For non-negative integers n, let

$$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + 2}}\pi } \right)} \over {\sum\limits_{k = 0}^n {{{\sin }^2}\left( {{{k + 1} \over {n + 2}}\pi } \right)} }}$$

Assuming cos^$$-1$$ x takes values in [0, $$\pi $$], which of the following options is/are correct?

Let f : R $$ \to $$ R be given by

$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define

$$F(x) = \int\limits_0^x {f(t)dt} $$, x > 0

Then which of the following options is/are correct?

Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x > 0

Let x₁ < x₂ < x₃ < ... < x_n < ... be all the points of local maximum of f and y₁ < y₂ < y₃ < ... < y_n < ... be all the points of local minimum of f.

Then which of the following options is/are correct?

Three lines $${L_1}:r = \lambda \widehat i$$, $$\lambda $$ $$ \in $$ R,

$${L_2}:r = \widehat k + \mu \widehat j$$, $$\mu $$ $$ \in $$ R and

$${L_3}:r = \widehat i + \widehat j + v\widehat k$$, v $$ \in $$ R are given.

For which point(s) Q on L₂ can we find a point P on L₁ and a point R on L₃ so that P, Q and R are collinear?