The curve $y=a x^3+b x^2+c x+5$ touches the $X$-axis at $P(-2,0)$. Then,$c=$

The triangle formed by the tangent to the curve $f(x) = x^2 + bx - b$ at the point $(1, 1)$ and the coordinate axes lies in the first quadrant. If its area is $2$,then the value of $b$ is

Let $M$ and $N$ be the number of points on the curve $y^{5}-9xy+2x=0$,where the tangents to the curve are parallel to the $x$-axis and $y$-axis,respectively. Then the value of $M + N$ equals $......$

If the relation $p$ (subnormal length) $= q$ (subtangent length)$^2$ holds true for the curve $b y^2 = (x+a)^3$,then the value of $\frac{p}{q}$ is equal to

If $\theta$ denotes the acute angle between the curves $y = 10 - x^2$ and $y = 2 + x^2$ at a point of their intersection,then $|\tan \theta |$ is equal to

If the tangent to the curve $y = f(x)$ at a point is perpendicular to the line $2x - 3y = 5$,then the value of $\frac{dy}{dx}$ at that point is: