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 $f^{\prime}(1)=$

The tangent to the curve $y=x^3+ax-b$ at the point $(1,-5)$ is perpendicular to the line $y-x+4=0$. Which one of the following points lies on the curve?

Prove that the curves $x=y^{2}$ and $xy=k$ cut at right angles if $8k^{2}=1$.

An angle between the curves $x^2 y = 1$ and $y(x^2 + 1) = 2$ is

The value of $c$ such that the straight line joining the points $(0,3)$ and $(5,-2)$ is tangent to the curve $y=\frac{c}{x+1}$ is

The area (in square units) of the triangle formed by the $X$-axis,the tangent,and the normal drawn at $(1, 1)$ to the curve $x^3 + y^3 = 2xy$ is