For the curve $4x^{5} = 5y^{4}$,the ratio of the cube of the subtangent at a point on the curve to the square of the subnormal at the same point is

The curve $y=ax^3+bx^2+cx+5$ touches the $X$-axis at $(-2,0)$ and cuts the $Y$-axis at a point $Q$ where its gradient is $3$. Then the values of $a, b, c$ respectively are:

If the locus of the points on the curve $x^3 y^2+\frac{x^2}{y}=5$ at which the tangent is parallel to the $X$-axis is $f(x, y)=0$,then the point that lies on this curve $f(x, y)=0$ is

Find the slope of the tangent to the curve $y = x^{3} - 3x + 2$ at the point whose $x$-coordinate is $3$.

The equation of the tangent to the curve $y = x + \frac{4}{x^{2}}$ that is parallel to the $x$-axis is:

If $P$ and $Q$ are two different points on the curve $y = x^3 - x$ such that the tangent at $P$ intersects the curve again at $Q$,then $\frac{m_{OQ} + 1}{m_{OP} + 1}$ is equal to,where $O$ is the origin and $m_{AB}$ represents the slope of the line segment $AB$.