If ${e^{f(x)}} = \frac{{10 + x}}{{10 - x}},\;x \in ( - 10,\;10)$ and $f(x) = kf\left( {\frac{{200x}}{{100 + {x^2}}}} \right)$,then $k = $

Consider the function $f(x) = x^3 - 8x^2 + 20x - 13$. The number of positive integers $x$ for which $f(x)$ is a prime number is:

Let $f: R \rightarrow R$ be defined by $f(x)=2x+3$. If $\alpha$ and $\beta$ are the roots of the equation $f(x^2)-2f(\frac{x}{2})-1=0$,then $\alpha^2+\beta^2=$

If $e^{f(x)}=\frac{10+x}{10-x}, x \in(-10,10)$ and $f(x)=k f\left(\frac{200 x}{100+x^2}\right)$,then $k$ is equal to

Let $S = \{1, 2, 3, 4, 5, 6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties:
$i$. $R$ has exactly $6$ elements.
$ii$. For each $(a, b) \in R$,we have $|a-b| \geq 2$.
Let $Y = \{R \in X : \text{The range of } R \text{ has exactly one element}\}$ and $Z = \{R \in X : R \text{ is a function from } S \text{ to } S\}$.
Let $n(A)$ denote the number of elements in a set $A$.
$(1)$ If $n(X) = {}^{m}C_{6}$,then the value of $m$ is. . . .
$(2)$ If the value of $n(Y) + n(Z)$ is $k^{2}$,then $|k|$ is. . . .

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 $(0,1)$?