Let a₁, a₂, a₃, .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b₁, b₂, b₃, .... be a sequence of positive integers in geometric progression with common ratio 2. If a₁ = b₁ = c, then the number of all possible values of c, for which the equality 2(a₁ + a₂ + ... + a_n) = b₁ + b₂ + ... + b_n holds for some positive integer n, is ...........

Let f : [0, 2] $$ \to $$ R be the function defined by

$$f(x) = (3 - \sin (2\pi x))\sin \left( {\pi x - {\pi \over 4}} \right) - \sin \left( {3\pi x + {\pi \over 4}} \right)$$

If $$\alpha ,\,\beta \in [0,2]$$ are such that $$\{ x \in [0,2]:f(x) \ge 0\} = [\alpha ,\beta ]$$, then the value of $$\beta - \alpha $$ is ..........

In a triangle PQR, let a = QR, b = RP, and c = PQ. If |a| = 3, |b| = 4

and $${{a\,.(\,c - \,b)} \over {c\,.\,(a - \,b)}} = {{|a|} \over {|a| + |b|}}$$, then the value of |a $$ \times $$ b|² is ......

For a polynomial g(x) with real coefficients, let m_g denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by

$$S = \{ {({x^2} - 1)^2}({a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}):{a_0},{a_1},{a_2},{a_3} \in R\} $$;

For a polynomial f, let f' and f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (m_f' + m_f''), where f $$ \in $$ S, is ..............