For x² $$ \ne $$ n$$\pi $$ + 1, n $$ \in $$ N (the set of natural numbers), the integral

$$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $$ is equal to :

(where c is a constant of integration)

Let $$\overrightarrow a $$ = $$\widehat i - \widehat j$$, $$\overrightarrow b $$ = $$\widehat i + \widehat j + \widehat k$$ and $$\overrightarrow c $$

be a vector such that $$\overrightarrow a $$ × $$\overrightarrow c $$ + $$\overrightarrow b $$ = $$\overrightarrow 0 $$

and $$\overrightarrow a $$ . $$\overrightarrow c $$ = 4, then |$$\overrightarrow c $$|² is equal to :

Let f : R $$ \to $$ R be a function defined as
$$f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x < 3} \cr {b + 5x} & ; & {3 \le x < 5} \cr {30} & ; & {x \ge 5} \cr } } \right.$$

Then, f is

5 students of a class have an average height 150 cm and variance 18 cm². A new student, whose height is 156 cm, joined them. The variance (in cm²) of the height of these six students is :