Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of $n$ for which the function $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$,is	(1)	8
(Q)	The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}$,is an increasing function on $\mathbb{R}$,is	(2)	9
(R)	The smallest natural number $n$ which is greater than 5,such that $x=3$ is a point of local minima of $h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)$,is	(3)	5
(S)	Number of $x_0 \in \mathbb{R}$ such that $l(x)=\sum_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right),x \in \mathbb{R}$,is NOT differentiable at $x_0$,is	(4)	6
		(5)	10

• A

  $(\mathrm{P}) \rightarrow(1),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(2),(\mathrm{S}) \rightarrow(5)$
• Correct answer
  B

  $(\mathrm{P}) \rightarrow(2),(\mathrm{Q}) \rightarrow(1),(\mathrm{R}) \rightarrow(4),(\mathrm{S}) \rightarrow(3)$
• C

  $(\mathrm{P}) \rightarrow(5),(\mathrm{Q}) \rightarrow(1),(\mathrm{R}) \rightarrow(4),(\mathrm{S}) \rightarrow(3)$
• D

  $(\mathrm{P}) \rightarrow(2),(\mathrm{Q}) \rightarrow(3),(\mathrm{R}) \rightarrow(1),(\mathrm{S}) \rightarrow(5)$

Solution

$\begin{aligned} & P(x)=10 x^3-45 x^2+60 x+35 \\ & P^{\prime}(x)=30(x-1)(x-2) \end{aligned}$ $\mathrm{P}(\mathrm{x})$ decreases in $[1,2]$ $\Rightarrow$ Range of $\mathrm{P}(\mathrm{x})=[55,60]$ $f(\mathrm{x})=\left[\frac{\mathrm{P}(\mathrm{x})}{\mathrm{n}}\right] \min$ value of $\mathrm{n}=9$ (Q) For $g(x)$ to be increasing $2 n^2-13 n-15 \geq 0$

(R) $\begin{aligned} & h(x)=\left(x^2-9\right)^n\left(x^2+2 x+3\right) \\ & h^{\prime}(x)=\left(x^2-9\right)^n(2 x+2)+\left(x^2+2 x+3\right) n\left(x^2-9\right)^{n-1} \cdot 2 x \\ & =\left(x^2-9\right)^{n-1}\left[2\left(x^2-9\right)(x+1)+2 n x\left(x^2+2 x+3\right)\right] \\ & =(x+3)^{n-1}(x-3)^{n-1} \cdot q(x)\end{aligned}$ Derivative must change sign at $x=3$ $\begin{aligned} & \because \mathrm{n}-1=\text { odd } \\ & \mathrm{n}=\text { even } \\ & \mathrm{n}=6 \end{aligned}$ (S) $\cos \left|\mathrm{x}-\mathrm{k}+\frac{1}{2}\right|$ is differentiable everywhere $\Rightarrow \sin |\mathrm{x}-\mathrm{k}|$ is NOT diff. at $\mathrm{k}=0,1,2,3,4$

Hence, the correct answer is option: 'B'.