Let $\mathrm{f}: N \rightarrow N$ be a function such that $\mathrm{f}(\mathrm{m}+\mathrm{n})=\mathrm{f}(\mathrm{m})+\mathrm{f}(\mathrm{n})$ for every $\mathrm{m}, \mathrm{n} \in N$. If $\mathrm{f}(6)=18$ then $\mathrm{f}(2) \cdot \mathrm{f}(3)$ is equal to :

Let $R _{1}$ and $R _{2}$ be two relations defined as follows :

$R _{1}=\left\{( a , b ) \in R ^{2}: a ^{2}+ b ^{2} \in Q \right\}$ and $R _{2}=\left\{( a , b ) \in R ^{2}: a ^{2}+ b ^{2} \notin Q \right\}$

where $Q$ is the set of all rational numbers. Then

If $P(S)$ denotes the set of all subsets of a given set $S, $ then the number of one-to-one functions from the set $S = \{ 1, 2, 3\}$ to the set $P(S)$ is

The number of bijective functions $f :\{1,3,5, 7, \ldots \ldots . .99\} \rightarrow\{2,4,6,8, \ldots \ldots, 100\}$, such that $f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \ldots f(99), \quad$ is

If for the function $f(x) = \frac{1}{4}{x^2} + bx + 10$ ; $f\left( {12 - x} \right) = f\left( x \right)\,\forall \,x\, \in \,R$ , then the value of $'b'$ is 

If $\theta$ is small $\&$ positive number then which of the following is/are correct ?