Parametric and Cartesian formsAQA A-Level Maths: Subtopic test

1. 1
   A curve is given by the parametric equations $x=2t+1$, $y=t^2-3$, where $t$ is a real number.

   (a)

   Find the value of $y$ at the point where $x=7$.

   [1 mark]

   • A$46$
   • B$3$
   • C$-3$
   • D$6$

   (b)

   Find the Cartesian equation of the curve.

   [1 mark]

   • A$y=\dfrac{(x-1)^2}{4}-3$
   • B$y=(x-1)^2-3$
   • C$y=\dfrac{(x+1)^2}{4}-3$
   • D$y=\dfrac{(x-1)^2}{2}-3$

   (c)

   Find the coordinates of the point on the curve where $y$ is least.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ is given by the parametric equations $x=4\cos\theta$, $y=3\sin\theta$ for $0\le\theta<2\pi$.

   (a)

   Find the Cartesian equation of $C$.

   [1 mark]

   • A$x^2+y^2=25$
   • B$\dfrac{x^2}{16}+\dfrac{y^2}{9}=1$
   • C$\dfrac{x^2}{4}+\dfrac{y^2}{3}=1$
   • D$16x^2+9y^2=1$

   (b)

   Find the coordinates of the point on $C$ where $\theta=\dfrac{\pi}{3}$.

   [1 mark]

   • A$\left(2\sqrt3,\dfrac32\right)$
   • B$\left(2,\dfrac32\right)$
   • C$\left(2,\dfrac{3\sqrt3}{2}\right)$
   • D$\left(4,3\right)$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve is given by the parametric equations $x=\dfrac{2}{t}$, $y=t+1$, where $t\neq0$.

   (a)

   Find the Cartesian equation of the curve, and state any restriction on $x$.

   [3 marks]

   (b)

   Find the coordinates of the points where the curve meets the line $y=x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ is given by the parametric equations $x=1+2\cos t$, $y=3\sin t-2$ for $0\le t<2\pi$.

   (a)

   (i) Show that the Cartesian equation of $C$ is $\dfrac{(x-1)^2}{4}+\dfrac{(y+2)^2}{9}=1$.
   (ii) State the coordinates of the centre of $C$, the greatest value of $y$ and the least value of $x$.

   [6 marks]

   (b)

   Find the exact $x$-coordinates of the points where $C$ meets the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions