The number of solutions of the equation $|x^2 - 2|x|| = 2^x$ is:

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements in the relation $R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + a_2 \text{ divides } b_2 + b_1\}$ 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 $(0,1)$?

Let $f$ and $g$ be increasing and decreasing functions,respectively,from $[0, \infty)$ to $[0, \infty)$. Let $h(x) = f(g(x))$. If $h(0) = 0$,then $h(x) - h(1)$ is:

If $f(x) = \log_{e}\left(\frac{1-x}{1+x}\right)$,$|x| < 1$,then $f\left(\frac{2x}{1+x^2}\right)$ is equal to

If $f(x) = 2\sin x$ and $g(x) = \cos^2 x$,then $(f + g)\left(\frac{\pi}{3}\right) = $