For any positive integer n, define

$${f_n}:(0,\infty ) \to R$$ as

$${f_n} = \sum\limits_{j = 1}^n {{{\tan }^{ - 1}}} \left( {{1 \over {1 + (x + j)(x + j - 1)}}} \right)$$

for all x$$ \in $$(0, $$\infty $$). (Here, the inverse trigonometric function tan^$$-$$1 x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$). Then, which of the following statement(s) is (are) TRUE?

Let T be the line passing through the points P($$-$$2, 7) and Q(2, $$-$$5). Let F₁ be the set of al pairs of circles (S₁, S₂) such that T is tangent to S₁ at P and tangent to S₂ at Q, and also such that S₁ and S₂ touch each other at a point, say M. Let E₁ be the set representing the locus of M as the pair (S₁, S₂) varies in F₁. Let the set of all straight line segments joining a pair of distinct points of E₁ and passing through the point R(1, 1) be F₂. Let E₂ be the set of the mid-points of the line segments in the set F₂. Then, which of the following statement(s) is (are) TRUE?

Let S be the set of all column matrices $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$ such that $${b_1},{b_2},{b_3} \in R$$ and the system of equations (in real variables)

$$\eqalign{ & - x + 2y + 5z = {b_1} \cr & 2x - 4y + 3z = {b_2} \cr & x - 2y + 2z = {b_3} \cr} $$

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$$$ \in $$S?

Consider two straight lines, each of which is tangent to both the circle x² + y² = (1/2) and the parabola y² = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is $$\sqrt 2 $$, then which of the following statement(s) is (are) TRUE?