If $$\beta $$ is one of the angles between the normals to the ellipse, x² + 3y² = 9 at the points (3 cos $$\theta $$, $$\sqrt 3 \sin \theta $$) and ($$-$$ 3 sin $$\theta $$, $$\sqrt 3 \,\cos \theta $$); $$\theta \in \left( {0,{\pi \over 2}} \right);$$ then $${{2\,\cot \beta } \over {\sin 2\theta }}$$ is equal to :

The function f defined by

f(x) = x³ $$-$$ 3x² + 5x + 7 , is :

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :

The tangent at the point (2, $$-$$2) to the curve, x²y² $$-$$ 2x = 4(1 $$-$$ y) does not pass through the point :