Slope of the normal to the curve $y = x^{2} - \frac{1}{x^{2}}$ at the point $(-1, 0)$ is:

What is the length of the normal at any point on the curve $y = \frac{1}{2}a(e^{x/a} + e^{-x/a})$?

At which point is the tangent to the curve $y^3 + 3x^2 = 12y$ vertical?

If the curve $y = ax^{2} + bx + c, x \in R,$ passes through the point $(1, 2)$ and the tangent line to this curve at the origin is $y = x$,then the possible values of $a, b, c$ are:

For $h, k \in N$, let $P(h, k)$ be the point of intersection of the curves $x^2 y - x^3 = 8$ and $y^3 - x y^2 = 32$. If $\theta$ is the acute angle between these two curves at $P$, then $\tan \theta =$

If the normal to the curve $y=f(x)$ at $(1,2)$ makes an angle $\frac{3 \pi}{4}$ with the positive $X$-axis, then $f^{\prime}(1)=$