The equation of the normal to the curve $y = \sin x$ at the point $(0, 0)$ is

The curves $y=4x^2+2x-8$ and $y=x^3-x+13$ touch each other at the point

Let $a, b, c \in \mathbb{R}$ be such that $2a + 3b + 6c = 0$ and $g(x)$ be the antiderivative of $f(x) = ax^2 + bx + c$. If the slopes of the tangents drawn to the curve $y = g(x)$ at $(1, g(1))$ and $(2, g(2))$ are equal,then

The angle between the curves $y^2=x$ and $x^2=y$ at the point $(1,1)$ is

Let $f(x) = \begin{cases} -x^2 & \text{for } x < 0 \\ x^2 + 8 & \text{for } x \ge 0 \end{cases}$. Then the $x$-intercept of the line that is tangent to the graph of $f(x)$ is

If $\theta$ is the angle between the curves $x^2-y^2=4$ and $y^2=3x$,then $\tan \theta=$