If for non-zero $$x,$$ $$af\left( x \right) + bf\left( {{1 \over x}} \right) = {1 \over x} - 25$$
where $$a \ne b$$ then $$\int\limits_1^2 {f\left( x \right)dx} \,$$ is

Let $$\,\,f\left( x \right) = {x^{ - \left( {1/3} \right)}}\,\,$$ and $${\rm A}$$ denote the area of the region bounded by $$f(x)$$ and the $$X-$$axis, when $$x$$ varies from $$-1$$ to $$1.$$ Which of the following statements is/are TRUE?
$${\rm I}.$$ $$f$$ is continuous in $$\left[ { - 1,1} \right]$$
$${\rm I}{\rm I}.$$ $$f$$ is not bounded in $$\left[ { - 1,1} \right]$$
$${\rm I}{\rm I}{\rm I}.$$ $${\rm A}$$ is nonzero and finite

$$\,\int\limits_{1/\pi }^{2/\pi } {{{\cos \left( {1/x} \right)} \over {{x^2}}}dx = } $$ __________.

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre specified pair of integers i, j with i < j. Given a shortcut i, j if you are at position i on the number line, you may directly move to j. suppose T(k) denotes the smallest number of steps needed to move from k to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular from 9 there is a shortcut to 15. Let y and z be such that T(9) = 1+ min(T(y),T(z)). Then the value of the product yz is _______.