A ladder 20 ft long leans against a vertical wall. The top end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is

The normal to the curve $$y = {x^2} - x + 1$$, drawn at the points with the abscissa $${x_1} = 0$$, $${x_2} = - 1$$ and $${x_3} = {5 \over 2}$$

Let, $$F(x) = {e^x},G(x) = {e^{ - x}}$$ and $$H(x) = G(F(x))$$, where x is a real variable. Then, $${{dH} \over {dx}}$$ at x = 0 is

The chord of the curve $$y = {x^2} + 2ax + b$$, joining the points where x = $$\alpha$$ and x = $$\beta$$, is parallel to the tangent to the curve at abscissa x is equal to