Let S be the set of all functions ƒ : [0,1] $$ \to $$ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $$ \in $$ (0,1), depending on ƒ, such that

The area (in sq. units) of the region

{(x,y) $$ \in $$ R² : x² $$ \le $$ y $$ \le $$ 3 – 2x}, is :

Let $$\overrightarrow a = \widehat i - 2\widehat j + \widehat k$$ and $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$ be two vectors. If $$\overrightarrow c $$ is a vector such that $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times \overrightarrow a $$ and $$\overrightarrow c .\overrightarrow a = 0$$, then $$\overrightarrow c .\overrightarrow b $$ is equal to

A capacitor is made of two square plates each of side 'a' making a very small angle $$\alpha $$ between them, as shown in figure. The capacitance will be close to :