Let $f(x) = (1 + {b^2}){x^2} + 2bx + 1$ and $m(b)$ the minimum value of $f(x)$ for a given $b$. As $b$ varies, the range of $m(b)$ is

Function ${\sin ^{ - 1}}\sqrt x $ is defined in the interval

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 ${f_k}\left( x \right) = \frac{1}{k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\;,x \in R$ and $k \ge 1$, then ${f_4}\left( x \right) - {f_6}\left( x \right)$ is equal to

A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series

$f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right)$ is equal to ....... .

Set $A$ has $3$ elements and set $B$ has $4$ elements. The number of injection that can be defined from $A$ to $B$ is