Greatest Integer, Fractional Part Function

For any real number x, the maximum value of $4–6x–x^2$ is ?

$f\left(x\right)=\sin \left[x\right]+\left[\sin x\right], 0<x<\frac{\pi }{2}$ (where [.]  represents the greatest integer function) can also be represented as

The value of $\left[\sin x\right]+\left[1+\sin x\right]+\left[2+\sin x\right]$ in $x\in \left(\pi ,\frac{3\pi }{2}\right]$ can be ([⋅] is the greatest integer function)

If $f\left(x\right)=\left\{\begin{array}{cc}\sqrt{2}:& x \mathrm{is} \mathrm{rational}\\ 1;& x \mathrm{is} \mathrm{irrational}\end{array}\right.$and $\varphi \left(x\right)=\left[f\left(x\right)\right], \left(\left[·\right]=\mathrm{G}.\mathrm{I}.\mathrm{F}.\right)$, then range of $\varphi \left(x\right)$ is,

If $f(x)=\cos [\pi] x+\cos [\pi x]$, where $[y]$ is the greatest integer not exceeding $y$, then find $f\left(\frac{\pi}{2}\right)$.

The area (in sq. units) enclosed by the solution set of $\left[x\right]\left[y\right]=3$ is equal to (where $\left[·\right]$ denotes greatest integer function)

Let $P=(5\sqrt{3}+8{)}^{2n+1}$ and $f=P-\left[P\right]$, where $\left[x\right]$ denotes the greatest integer not greater than $x.P.f={11}^{2n+1}$. 

If $R=(2+\sqrt{3}{)}^n$, then show that $R\{1-R+[R\left]\right\}=1, \left[x\right]$ denotes the greatest integer not greater than $x$.

Let $\left[x\right]$ be the greatest integer less than or equal to $x$, for a real number $x$. Then the equation $\left[x^2\right]=x+1$ has

If $\left[.\right]$ denotes the greatest integer function, then the value of $\left[\frac{1}{2}\right]+\left[\frac{1}{2}+\frac{1}{100}\right]+\left[\frac{1}{2}+\frac{2}{100}\right]+$$\dots .+\left[\frac{1}{2}+\frac{99}{100}\right]$ is

If $\sum _{n=1}^k\left[\frac{1}{3}+\frac{n}{90}\right]=21,$ where $\left[x\right]$ denotes the integral part of $x,$ then $k$ is equal to
 

If $f: R\to R$ is defined by $f\left(x\right)=x-\left[x\right]+\frac{1}{4}$ where $\left[x\right]$ is the greatest integer not exceeding $x\mathit{,}$ then $\left\{x\in R: f\left(x\right)=\frac{1}{4}\right\}$ is equal to

If in greatest integer function, the domain is a set of real numbers, then range will be set of

Find $[2.75],$ if $\left[x\right]$ denotes greatest integer not greater than $x$?

Let $f:[0,\mathrm{ }\mathrm{\infty })\to N$ be defined by $f\left(x\right)=\left\{\begin{array}{ll}0,& 0\le x<1\\ n,& n\le x<n+1,\mathrm{ }n\in N\end{array}\right.$, then which of the following option is correct regarding $f$

Given  $0<x<1000$ and $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{5}\right]=\frac{31}{30}x$ where $[ .]$ is GIF. If the number of possible values of $x$ is $a$ , then $a-30$ is equal to

If $[.]$ represents the greatest integer function, where $f\left(x\right)=\cos \left[{\pi }^2\right]x+\cos \left[-{\pi }^2\right]x$, then find the value of $f\left(\frac{\pi }{2}\right)+f\left(\pi \right)+\frac{1}{2}f(-\pi )+\sqrt{2}f\left(\frac{\pi }{4}\right)$.

Consider the equation $-3(x-[x]{)}^2+2(x-\left[x\right])+a^2=0$ where [] represents the greatest integer function. For how many integer values of a, the equation has no integral solution?

$\left[x\right]$ and $\left\{x\right\}$ represent the greatest integer function and fractional part function respectively. Let $f\left(x\right)=\left[x\right]+\sum _{i=1}^{2020}\frac{\{x+r\}}{2020}.$ Find the value of $10f(-1000)$

What is the number of integral values of $n$ (where $n\ge 2$ ) such that the equation $2n\left\{x\right\}=3x+2\left[x\right]$ has exactly $5$ solutions (where $[.]$ denotes the greatest integer function and $\left\{x\right\}$ is fractional part of $x$)?