Consider the differential equation given below. $${{dy} \over {dx}} + {x \over {1 - {x^2}}}y = x\sqrt y $$

The integrating factor of the differential equation is

A real 2 $$\times$$ 2 non-singular matrix A with repeated eigen value is given as

$$A = \left[ {\matrix{ x & { - 3.0} \cr {3.0} & {4.0} \cr } } \right]$$

where x is a real positive number. The value of x (rounded off to one decimal) is _______

In a high school having equal number of boy students and girl students, 75% of the students study Science and the remaining 25% students study Commerce. Commerce students are two times more likely to be a boy than are Science students. The amount of information gained in knowing that a randomly selected girl student studies Commerce (rounded off to three decimal place) is ______ bits.

Consider the integral

$$\oint {{{\sin (x)} \over {{x^2}({x^2} + 4)}}dx} $$

where C is counter-clockwise oriented circle defined as |x $$-$$ i| = 2. The value of the integral is