Lesson 9.8

9.8 Modelling damped harmonic motion Quiz: Pearson Edexcel Further Maths, Unit 9

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson 9.8, Modelling damped harmonic motion: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 9: Differential equations, written with Revision Ninja.

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The 20 questions

  1. Which equation models damped oscillation of a mass m with resistance proportional to speed, spring constant k and damping coefficient c?

    • m x'' + k x = 0
    • m x' + c x + k = 0
    • m x'' - c x' + k x = 0
    • m x'' + c x' + k x = 0

  2. In damped harmonic motion, damping resistance is directly proportional to which quantity?

    • Velocity
    • Displacement
    • Mass
    • Acceleration

  3. What type of auxiliary roots produce underdamped oscillatory motion?

    • Repeated real roots
    • Distinct real roots
    • Pure imaginary roots
    • Complex roots

  4. What type of auxiliary roots characterises overdamped harmonic motion?

    • Pure imaginary
    • Distinct negative real
    • Repeated negative real
    • Complex conjugate

  5. Which type of auxiliary equation roots corresponds to critical damping?

    • Purely imaginary roots
    • Repeated real roots
    • Two distinct real
    • Complex conjugate roots

  6. For x'' + 2x' + 2x = 0, the amplitude envelope is proportional to which expression?

    • e^(-2t)
    • e^(-t)
    • e^t
    • 1/t

  7. For x'' + 6x' + 5x = 0, what type of damping occurs?

    • Undamped
    • Underdamped
    • Critically damped
    • Overdamped

  8. What type of damping occurs in the differential equation x'' + 4x' + 4x = 0?

    • Overdamping
    • Critical damping
    • No damping
    • Underdamping

  9. For x'' + 2x' + 10x = 0, what is the angular frequency of the oscillation?

    • sqrt(10)
    • 10
    • 1
    • 3

  10. Given x'' + 2x' + 5x = 0 with x(0) = 1 and x'(0) = 0, find the constant B in x = e^(-t)(cos 2t + B sin 2t).

    • 2
    • 1
    • 1/2
    • -1/2

  11. For a lightly damped oscillator x = e^(-t)(A cos t + B sin t), how long does the amplitude take to fall to e^(-2) of its initial value?

    • 1
    • 4
    • 2
    • ln 2

  12. For x'' + 4x' + 3x = 0 with x(0) = 0 and x'(0) = 1, what is the constant A in x = A e^(-t) + B e^(-3t)?

    • -1/2
    • 1
    • 3
    • 1/2

  13. In forced damped harmonic motion, what does the particular integral represent?

    • Steady-state motion
    • Transient motion
    • Initial displacement
    • Natural frequency

  14. For x'' + 2x' + 2x = 0, what is the period of the damped oscillation?

    • 2 pi
    • 4 pi
    • pi/2
    • pi

  15. For x'' + 2x' + 10x = 0, after what time does the amplitude fall to 10% of its initial value?

    • ln 2, about 0.69
    • ln 10, about 2.30
    • 0.1
    • 10

  16. A damped oscillator has amplitude proportional to e^(-t). After what time does the amplitude halve?

    • ln 2
    • 2 ln 2
    • 1/2
    • 1

  17. What type of auxiliary equation roots produces oscillations in underdamped motion?

    • Distinct real roots
    • Complex conjugate roots
    • Zero roots
    • Repeated real roots

  18. What mathematical feature in a differential equation represents forced vibration?

    • Complex roots
    • Non-zero right-hand side
    • Zero right-hand side
    • Negative damping term

  19. What type of motion is described by the equation x'' + 5x' + 4x = 0?

    • Underdamped motion
    • Simple harmonic motion
    • Overdamped motion
    • Critically damped motion

  20. What damping occurs when the auxiliary equation discriminant is strictly positive?

    • Overdamping
    • Forced damping
    • Underdamping
    • Critical damping

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