Suppose a, b denote the distinct real roots of the quadratic polynomial x² + 20x $$-$$ 2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x² $$-$$ 20x + 2020. Then the value of

ac(a $$-$$ c) + ad(a $$-$$ d) + bc(b $$-$$ c) + bd(b $$-$$ d) is

If the function f : R $$ \to $$ R is defined by f(x) = |x| (x $$-$$ sin x), then which of the following statements is TRUE?

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

f(x) = e^{x $$-$$ 1} $$-$$ e^{$$-$$|x $$-$$ 1|}

and g(x) = $${1 \over 2}$$(e^{x $$-$$ 1} + e^{1 $$-$$ x}).

The the area of the region in the first quadrant bounded by the curves y = f(x), y = g(x) and x = 0 is

Let a, b and $$\lambda $$ be positive real numbers. Suppose P is an end point of the latus return of the
parabola y² = 4$$\lambda $$x, and suppose the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is