$[t]$ denotes the greatest integer function and $[t-m] = [t] - m$ when $m \in \mathbb{Z}$. If $k = 2[2x - 1] - 1$ and $3[2x - 2] + 1 = 2[2x - 1] - 1$,then the range of $f(x) = [k + 5x]$ is

If $f(x) = \frac{2^{2x}}{2^{2x} + 2}$,$x \in R$,then $f\left(\frac{1}{2023}\right) + f\left(\frac{2}{2023}\right) + \dots + f\left(\frac{2022}{2023}\right)$ is equal to

Let $N$ be the set of all natural numbers and $f: N \rightarrow N$ be such that $1990 < f(1990) < 2100$ and satisfies the equation $x-f(x)=19[\frac{x}{19}]-90[\frac{f(x)}{90}]$,where $[y]$ denotes the greatest integer less than or equal to $y$. Then the number of possible values of $f(1990)$ is

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x) = \alpha x^5 + \beta x^3 + \gamma x, x \in \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \dots, a_n$ are in arithmetic progression with mean zero,then the value of $f(g(\frac{1}{n} \sum_{i=1}^{n} f(a_i)))$ is equal to.

Let $f: R \rightarrow (0,1)$ be a continuous function. Then,which of the following function$(s)$ has(have) the value zero at some point in the interval $(0,1)$?

The probability that a relation $R$ from $\{x, y\}$ to $\{x, y\}$ is both symmetric and transitive is equal to: