Relations and F...

If $$f(x)={ x }^{ 3 }+{ x }^{ 2 }{ f }^{ ' }(1)+x{ f }^{ '' }(2)+{ f }^{ ''' }(3)$$ for all $$x\in R$$, then f(x) =

If $$\frac { { x }^{ 4 } }{ \left( x-a \right) \left( x-b \right) \left( x-c \right)  } $$
$$=p(x)+\frac { A }{ x-a } +\frac { B }{ x-b } +\frac { C }{ x-c } then\quad p(x)=$$

Let $$f_k(x)=\dfrac{1}{k}(\sin^kx+\cos^kx)$$ where $$x\in R$$ and $$k\geq 1$$. Then $$f_4(x)-f_6(x)=?$$

If  $$f(x)=x^{ { 2 } }+2x+2;x\geq -1=-x^{ { 3 } }+3x+1;x<-1,$$  then the value of  $$f ( f ( - 2 ) )$$  is :

Let $$f(z)\equiv z^{3}+iz^{2}+iz-1$$. Among the roots of the equation $$f(z)=0$$

Let $$F\left( x \right) \begin{vmatrix} 1 & 1+\sin { x }  & 1+\sin { x } +\cos { x }  \\ 2 & 3+2\sin { x }  & 4+3\sin { x } +2\cos { x }  \\ 3 & 6+3\sin { x }  & 10+6\sin { x } +3\cos { x }  \end{vmatrix}, x\in R$$; then $$F'\left(\dfrac{\pi}{2}\right)$$ is equla to:

$$f (x) = x^4 - 10x^3 + 35x^2 - 50x + c$$ is a constant. the number of real roots of . f (x) = 0 and 
f'' (x) = 0 are respectively 

The function $$f$$ satisfies the functional equation $$3f(x) + 2 f\left (\dfrac {x + 59}{x - 1}\right ) = 10x + 30$$ for all real $$x\neq 1$$. The value of $$f(7)$$ is

If $$f(x)=\dfrac{x^2}{2}+3x$$ where $$x=2, \delta x=0.05$$ then $$df=?$$

If a relation R is defined on the set Z of integers as follows : (a,b) $$\epsilon \quad R\Leftrightarrow { a }^{ 2 }+{ b }^{ 2 }=25,$$ , Then domain (R)=