Parametric functions Knowledge-Based Download PDF

Questions

1. 1.

   A curve is defined parametrically by $x = t^2 - 1$ and $y = t^3 - 3t$. What are the coordinates of the self-intersection point of this curve?

   • A) $(0, 0)$
   • B) $(-1, 0)$
   • C) $(2, 0)$
   • D) $(3, 0)$
2. 2.

   A curve is defined parametrically by $x = 3\cos t$ and $y = 5\sin t$. What is the distance from the origin to the point on the curve corresponding to $t = \frac{\pi}{3}$?

   • A) $\sqrt{13}$
   • B) 4
   • C) $\sqrt{21}$
   • D) $\sqrt{34}$
3. 3.

   A particle moves in the $xy$-plane with its position given by the parametric equations $x = t + 2$ and $y = t^2 - 5$, where $t$ is a real parameter. Which point represents the intersection of the particle's path with the line $y = 3x - 1$ for $t > 0$?

   • A) $(0, -1)$
   • B) $(4, -1)$
   • C) $(7, 20)$
   • D) $(5, 14)$
4. 4.

   A curve is defined parametrically by the equations $x = t^3 - 4t$ and $y = t^2 - 1$. Which point represents the intersection of the curve with the $y$-axis for $t > 0$?

   • A) $(0, -1)$
   • B) $(-3, 0)$
   • C) $(0, 3)$
   • D) $(0, 5)$
5. 5.

   A curve is defined parametrically by the equations $x = \ln(2t)$ and $y = 4t^2$ for $t > 0$. Which of the following represents the Cartesian equation of this curve?

   • A) $y = 2e^{2x}$
   • B) $y = e^{2x}$
   • C) $y = 4e^{2x}$
   • D) $y = e^x$