Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y $$ \subseteq $$ X, Z $$ \subseteq $$ X and Y $$ \cap $$ Z is empty, is :

Let x₁, x₂,........., x_n be n observations, and let $$\overline x $$ be their arithematic mean and $${\sigma ^2}$$ be their variance.

Statement 1 : Variance of 2x₁, 2x₂,......., 2x_n is 4$${\sigma ^2}$$.
Statement 2 : : Arithmetic mean of 2x₁, 2x₂,......, 2x_n is 4$$\overline x $$.

If $$f:R \to R$$ is a function defined by

$$f\left( x \right) = \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi $$,

where [x] denotes the greatest integer function, then $$f$$ is

Consider the function, $$f\left( x \right) = \left| {x - 2} \right| + \left| {x - 5} \right|,x \in R$$

Statement - 1 : $$f'\left( 4 \right) = 0$$

Statement - 2 : $$f$$ is continuous in [2, 5], differentiable in (2, 5) and $$f$$(2) = $$f$$(5)