If two tangents drawn from a point P to the
parabola y² = 16(x $$-$$ 3) are at right angles, then the locus of point P is :

If the solution curve of the differential equation (2x $$-$$ 10y³)dy + ydx = 0, passes through the points (0, 1) and (2, $$\beta$$), then $$\beta$$ is a root of the equation :

Let [$$\lambda$$] be the greatest integer less than or equal to $$\lambda$$. The set of all values of $$\lambda$$ for which the system of linear equations
x + y + z = 4,
3x + 2y + 5z = 3,
9x + 4y + (28 + [$$\lambda$$])z = [$$\lambda$$] has a solution is :

The set of all values of K > $$-$$1, for which the equation $${(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0$$ has real roots, is :