The magnitude of magnetic flux density (B) in micro Teslas ( µT ) at the center of a loop of wire wound as a regular hexagon of side length 1 m carrying a current (I = 1A), and placed in vacuum as shown in the figure is __________.

The matrix $$A = \left[ {\matrix{ {{3 \over 2}} & 0 & {{1 \over 2}} \cr 0 & { - 1} & 0 \cr {{1 \over 2}} & 0 & {{3 \over 2}} \cr } } \right]$$ has three distinct eigen values and one of its eigen vectors is $$\left[ {\matrix{ 1 \cr 0 \cr 1 \cr } } \right].$$ Which one of the following can be another eigen vector of $$A$$?

A function $$f(x)$$ is defined as
$$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$

Which one of the following statements is TRUE?

Let $${\rm I} = c\int {\int {_Rx{y^2}dxdy,\,\,} } $$ where $$R$$ is the region shown in the figure and $$c = 6 \times {10^{ - 4}}.\,\,$$ The value of $${\rm I}$$ equals ___________.