If $$\alpha $$ and $$\beta $$ be the coefficients of x⁴ and x² respectively in the expansion of
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then

Let ƒ : (1, 3) $$ \to $$ R be a function defined by
$$f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}$$ , where [x] denotes the greatest integer $$ \le $$ x. Then the range of ƒ is

Let S be the set of all functions ƒ : [0,1] $$ \to $$ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $$ \in $$ (0,1), depending on ƒ, such that

An object is gradually moving away from the focal point of a concave mirror along the axis of the mirror. The graphical representation of the magnitude of linear magnification (m) versus distance of the object from the mirror (x) is correctly given by :
(Graphs are drawn schematically and are not to scale)