If $f(x) = \sin \log x$,then the value of $f(xy) + f\left( \frac{x}{y} \right) - 2f(x) \cos \log y$ is equal to

Let $R$ denote the set of all real numbers and $R^{+}$ denote the set of all positive real numbers. For the subsets $A$ and $B$ of $R$,define $f: A \rightarrow B$ by $f(x) = x^2$ for $x \in A$. Match the following lists:
| Column $I$ | Column $II$ |
| :--- | :--- |
| $A$. $f$ is one-one and onto,if | $1$. $A = R^{+}, B = R$ |
| $B$. $f$ is one-one but not onto,if | $2$. $A = B = R$ |
| $C$. $f$ is onto but not one-one,if | $3$. $A = R, B = R^{+}$ |
| $D$. $f$ is neither one-one nor onto,if | $4$. $A = B = R^{+}$ |

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

For a real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. For $x \in \mathbb{R}$,let $f(x) = [x] \sin(\pi x)$. Then,

The function $f(x) = \sec \left[ \log \left( x + \sqrt{1 + x^2} \right) \right]$ is . . . . . . function.

If the number of elements in the sets $G$ and $A$ are $3$ and $4$ respectively,then match the items of List-$I$ with those of List-$II$.

List-$I$	List-$II$
$A$. The number of non-bijective functions from $G \times G$ to $G$	$I$. $24$
$B$. The number of bijective functions from $A$ to $A$	$II$. $0$
$C$. The number of functions from $G$ to $G \times A$	$III$. $1728$
$D$. The number of surjective functions from $A$ to $A \times A$	$IV$. $12$
	$V$. $19683$