Parametric equations in modellingAQA A-Level Maths: Subtopic test

1. 1
   A ball is kicked from level ground. At time $t$ seconds after the kick, its horizontal distance from the starting point is $x=12t$ metres and its height is $y=16t-5t^2$ metres. The model applies until the ball lands.

   (a)

   Find the time the ball is in the air.

   [1 mark]

   • A$1.6$ s
   • B$3.2$ s
   • C$\dfrac{5}{16}$ s
   • D$16$ s

   (b)

   Find the horizontal distance travelled by the ball before it lands.

   [1 mark]

   • A$38.4$ m
   • B$19.2$ m
   • C$51.2$ m
   • D$12$ m

   (c)

   Find the greatest height reached by the ball.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A Ferris wheel has its centre $12$ m above the ground. The position of a passenger pod is modelled by $x=10\sin(0.2t)$, $y=12-10\cos(0.2t)$, where $x$ is the horizontal distance in metres from the vertical line through the centre, $y$ is the height above the ground in metres and $t$ is the time in seconds after the pod passes its lowest point. The angle $0.2t$ is in radians.

   (a)

   Find the height of the pod above the ground when $t=0$.

   [1 mark]

   • A$12$ m
   • B$22$ m
   • C$10$ m
   • D$2$ m

   (b)

   Find the time taken for the pod to make one complete revolution.

   [1 mark]

   • A$5\pi$ s
   • B$0.4\pi$ s
   • C$10\pi$ s
   • D$2\pi$ s

   (c)

   State the greatest height of the pod above the ground, and justify your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A runner follows an oval track. Relative to the centre of the track, the runner's position at time $t$ seconds is $x=80\cos\theta$, $y=50\sin\theta$, where $\theta=0.1t$ radians and $x$ and $y$ are in metres.

   (a)

   Show that the runner's path has the Cartesian equation $\dfrac{x^2}{6400}+\dfrac{y^2}{2500}=1$.

   [3 marks]

   (b)

   The runner starts at $t=0$. Find the first time at which $x=40$. Give your answer in exact form and to $3$ significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two model aircraft fly in a vertical plane for $0\le t\le10$, where $t$ is the time in seconds. At time $t$, aircraft $A$ is at $x=5t$, $y=20-2t$ and aircraft $B$ is at $x=10+4t$, $y=3t-7$, where $x$ is the horizontal distance and $y$ is the height, both in metres.

   (a)

   Find the Cartesian equations of the paths of $A$ and $B$, and hence find the coordinates of the point where the two paths cross.

   [6 marks]

   (b)

   Show that the aircraft do not collide at the point where their paths cross, and comment on whether the model predicts a safe separation.

   [6 marks]

   Total for question 4: 12 marks

End of questions