For the curve $y=4x^{3}-2x^{5}$,find all the points at which the tangent passes through the origin.

The values of parameter $a$ such that the line $(\log_{2}(1 + 5a - a^{2}))x - 5y - (a^{2} - 5) = 0$ is a normal to the curve $xy = 1$ may lie in the interval

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 $x$ equal to:

If the tangent to the curve $y = f(x) = x \log_{e} x$ $(x > 0)$ at a point $(c, f(c))$ is parallel to the line segment joining the points $(1, 0)$ and $(e, e)$,then $c$ is equal to:

The distance,from the origin,of the normal to the curve,$x = 2\cos t + 2t\sin t, y = 2\sin t - 2t\cos t$ at $t = \frac{\pi}{4}$,is

The tangent to the curve $y=e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$: