What is the sum of the intercepts on the coordinate axes of the tangent to the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$ at the point $(x_1, y_1)$?

The tangent to the curve $y = 2x^2 - x + 1$ at a point $P$ is parallel to the line $y = 3x + 4$. The coordinates of $P$ are:

If $\theta$ denotes the acute angle between the curves $y=10-x^2$ and $y=2+x^2$,at a point of the intersection,then $|\tan \theta|$ is equal to

The acute angle between the curves $y=3x^2-2x-1$ and $y=x^3-1$ at their point of intersection which lies in the first quadrant is

The coordinates of a point on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$ where the tangent is inclined at an angle $\frac{\pi}{4}$ to the positive $X$-axis,are

If $A = \{P(\alpha, \beta) \mid \text{the tangent drawn at } P \text{ to the curve } y^3 - 3xy + 2 = 0 \text{ is a horizontal line}\}$ and $B = \{Q(a, b) \mid \text{the tangent drawn at } Q \text{ to the curve } y^3 - 3xy + 2 = 0 \text{ is a vertical line}\}$,then $n(A) + n(B) = $