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(B) $(P)$ Let $P(x) = 10x^3 - 45x^2 + 60x + 35$. Then $P'(x) = 30x^2 - 90x + 60 = 30(x-1)(x-2)$.Since $P'(x) \leq 0$ for $x \in [1, 2]$,$P(x)$ decreas

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$(P) \rightarrow (2), (Q) \rightarrow (1), (R) \rightarrow (4), (S) \rightarrow (3)$

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(B) $(P)$ Let $P(x) = 10x^3 - 45x^2 + 60x + 35$. Then $P'(x) = 30x^2 - 90x + 60 = 30(x-1)(x-2)$.Since $P'(x) \leq 0$ for $x \in [1, 2]$,$P(x)$ decreases from $P(1) = 60$ to $P(2) = 55$. For $f(x) = [P(x)/n]$ to be continuous,the range of $P(x)/n$ must not contain any integer. The range is $[55/n, 60/n]$. The length of this interval is $5/n$. For this to contain no integers,we need $5/n  5$. Also,for the function to be continuous,the values must be between two consecutive integers. Checking $n=9$,range is $[55/9, 60/9] = [6.11, 6.66]$,which contains no integer. Thus,$n=9$ is the minimum value.$(Q)$ For $g(x)$ to be increasing,$g'(x) = (2n^2 - 13n - 15)(3x^2 + 3) \geq 0$. Since $3x^2+3 > 0$,we need $2n^2 - 13n - 15 \geq 0$. Solving $(2n+2)(n-7.5) \geq 0$,we get $n \geq 7.5$. The smallest natural number is $n=8$.$(R)$ $h'(x) = n(x^2-9)^{n-1}(2x)(x^2+2x+3) + (x^2-9)^n(2x+2) = (x^2-9)^{n-1} [2nx(x^2+2x+3) + 2(x+3)(x-3)]$. For $x=3$ to be a local minimum,the derivative must change sign from negative to positive. This requires $(x-3)^{n-1}$ to change sign,so $n-1$ must be odd,meaning $n$ is even. The smallest even $n > 5$ is $n=6$.$(S)$ $l(x) = \sum_{k=0}^4 (\sin|x-k| + \cos|x-k+1/2|)$. The term $\sin|x-k|$ is not differentiable at $x=k$. There are $5$ such points $(k=0, 1, 2, 3, 4)$. Thus,there are $5$ points of non-differentiability.

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 $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right), x \in R$,is an increasing function on $R$,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 $h(x)=\left(x^2-9\right)^{n}\left(x^2+2 x+3\right)$,is	$(3)$ $5$
$(S)$ Number of $x_0 \in R$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right), x \in R$ is not differentiable at $x_0$,is	$(4)$ $6$
	$(5)$ $10$

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