Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of al prime factors of $2310$ and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is :

The period of the function $f(x) = \log \cos 2x + \sin 4x$ is :-

For a real number $x,\;[x]$ denotes the integral part of $x$. The value of $\left[ {\frac{1}{2}} \right] + \left[ {\frac{1}{2} + \frac{1}{{100}}} \right] + \left[ {\frac{1}{2} + \frac{2}{{100}}} \right] + .... + \left[ {\frac{1}{2} + \frac{{99}}{{100}}} \right]$ is

Show that the function $f: R \rightarrow R$ defined as $f(x)=x^{2},$ is neither one-one nor onto.

Domain of function $f(x) = log|5{x} - 2x|$ is $x \in R - A$, then $n(A)$ is (where $\{.\}$ denotes fractional part function)

Let $\quad E_1=\left\{x \in R : x \neq 1\right.$ and $\left.\frac{x}{x-1}>0\right\}$ and $\quad E_2=\left\{x \in E_1: \sin ^{-1}\left(\log _e\left(\frac{x}{x-1}\right)\right)\right.$ is a real number $\}$.

(Here, the inverse trigonometric function $\sin ^{-1} x$ assumes values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ )

Let $f : E _1 \rightarrow R$ be the function defined by $f(x)=\log _c\left(\frac{x}{x-1}\right)$ and $g: E_2 \rightarrow R$ be the function defined by $g(x)=\sin ^{-1}\left(\log _e\left(\frac{x}{x-1}\right)\right)$

The correct option is: