For x $$ \in $$ R, x $$ \ne $$ 0, if y(x) is a differentiable function such that

x $$\int\limits_1^x y $$ (t) dt = (x + 1) $$\int\limits_1^x ty $$ (t) dt,  then y (x) equals :

(where C is a constant.)

A hyperbola whose transverse axis is along the major axis of the conic, $${{{x^2}} \over 3} + {{{y^2}} \over 4} = 4$$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $${3 \over 2},$$ then which of the following points does NOT lie on it?

The solution of the differential equation

$${{dy} \over {dx}}\, + \,{y \over 2}\,\sec x = {{\tan x} \over {2y}},\,\,$$

where 0 $$ \le $$ x < $${\pi \over 2}$$, and y (0) = 1, is given by :

ABC is a triangle in a plane with vertices

A(2, 3, 5), B(−1, 3, 2) and C($$\lambda $$, 5, $$\mu $$).

If the median through A is equally inclined to the coordinate axes, then the value of ($$\lambda $$³ + $$\mu $$³ + 5) is :