For some $a, b, c \in N$,let $f(x)=ax-3$ and $g(x)=x^b+c$,$x \in R$. If $(fog)^{-1}(x)=\left(\frac{x-7}{2}\right)^{1/3}$,then $(fog)(ac) + (gof)(b)$ is equal to $..........$

Let $f(x) = -1 + |x - 2|$ and $g(x) = 1 - |x|$. Then the set of all points where $fog$ is discontinuous is

If the function $f:(-\infty,-1] \rightarrow(a, b]$ defined by $f(x)=e^{x^3-3 x+1}$ is one-one and onto,then the distance of the point $P(2 b+4, a+2)$ from the line $x+e^{-3} y=4$ is:

Let $S = \mathbb{N} \cup \{0\}$. Define a relation $R$ from $S$ to $\mathbb{R}$ by: $R = \{(x, y) : \log_e y = x \log_e \left(\frac{2}{5}\right), x \in S, y \in \mathbb{R}\}$. Then,the sum of all the elements in the range of $R$ is equal to

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:

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