Functions Involving Parameters, Vectors, and Matrices

Which of the following equations represents a circle?

What is the standard form equation of an ellipse given by $4x^2 + 9y^2 = 36$?

Find the inverse of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

The position of a particle is given by $x(t) = \cos(t)$ and $y(t) = \sin(t)$. Find the arc length of the curve from $t = 0$ to $t = 2\pi$.

Given vectors $\mathbf{u} = \langle 1, 2 \rangle$ and $\mathbf{v} = \langle 3, -1 \rangle$, find $\mathbf{u} + \mathbf{v}$.

A particle's position is given by $\mathbf{r}(t) = \langle t^3, t^2 \rangle$. Find the velocity vector $\mathbf{v}(t)$.

What is the determinant of the matrix $\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$?

A particle's position is given by the parametric equations $x(t) = t + 1$ and $y(t) = t - 1$. What is the particle's position at $t = 1$?

Given the parametric equations $x(t) = 2t$ and $y(t) = t^2$, find $\frac{dy}{dx}$.

Parametrize the conic section $x^2 + y^2 = 16$.