Let $f(x) = \log_e(\sin x)$ for $0 < x < \pi$ and $g(x) = \sin^{-1}(e^{-x})$ for $x \ge 0$. If $\alpha$ is a positive real number such that $a = (fog)'(\alpha)$ and $b = (fog)(\alpha)$,then which of the following is true?

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be defined as $f(x) = \begin{cases} x+a, & x < 0 \\ |x-1|, & x \geq 0 \end{cases}$ and $g(x) = \begin{cases} x+1, & x < 0 \\ (x-1)^2+b, & x \geq 0 \end{cases}$ where $a, b$ are non-negative real numbers. If $(g \circ f)(x)$ is continuous for all $x \in R$,then $a+b$ is equal to ......

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Define $f: S \rightarrow S$ as $f(n) = \begin{cases} 2n, & \text{if } n = 1, 2, 3, 4, 5 \\ 2n - 11, & \text{if } n = 6, 7, 8, 9, 10 \end{cases}$. Let $g: S \rightarrow S$ be a function such that $f \circ g(n) = \begin{cases} n + 1, & \text{if } n \text{ is odd} \\ n - 1, & \text{if } n \text{ is even} \end{cases}$. Then $g(10) \cdot (g(1) + g(2) + g(3) + g(4) + g(5))$ is equal to

Let $f(x) = x^3$ and $g(x) = 3^x$,then the quadratic equation whose roots are solutions of the equation $(f \circ g)(x) = (g \circ f)(x)$ (for $x \neq 0$) is

Let $f(x)=3+2x$ and $g_n(x)=(f \circ f \circ f \circ \dots \text{n times})(x)$. For all $n \in N$,if all the lines $y=g_n(x)$ pass through a fixed point $(\alpha, \beta)$,then $\alpha+\beta=$

If $f(x)=3x-2$ and $g(x)=x^2$,then $f \circ g(x) = \_\_\_\_$