To find ratio of Length and Breadth of a Rectangle Given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of the sheet, what is the ratio of length to breadth of the sheet?

It is just square-root of Golden Ratio.  
Let $ 2a, 2b $ be length and breadth. Then by folding the slope of diagonal= b/a ;
From your condition we have
$$ 2 b = 2 a \cos \theta $$
$$ 2 \sin^2\theta + \sin \theta -1 =0 \rightarrow \sin \theta = \frac{\sqrt5 -1}{2} = \frac{1}{\phi } $$
$$ \frac{a}{b} = \sec \theta = \sqrt \phi. $$
Nice and simple problem.
