If the range of $f(x) = \frac{2x^2-14x^2-8x+49}{x^4-7x^2-4x+23}$ is ($a, b$], then ($a +b$) is

The function $f\left( x \right) = \left| {\sin \,4x} \right| + \left| {\cos \,2x} \right|$, is a periodic function with period

Let $f : R \to R$ be a function defined by $f(x) =  - \frac{{|x{|^3} + |x|}}{{1 + {x^2}}}$; then the graph of $f(x)$ is lies in the :-

Let $f(x)$ and $g(x)$ be two functions given by $f\left( x \right) = \frac{{2\sin \pi x}}{x}$ and $g\left( x \right) = f\left( {1 - x} \right) + f\left( x \right).$ If $g\left( x \right) = kf(\frac{x}{2})f\left( {\frac{{1 - x}}{2}} \right)$,then the value of $k$ is

Let $f\left( n \right) = \left[ {\frac{1}{3} + \frac{{3n}}{{100}}} \right]n$ , where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\sum\limits_{n = 1}^{56} {f\left( n \right)} $ 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