Relations and F...

Let $$f(x)=\cot ^{ -1 }{ \left( \cfrac { 1-{ x }^{ 2 } }{ 2x }  \right)  } +\cot ^{ -1 }{ \left( \cfrac { 1-3{ x }^{ 2 } }{ 3x-{ x }^{ 3 } }  \right)  } -\cot ^{ -1 }{ \left( \cfrac { 1-6{ x }^{ 2 }+{ x }^{ 4 } }{ 4x-4{ x }^{ 3 } }  \right)  } $$, the $$F'(x)$$ equals

If $$f(x)=\left| x \right| ,x\in R$$, then

If $$g(x)={ x }^{ 2 }+x-2$$ and $$\cfrac { 1 }{ 2 } (g\circ f)(x)=2{ x }^{ 2 }-5x+2$$, then $$f(x)$$ is

If $$(ax^2 + bx + c)y +a'x^2+b'x+c=0$$, then the condition that x may be a rational function of y is

If $$f(x)=\sin ^{ 2 }{ x } +\sin ^{ 2 }{ \left( x+\cfrac { \pi  }{ 3 }  \right)  } +\cos { x } \cos { \left( x+\cfrac { \pi  }{ 3 }  \right)  } $$ and $$g\left( \cfrac { 5 }{ 4 }  \right) =1$$, then $$g\circ f(x)$$ is equal to

If $$f(x)=ax+b $$ and $$g(x)=cx+d$$, then $$f\left( g(x) \right) =g\left( f(x) \right) \Leftrightarrow$$

The inverse of the function $$f(x) = \log (x^{2} + 3x + 1), x\epsilon [1, 3]$$, assuming it to be an onto function, is

If $$\ast$$ is defined by $$a\ast b = a - b^{2}$$ and $$\oplus$$ is defined by $$a\oplus b = a^{2} + b$$, where $$a$$ and $$b$$ are integers, then $$(3\oplus 4)\ast 5$$ is equal to

If $$f\left( x \right) $$ and $$g\left( x \right) $$ are two functions with $$g\left( x \right) =x-\dfrac { 1 }{ x } $$ and $$f\circ g\left( x \right) ={ x }^{ 3 }-\dfrac { 1 }{ { x }^{ 3 } } $$, then $$f^{ ' }\left( x \right) $$ is equal to

 If $$f:R\rightarrow R$$, $$g:R\rightarrow R$$ are defined by$$ f(x)=5x-3$$, $$g(x)=x^{2}+3$$, then $$(gof^{-1})(3)$$=