The number of those tangents to the curve $y^2 - 2x^3 - 4y + 8 = 0$ which pass through the point $(1, 2)$ is

If the angle between the curves $y=e^{2(1+x)-4}$ and $x^2 y=1$ at the point $(1,1)$ is $\theta$,then $|\sin \theta|+|\cos \theta|=$

Let $C$ be a curve given by $y(x)=1+\sqrt{4x-3}$,$x>\frac{3}{4}$. If $P$ is a point on $C$ such that the tangent at $P$ has a slope of $\frac{2}{3}$,then a point through which the normal at $P$ passes is:

$A(1, -3)$ and $B(4, 3)$ are two points on the curve $y = x - \frac{4}{x}$. The points on the curve,the tangents at which are parallel to the chord $AB$,are

The curve given by $x + y = e^{xy}$ has a tangent parallel to the $y-$ axis at the point

If the tangent drawn at the point $(x_1, y_1)$,where $x_1, y_1 \in \mathbb{N}$,on the curve $y = x^4 - 2x^3 + x^2 + 5x$ passes through the origin,then $x_1 + y_1 =$