Parametric equations in modellingEdexcel A-Level Maths: Subtopic test

1. 1
   A drone moves in a straight line at constant velocity. Relative to a fixed origin it is at the point $(2,10)$ when $t=0$ and at the point $(14,34)$ when $t=4$, where $t$ is the time in seconds and distances are in metres.

   (a)

   Which of the following gives the position of the drone at time $t$ seconds?

   [1 mark]

   • A$x=2+3t,\ y=10+6t$
   • B$x=2+12t,\ y=10+24t$
   • C$x=14+3t,\ y=34+6t$
   • D$x=3+2t,\ y=6+10t$

   (b)

   Find the speed of the drone.

   [1 mark]

   • A$9$ m s$^{-1}$
   • B$45$ m s$^{-1}$
   • C$3\sqrt5$ m s$^{-1}$
   • D$6$ m s$^{-1}$

   (c)

   The drone reaches the line $x=20$. Find the time at which this happens and the $y$-coordinate of the drone at that time.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A seat on a Ferris wheel moves so that at time $t$ minutes, $t\ge0$, its position is modelled by $x=15\sin\left(\frac{\pi t}{2}\right)$, $y=16-15\cos\left(\frac{\pi t}{2}\right)$, where $x$ is the horizontal distance in metres from the vertical line through the centre of the wheel and $y$ is the height in metres above the ground.

   (a)

   Find the greatest height of the seat above the ground.

   [1 mark]

   • A$15$ m
   • B$16$ m
   • C$30$ m
   • D$31$ m

   (b)

   How long does the seat take to complete one full revolution?

   [1 mark]

   • A$2$ minutes
   • B$4$ minutes
   • C$8$ minutes
   • D$\frac{\pi}{2}$ minutes

   (c)

   Find a Cartesian equation for the path of the seat.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball is kicked from the ground at time $t=0$ seconds. Until it returns to the ground, its position in metres is modelled by $x=12t$, $y=9t-5t^2$, where $x$ is the horizontal distance from the kick and $y$ is the height above the ground.

   (a)

   Find the time for which the ball is in the air, and the horizontal distance it travels in this time.

   [3 marks]

   (b)

   Find the greatest height reached by the ball.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two tugs, $A$ and $B$, move in straight lines at constant velocity. Relative to the harbour as origin, with distances in km and time $t$ in hours, tug $A$ is at $(0,2)$ when $t=0$ and at $(12,8)$ when $t=4$, and tug $B$ is at $(14,-5)$ when $t=0$ and at $(6,5)$ when $t=4$.

   (a)

   Find parametric equations for the positions of tug $A$ and of tug $B$. Show that the paths of the tugs cross at the point $(6,5)$, but that the tugs do not collide.

   [6 marks]

   (b)

   Find the minimum distance between the two tugs, and the time at which it occurs.

   [6 marks]

   Total for question 4: 12 marks

End of questions