I9-I11 Coupled equations and damped motion models Quiz: AQA Further Maths, Unit 2

The 20 questions

1. Which pair of equations is a predator-prey model with one independent and two dependent variables?

   • dx/dt = a x - b x y and dy/dt = -c y + d x y
   • dx/dt = a x + b y and dy/dt = c x
   • dx/dt = a x y and dy/dt = c
   • d^2x/dt^2 = a x and dy/dt = c

2. What is the non-zero equilibrium of dx/dt = a x - b x y, dy/dt = -c y + d x y?

   • (0, 0) only
   • (a/c, b/d)
   • (c/d, a/b)
   • (a/b, c/d)

3. Coupled first order equations involve which of the following?

   • Two dependent variables, their first derivatives and one independent variable
   • Two independent variables
   • No derivatives at all
   • One dependent variable and a second derivative

4. If dv/dt = -(k/m) v, with v(0) = v0, what is v(t)?

   • v0 e^(kt/m)
   • v0 e^(-mt/k)
   • v0 - kt/m
   • v0 e^(-kt/m)

5. For motion under gravity with linear resistance, m dv/dt = mg - kv, what is the terminal velocity?

   • k/(mg)
   • mk/g
   • mg/k
   • g/k

6. Using Newton's second law with T = kx, which differential equation describes the motion of a mass m?

   • x'' = -k m x
   • m x'' = k x
   • m x'' = -k x
   • m x' = -k x

7. For dx/dt = 2x - xy and dy/dt = -y + xy, what is the non-zero equilibrium?

   • (1, 2)
   • (0, 0)
   • (2, 2)
   • (2, 1)

8. For dx/dt = 3x - 2xy and dy/dt = -y + xy, what is the non-zero equilibrium?

   • (1, 3/2)
   • (3/2, 1)
   • (1, 3)
   • (3, 2)

9. A body of mass 2 kg has resistance -4v and starts with speed 10 m/s. What is v(t)?

   • 10 e^(-2t)
   • 10 e^(-4t)
   • 10 e^(2t)
   • 5 e^(-2t)

10. A body of mass 0.5 kg falls with weight mg, g = 10 m/s^2, and linear resistance k = 2 N s/m. What is the terminal velocity in m/s?

    • 25
    • 0.1
    • 2.5
    • 10

11. Which pair of functions solves dx/dt = y and dy/dt = -x with x(0) = 1 and y(0) = 0?

    • x = sin(t), y = cos(t)
    • x = cos(t), y = sin(t)
    • x = e^t, y = e^(-t)
    • x = cos(t), y = -sin(t)

12. If dx/dt = 2x and x(0) = 5, what is x(t)?

    • 5 e^(-2t)
    • 10 e^t
    • 5 e^(2t)
    • 5 + 2t

13. What is the non-zero equilibrium of dx/dt = x(1 - y), dy/dt = y(x - 2)?

    • (0, 2)
    • (1, 2)
    • (1, 1)
    • (2, 1)

14. A mass m = 1 kg is on a spring with k = 9 N/m. What is the period of the motion?

    • 3 pi/2
    • 2 pi/3
    • pi/3
    • 2 pi

15. A body has m = 4 kg and resistance coefficient k = 2 in m dv/dt = -kv. What is the time constant m/k?

    • 2 s
    • 6 s
    • 0.5 s
    • 8 s

16. Solve m dv/dt = mg - kv with v(0) = 0.

    • v = (mg/k)(1 - e^(kt/m))
    • v = mg(1 - e^(-kt/m))
    • v = (mg/k) e^(-kt/m)
    • v = (mg/k)(1 - e^(-kt/m))

17. For dx/dt = 3x - 2xy, what is x(t) for all t if x(0) = 0?

    • 0
    • 3t
    • e^(3t)
    • 1

18. For a falling body with terminal velocity 20 m/s and k/m = 0.5 per second, what is g?

    • 2
    • 0.5
    • 10
    • 40

19. A particle has velocity v = 10 e^(-2t) m/s. What is the total distance travelled from t = 0 to infinity?

    • 10 m
    • 2.5 m
    • 5 m
    • 20 m

20. For the coupled system dx/dt = y and dy/dt = -4x, which second order equation does x satisfy?

    • x'' + 4x = 0
    • x'' + 2x = 0
    • x'' - 4x = 0
    • x' + 4x = 0

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