Parametric equationsEdexcel International A Level Maths: Subtopic test

1. 1
   A curve $C$ has parametric equations $x=3t-1$, $y=t^2+2$, where $t$ is a real parameter.

   (a)

   Find the coordinates of the point on $C$ where $t=-1$.

   [1 mark]

   • A$(-4,1)$
   • B$(-3,3)$
   • C$(-2,3)$
   • D$(-4,3)$

   (b)

   Which is a Cartesian equation of $C$?

   [1 mark]

   • A$y=\frac{(x+1)^2}{9}+2$
   • B$y=\frac{(x+1)^2}{3}+2$
   • C$y=\frac{(x-1)^2}{9}+2$
   • D$y=9(x+1)^2+2$

   (c)

   Find the coordinates of the points where $C$ meets the line $y=11$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $D$ has parametric equations $x=3\sin t$, $y=2\cos t$, for $0\leq t<2\pi$.

   (a)

   Find the coordinates of the point on $D$ where $t=\frac{\pi}{6}$.

   [1 mark]

   • A$\left(\sqrt3,\frac32\right)$
   • B$\left(\frac{3\sqrt3}{2},1\right)$
   • C$\left(\frac32,\sqrt3\right)$
   • D$\left(\frac32,1\right)$

   (b)

   Which is a Cartesian equation of $D$?

   [1 mark]

   • A$\frac{x^2}{3}+\frac{y^2}{2}=1$
   • B$\frac{x^2}{9}+\frac{y^2}{4}=1$
   • C$9x^2+4y^2=1$
   • D$x^2+y^2=13$

   (c)

   Find the coordinates of the points where $D$ meets the $y$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball is thrown so that, $t$ seconds later, its horizontal distance is $x=20t$ metres and its height is $y=15t-5t^2$ metres, for $t\geq0$ until it lands.

   (a)

   Find a Cartesian equation of the path of the ball, in the form $y=f(x)$.

   [3 marks]

   (b)

   Find the horizontal distance the ball travels before it lands, and the greatest height it reaches.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Parametric equations describe curves in the $(x,y)$ plane using a third variable, called the parameter.

   (a)

   The curve $C$ has parametric equations $x=2+3\sin\theta$, $y=1+3\cos\theta$, for $0\leq\theta<2\pi$.
   (i) Show that $C$ has Cartesian equation $(x-2)^2+(y-1)^2=9$.
   (ii) The line $y=x-4$ meets $C$ at two points. Find their coordinates.

   [6 marks]

   (b)

   The curve $H$ has parametric equations $x=2t$, $y=\frac{2}{t}$, for $t\neq0$.
   (i) Find a Cartesian equation of $H$.
   (ii) The line $x+y=5$ meets $H$ at the points $P$ and $Q$. Find the value of $t$ at each point, and the coordinates of $P$ and $Q$.

   [6 marks]

   Total for question 4: 12 marks

End of questions