At which points on the curve $y = x^3 - 3x^2 - 9x + 7$ is the tangent parallel to the $x$-axis?

If the line $y=-4x+b$ is tangent to the curve $y=\frac{1}{x}$,then $b$ equals

Tangents are drawn to the curve $y = \sin x$ from the origin. The locus of the points of contact is

$A$ curve is represented by the equations $x = \sec^2 t$ and $y = \cot t$,where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t = \pi/4$ meets the curve again at the point $Q$,then the $x$-coordinate of $Q$ is equal to

If a tangent to the curve $y = 6x - x^2$ is parallel to the line $4x - 2y - 1 = 0$,then the point of tangency on the curve is

The sum of the intercepts made by a tangent drawn to the curve $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$ at the point $(a, b)$ on the coordinate axes is: