If the tangent drawn to the curve $y=x^3$ at a point $(\alpha, \beta)$ cuts the curve again at another point $(\alpha_1, \beta_1)$,then $\frac{\beta_1}{\beta}=$

The line which is parallel to the $x$-axis and intersects the curve $y = \sqrt{x}$ at an angle of $45^\circ$ is given by:

The slope of the tangent to the curve $y = \int_0^x \frac{1}{1+t^3} dt$ at the point where $x = 1$ is

If the angle made by the tangent at the point $(x_{0}, y_{0})$ on the curve $x=12(t+\sin t \cos t)$,$y =12(1+\sin t )^{2}$,$0 < t < \frac{\pi}{2}$,with the positive $x$-axis is $\frac{\pi}{3}$,then $y_{0}$ is equal to

If the tangent at the point $P$ with coordinates $(h, k)$ on the curve $y^{2}=2x^{3}$ is perpendicular to the straight line $4x=3y$, then

The coordinates of the point on the curve $y=x^2-3x+2$ where the tangent is perpendicular to the straight line $y=x$ are