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 _________.

Orange light of wavelength 6000 $$ \times $$ 10^–10 m illuminates a single
slit of width 0.6 $$ \times $$ 10^–4 m. The maximum possible number of diffraction minima produced on both sides of the central maximum is ___________.

The change in the magnitude of the volume of an ideal gas when a small additional pressure $$\Delta $$P is applied at a constant temperature, is the same as the change when the temperature is reduced by a small quantity $$\Delta $$T at constant pressure. The initial temperature and pressure of the gas were 300 K and 2 atm. respectively.
If |$$\Delta $$T| = C|$$\Delta $$P| then value of C in (K/atm.) is _________.