Mathematics Tes...

Statement 1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P.

Statement 2: The system of parabolas y² = 4ax satisfies a differential equation of degree 1 and order 1.

If each of the lines 5x + 8y = 13 and 4x – y = 3 contains a diameter of the circle x² + y² – 2(a² – 7a + 11) x – 2 (a² – 6a + 6) y + b³ + 1 = 0, then:

Statement 1: The equation x log x = 2 – x is satisfied by at least one value of x lying between 1 and 2.

Statement 2: The function f (x) = x log x is an increasing function in [1, 2] and g(x) = 2 – x is decreasing function in [1, 2] and the graphs represented by these functions intersect at a point in [1, 2].