An N-type semiconductor having uniform doping is biased as shown in the figure.

If E_C is the lowest energy level of the conduction band, E_V is the highest energy level of the valance band and E_F is the Fermi level, which one of the following represents the energy band diagram for the biased N-type semiconductor?

The series $$\sum\limits_{n = 0}^\infty {{1 \over {n!}}\,} $$ converges to

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenuse and the side is

The magnitude of the gradient for the function $$f\left( {x,y,z} \right) = {x^2} + 3{y^2} + {z^3}\,\,$$ at the point $$(1,1,1)$$ is _________.