Let $$f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R$$. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ____________.

Let l₁ be the line in xy-plane with x and y intercepts $${1 \over 8}$$ and $${1 \over {4\sqrt 2 }}$$ respectively, and l₂ be the line in zx-plane with x and z intercepts $$ - {1 \over 8}$$ and $$ - {1 \over {6\sqrt 3 }}$$ respectively. If d is the shortest distance between the line l₁ and l₂, then d^$$-$$2 is equal to _______________.

Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Two identical balls A and B thrown with same velocity 'u' at two different angles with horizontal attained the same range R. IF A and B reached the maximum height h₁ and h₂ respectively, then $$R = 4\sqrt {{h_1}{h_2}} $$

Reason R : Product of said heights.

$${h_1}{h_2} = \left( {{{{u^2}{{\sin }^2}\theta } \over {2g}}} \right)\,.\,\left( {{{{u^2}{{\cos }^2}\theta } \over {2g}}} \right)$$

Choose the correct answer :

Two buses P and Q start from a point at the same time and move in a straight line and their positions are represented by $${X_P}(t) = \alpha t + \beta {t^2}$$ and $${X_Q}(t) = ft - {t^2}$$. At what time, both the buses have same velocity?