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 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 $f$

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 $x$)?

If the number of solutions of the equation $e^{2x}+e^x-2=\left[\left\{x^2+11x+10\right\}\right]$ is $n$, where $\{.\}$ is the fractional part and $[.]$ is the greatest integer. Then, $12n$ is equal to:

If $z$ is a complex number and the minimum value of $\left|z\right|+|z–1|+|2z–3|$ is $\lambda$ and if $y=2\left[x\right]+3=3[x–\lambda ]$ . Then find the value of$[x+y]$ (where [.] denotes the greatest integer function)

If $f\left(x\right)=\left[x^2\right]-{\left[x\right]}^2$ where $\left[x\right]$ denotes the greatest integer $\le x,$ and $x\in \left[\mathrm{0, 2}\right],$ then the set of values of $f\left(x\right)$ are

On the real line $R$, we define two functions $f$ and $g$ as follows:

$f\left(x\right)=\mathrm{m}\mathrm{i}\mathrm{n}\{x-\left[x\right],1-x+\left[x\right]\}$

$g\left(x\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{x-\left[x\right],1-x+\left[x\right]\right\}$

Where $\left[x\right]$ denotes the largest integer not exceeding $x$ . The positive integer $n$ for which ${\int }_0^n\left(g\left(x\right)-f\left(x\right)\right) dx=100$ is

For a real number $r$ let $\left[r\right]$ denote the largest integer less than or equal to $r.$ Let $a>1$ a real number which is not an integer and $k$ be the smallest positive integer such that $\left[a^k\right]>{\left[a\right]}^k$ Then which of the following statement is always true?

For a real number $r$ we denote by $\left[r\right]$ the largest integer less than or equal to $r$. If $x, y$ are real numbers with $x, y \ge 1$ then which of the following statements is always true?

If $f\left(x\right)=\cos \left\{\frac{\pi }{2}\left[x\right]-x^3\right\}, -1<x<2$ and $\left[x\right]$ is the greatest integer less than or equal to $x$, then $f^{\text{'}}\left(\sqrt[3]{\frac{\pi }{2}}\right)$ is