If $f(x) = x \left( \frac{1}{x-1} + \frac{1}{x} + \frac{1}{x+1} \right)$ for $x > 1$,then:

Let $f(x) = \frac{\sqrt{x - 2\sqrt{x - 1}}}{\sqrt{x - 1} - 1}$. Then:

Consider the function $f: [\frac{1}{2}, 1] \rightarrow \mathbb{R}$ defined by $f(x) = 4\sqrt{2}x^3 - 3\sqrt{2}x - 1$. Consider the following statements:
$(I)$ The curve $y = f(x)$ intersects the $x$-axis exactly at one point.
$(II)$ The curve $y = f(x)$ intersects the $x$-axis at $x = \cos \frac{\pi}{12}$.
Then:

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) = \text{the sum of positive divisors of } n$,then $f(2^k \cdot 3)$,where $k$ is a positive integer,is

Which of the following statements is false?

If $f(x) = \cos(\log x)$,then the value of $f(x^2) \cdot f(y^2) - \frac{1}{2} \left[ f\left(\frac{x^2}{y^2}\right) + f(x^2 y^2) \right]$ is: