }  \right) ={ x }^{ 3 }+\cfrac { 1 }{ { x }^{ 2 } } -4\left( { x }^{ 3 }+\cfrac { 1 }{ { x }^{ 2 } }  \right) +13$$ then the value of $$f(2+\sqrt { 3 } )=$$

Let $$\frac { d }{ dx } f(x)=\frac { { e }^{ sin\quad x } }{ x } .x>0.If\int _{ x }^{ 4 }{ \frac { 3 }{ x } { e }^{ sin{ x }^{ 3 } } } dx=F(k)-F(I)$$ then one of the possible value of K is

If $$f\left( x \right) = \cos \left[ {{\pi ^{^2}}} \right]x + \cos \left[ { - {\pi ^{^2}}} \right]x,$$ then 

If $$f(x)=cos(log,x),\quad then\quad f(x)f(x)-\cfrac { 1 }{ 2 } \left[ f\left( \cfrac { x }{ y }  \right) +f(xy) \right] $$ has value

Let $$A=\left\{ 1,2,3 \right\} ,\quad B=\left\{ 1,3,5 \right\} .$$ If relation R from A to B is given by $$R=\left\{ \left( 1,3 \right) ,\left( 2,5 \right) ,\left( 3,3 \right)  \right\} .\quad Then\quad { R }^{ 1 }\quad is$$

If $$2f\left( x \right) - 3f\left( {\frac{1}{x}} \right) = {x^2},x$$ is not equal to zero, then $$f(2)$$ is equal to - 

For each positive integer $$n$$,let $$f\left( n+1 \right)=n{ \left( -1 \right)  }^{ n+1 }-2f\left( n \right)$$ and $$f\left( 1 \right)=f\left( 2010 \right)$$. Then $$\sum _{ K=1 }^{ 2009 }{ f\left( K \right) }$$ is equal to

Let $$f\left( x \right) = \frac{{\alpha x}}{{x + 1}},\,x \ne  - 1$$. then for what value of $$f\left( {f\left( x \right)} \right) = x?$$ 

If  $$f ( x ) = \cos ( \log x ) ,$$  then the value of  $$f ( x ) \cdot f ( y ) - \dfrac { 1 } { 2 } \left\{ f ( x y ) + f \left( \dfrac { x } { y } \right) \right\}$$  equals

Let $$f(x)=(1+\sin x)(2+{\sin}^{2}x)....+(n+{\sin}^{n}x)$$, then $$f'(0)$$ is equal to