What is the equation of the normal to the curve $y = x \log x$ which is parallel to the line $2x - 2y + 3 = 0$?

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

Let $f(x)$ be a differentiable function,$A(0, \alpha)$ and $B(8, \beta)$ be two points on the curve $y=f(x)$. Given $f(0)=2$ and $f^{\prime}(4)=\frac{-3}{4}$. If the chord $AB$ of the curve is parallel to the tangent drawn at the point $(4, f(4))$,then $\beta=$

If the tangent of the curve $4y^3 = 3ax^2 + x^3$ drawn at the point $(a, a)$ forms a triangle of area $\frac{25}{24}$ sq. units with the coordinate axes,then $a =$

The angle between the tangents to the curve $y = x^2 - 5x + 6$ at the points $(2, 0)$ and $(3, 0)$ is:

Find the slope of the tangent to the curve $y=\frac{x-1}{x-2}, x \neq 2$ at $x=10$.