$f(x)=4 \log _{e}(x-1)-2 x^{2}+4 x+5, x>1$,which one of the following is $NOT$ correct?

For the function $f(x) = x^4 (12 \ln x - 7)$,which of the following statements is true?

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

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	$(4)$ $6$
	$(5)$ $10$

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 $b.$

Let $f(x) = 2x + \tan^{-1} x$ and $g(x) = \log_e(\sqrt{1+x^2} + x)$,$x \in [0, 3]$. Then: