For x > 1, if (2x)^2y = 4e^2x$$-$$2y,

then (1 + log_e 2x)² $${{dy} \over {dx}}$$ is equal to :

The area (in sq. units) of the region bounded by the parabola, y = x² + 2 and the lines, y = x + 1, x = 0 and x = 3, is

Let C₁ and C₂ be the centres of the circles x² + y² – 2x – 2y – 2 = 0 and x² + y² – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC₁QC₂ is :

$$\mathop {\lim }\limits_{x \to \pi /4} {{{{\cot }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$$ is :