If for a > 0, the feet of perpendiculars from the points A(a, $$-$$2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, $$-$$a, $$-$$1) and D respectively, then the length of line segment CD is equal to :

Let [ x ] denote greatest integer less than or equal to x. If for n$$\in$$N,

$${(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}} $$,

then $$\sum\limits_{j = 0}^{\left[ {{{3n} \over 2}} \right]} {{a_{2j}} + 4} \sum\limits_{j = 0}^{\left[ {{{3n - 1} \over 2}} \right]} {{a_{2j}} + 1} $$ is equal to :

Consider three observations a, b, and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

The range of a$$\in$$R for which the

function f(x) = (4a $$-$$ 3)(x + log_e 5) + 2(a $$-$$ 7) cot$$\left( {{x \over 2}} \right)$$ sin²$$\left( {{x \over 2}} \right)$$, x $$\ne$$ 2n$$\pi$$, n$$\in$$N has critical points, is :