If    $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3},$$ then 'a' is equal to :

The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $$2\sqrt 2 $$ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represented by :

If   $$2\int\limits_0^1 {{{\tan }^{ - 1}}xdx = \int\limits_0^1 {{{\cot }^{ - 1}}} } \left( {1 - x + {x^2}} \right)dx,$$

then $$\int\limits_0^1 {{{\tan }^{ - 1}}} \left( {1 - x + {x^2}} \right)dx$$ is equalto :

For x $$ \in $$ R, x $$ \ne $$ 0, Let f₀(x) = $${1 \over {1 - x}}$$ and
f_n+1 (x) = f₀(f_n(x)), n = 0, 1, 2, . . . .

Then the value of f₁₀₀(3) + f₁$$\left( {{2 \over 3}} \right)$$ + f₂$$\left( {{3 \over 2}} \right)$$ is equal to :