Functions Involving Parameters, Vectors, and Matrices

For what parameter 'k' in $k \sin(\theta) \cos(\theta) = \sin(\theta) - k$ does $\theta$ have exactly two distinct solutions on the interval $[0, \pi]$?

Given that an investment's value over time follows the rational function modeled by $V(t) = \frac{1000}{1+9e^{-0.6t}}$, where $V(t)$ represents dollar value and $t$ years since initial investment, when will the investment first exceed $900$?

Given the rectangular hyperbola $xy=c^2$, what is an expression for ‘x’ when ‘y’ is parametized such that ‘y=t’?

Which set represents a simple parametrization for the line $y=x$?

How Might One Rewrite The Locus Of Points Equidistant From Point $(0,-a)$ And Line $Y=a$ Using A Parameter $T$?

If the position of an object is given by the implicit equation $xy + \ln\left(\frac{y}{x}\right)=3$ where $x >0$ and $y>0$, which set of parametric equations best describes its motion if it follows the curve at constant horizontal velocity?

Given the equation $x = a \cos(t)$ where $t$ ranges between $[0, \pi]$, how would you express it in terms of a variable $U$ defined as $U = t^3$?

If we parametrize the circle $x^2 + y^2 = 25$ using the parameter $t$ for angle measure in radians, which set of equations correctly represents the $x$ and $y$ coordinates in terms of $t$?

Given an ellipse defined by the equation $\frac{x^2}{9} + \frac{y^2}{16} = 1$, which parametric equations correctly describe this ellipse when parameterized by angle measure $t$?

If the function $f(x) = \frac{2x^3 - 8}{x - 2}$ is redefined at $x = 2$ to make it continuous, what should $f(2)$ equal?