Column – I		Column – II
A)	If  $f\left(x\right)=\underset{n\to \infty }{\lim }\left(\sum _{r=1}^{n}r{\cos }^{r}x\right)$ where $x\in R-\{n\pi ,n\in I\}$ ,then the value of $\left[\underset{x\to 0}{\lim }{\left({(1-\cos x)}^{2}f\left(x\right)\right)}^{\frac{1}{\cos x-1}}\right]$  is (where [ . ] is greatest integer function)	P)	2
B)	If the function  $f\left(x\right)$  is given by $f\left(x\right)=x^8-22x^5-6x^3+3x^2+55$  then the number of integral solutions of the equation $f\left(x\right)=0$  is	Q)	0
C)	The number of local maxima of function $f\left(x\right)=x^2+4\cos x+5$ is k and $g\left(x\right)=2{\left|x\right|}^3+3x^2-12\left|x\right|+1$  ,where $x\in [-1,2]$  then greatest value of  $g\left(x\right)$  is m then m-k is	R)	4
D)	If  $f:R-\{-1\}\to R$ and $f\left(x\right)$ (which is not a constant function and $f\left(x\right)\ne x,x\ne 0,-2$  )is a differentiable function satisfies $f(x+f\left(y\right)+xf\left(y\right))=y+f\left(x\right)+yf\left(x\right),\forall x,y\in R-\{-1\}$ then the value of  2023 $\{1+f\left(2022\right)\}$  is	S)	1