Let $$\lambda \ne 0$$ be in R. If $$\alpha $$ and $$\beta $$ are the roots of the
equation, x² - x + 2$$\lambda $$ = 0 and $$\alpha $$ and $$\gamma $$ are the roots of
the equation, $$3{x^2} - 10x + 27\lambda = 0$$, then $${{\beta \gamma } \over \lambda }$$ is equal to:

The solution of the differential equation

$${{dy} \over {dx}} - {{y + 3x} \over {{{\log }_e}\left( {y + 3x} \right)}} + 3 = 0$$ is:

(where c is a constant of integration)

The function
$$f(x) = \left\{ {\matrix{ {{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} \cr {{1 \over 2}\left( {\left| x \right| - 1} \right),} & {\left| x \right| > 1} \cr } } \right.$$ is :

The distance between an object and a screen is 100 cm. A lens can produce real image of the object on the screen for two different positions between the screen and the object. The distance between these two positions is 40 cm. If the power of the lens is close to $$\left( {{N \over {100}}} \right)D$$ where N is an integer, the value of N is _________.