No Author Solutions for Exercise 1: EXERCISE

Let $A$ and $B$ be two sets defined as $A=\left\{\right(x, y):|x-3|<1$ and $|y-3|<1\}$ and $B=\left\{(x, y):4x^2+9y^2-32x-54y+109\le 0\right\}.$ Then,

Let $f : (4, 6)\to (6, 8)$ be a function defined by $f\left(x\right)=x+\left[\frac{x}{2}\right],$ where $[ ]$ denotes the greatest integer function, then $f^{-1}\left(x\right)$ is equal to

If $f\left(x\right)=\cos \left[\pi \right]x+\cos \left[\pi x\right],$ where $[ ]$ is greatest integer less than or equal to $x,$ then $f\left(\frac{\pi }{2}\right)$ is equal to

If $f\left(x\right)$ and $g\left(x\right)$ are periodic functions with periods $7$ and $11,$ respectively. Then, the period of $F\left(x\right)=f\left(x\right)·g\left(\frac{x}{5}\right)+g\left(x\right)·f\left(\frac{x}{3}\right)$ is

Let $f:R\to R$ be a function defined by $f\left(x\right)=\frac{x-m}{x-n}$, where $m\ne n$, then

If $\left[x\right]$ and $\left\{x\right\}$ represent the integral and fractional parts of $x$ respectively, then the value of $\sum _{r=1}^{2000}\frac{\{x+r\}}{2000}$ is

If $f\left(x\right)=\frac{1}{\sqrt{|x-1|-\left[x\right]}},$ where $[ ]$ denotes the greatest integer function, then the domain of $f\left(x\right)$ is

Statement I : The functions $f\left(x\right)=x^2-x+1, x\ge \frac{1}{2}$ and $g\left(x\right)=\frac{1}{2}+\sqrt{\left(x-\frac{3}{4}\right)},$ then the number of solutions of the equation $f\left(x\right)-g\left(x\right)$ is two.
Statement II : $f\left(x\right)$ and $g\left(x\right)$ are inverse of each other.