The normal to the curve
$$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point
$$\theta\, '$$ is such that

Let f be differentiable for all x. If f(1) = -2 and f'(x) $$ \ge $$ 2 for
x $$ \in \left[ {1,6} \right]$$, then

Area of the greatest rectangle that can be inscribed in the
ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \mathrm{s}$. Then the lizard will catch the insect after :