If n is the number of irrational terms in the
expansion of $${\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}$$, then (n $$-$$ 1) is divisible by :

The locus of the midpoints of the chord of the circle, x² + y² = 25 which is tangent to the hyperbola, $${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$$ is :

Let the functions f : R $$ \to $$ R and g : R $$ \to $$ R be defined as :

$$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right.$$ and

$$g(x) = \left\{ {\matrix{ {{x^3},} & {x < 1} \cr {3x - 2,} & {x \ge 1} \cr } } \right.$$

Then, the number of points in R where (fog) (x) is NOT differentiable is equal to :

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 :