Find the equation of the normal to the curve $y = x^3$ at the point $P(1, 1)$.

If the slope of the tangent drawn at any point $(x, y)$ to the curve $y=f(x)$ is $3x^2-5$ and $f(1)=2$,then the tangent at $(1, 2)$ to the curve $y=f(x)$ intersects the curve at the point

The abscissa of the point on the curve $y=a\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$ where the tangent is parallel to the $X$-axis is

The lengths of tangent,subtangent,normal and subnormal for the curve $y=x^2+x-1$ at $(1,1)$ are $A, B, C$ and $D$ respectively,then their increasing order is

If the tangent drawn at a point $P$ on the curve $y=3x^2-5x+7$ is parallel to its chord joining the points $(1, y_1)$ and $(2, y_2)$ on it,then the $x$-coordinate of the point $P$ is

The equation of the tangent to the curve $y = 2 \cos x$ at $x = \frac{\pi}{4}$ is