Let $f(x) = \sin x + (x^3 - 3x^2 + 4x - 2) \cos x$ for $x \in (0, 1)$. Consider the following statements:
$I.$ $f$ has a zero in $(0, 1)$.
$II.$ $f$ is monotone in $(0, 1)$.
Then,

Let $R$ 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 $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right), x \in R$,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	$(4)$ $6$
	$(5)$ $10$

The function $f(x) = x^{3} - 6x^{2} + ax + b$ is such that $f(2) = f(4) = 0$. Consider two statements.
$(S_1)$ There exists $x_{1}, x_{2} \in (2, 4)$,$x_{1} < x_{2}$,such that $f^{\prime}(x_{1}) = -1$ and $f^{\prime}(x_{2}) = 0$.
$(S_2)$ There exists $x_{3}, x_{4} \in (2, 4)$,$x_{3} < x_{4}$,such that $f$ is decreasing in $(2, x_{4})$,increasing in $(x_{4}, 4)$ and $2f^{\prime}(x_{3}) = \sqrt{3}f(x_{4})$.
Then

Let $f(x)=2^x-x^2, x \in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f^{\prime}(x)$ intersect the $x$-axis,then the value of $m+n$ is

If $(x-a)^{2}+(y-b)^{2}=c^{2},$ for some $c > 0,$ prove that $\frac{\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{\frac{3}{2}}}{\frac{d^{2} y}{d x^{2}}}$ is a constant independent of $a$ and $b.$

The number of solutions of the equation $2e^{|x|} \tan^{-1}|x| = 1$ is -