The set of all values of $$a$$ for which $$\mathop {\lim }\limits_{x \to a} ([x - 5] - [2x + 2]) = 0$$, where [$$\alpha$$] denotes the greatest integer less than or equal to $$\alpha$$ is equal to

The locus of the mid points of the chords of the circle $${C_1}:{(x - 4)^2} + {(y - 5)^2} = 4$$ which subtend an angle $${\theta _i}$$ at the centre of the circle $$C_1$$, is a circle of radius $$r_i$$. If $${\theta _1} = {\pi \over 3},{\theta _3} = {{2\pi } \over 3}$$ and $$r_1^2 = r_2^2 + r_3^2$$, then $${\theta _2}$$ is equal to :

Let $$y=y(x)$$ be the solution of the differential equation $$(x^2-3y^2)dx+3xy~dy=0,y(1)=1$$. Then $$6y^2(e)$$ is equal to

If $$f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}$$, then $$f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \,...\, + \,f\left( {{{2022} \over {2023}}} \right)$$ is equal to