Lesson I2

I2 Iterative methods and Newton-Raphson Quiz: AQA Maths, Unit 9

20 questions

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Lesson I2, Iterative methods and Newton-Raphson: 20 multiple choice questions for the AQA Maths (7357), Unit 9: Numerical methods, written with Revision Ninja.

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The 20 questions

  1. What is the Newton-Raphson iteration for solving f(x) = 0?

    • x(n+1) = f(x(n))/f'(x(n))
    • x(n+1) = x(n) - f(x(n))/f'(x(n))
    • x(n+1) = x(n) - f'(x(n))/f(x(n))
    • x(n+1) = x(n) + f(x(n))/f'(x(n))

  2. Geometrically, what line is used at each step in the Newton-Raphson method?

    • Secant line
    • Asymptote
    • Normal line
    • Tangent line

  3. A fixed-point iteration x(n+1) = g(x(n)) converges to a value alpha. Which equation does alpha satisfy?

    • x = g(x)
    • g(x) = 0
    • g'(x) = 0
    • x^2 = g(x)

  4. At what type of point does the Newton-Raphson method fail due to division by zero?

    • Stationary point
    • Root of function
    • Discontinuity
    • Point of inflection

  5. What line is y = g(x) plotted against in a cobweb diagram?

    • y = -x
    • y = 0
    • x = 0
    • y = x

  6. f(x) = x^2 - 2 with x0 = 1 is used in Newton-Raphson. What is x1?

    • 1.4
    • 1.25
    • 2
    • 1.5

  7. Using the same f(x) = x^2 - 2 and x0 = 1, what is x2 to 4 decimal places?

    • 1.4167
    • 1.4286
    • 1.4
    • 1.4333

  8. For fixed-point iteration x(n+1) = cos(x(n)) starting from x0 = 0.5, what is x1 to 4 decimal places?

    • 1.0000
    • 0.8776
    • 0.4794
    • 0.5403

  9. Newton-Raphson is applied to f(x) = x^3 - 5 starting at x0 = 2. What is x1?

    • 2.25
    • 1.8
    • 1.75
    • 1.5

  10. Newton-Raphson is applied to f(x) = ln x + x - 2 starting at x0 = 2. What is x1 to 3 decimal places?

    • 1.5
    • 2.462
    • 1.538
    • 1.462

  11. Which rearrangement gives a fixed-point iteration for solving x^3 + x - 3 = 0?

    • x = 3 + x^3
    • x = 3 - x^3
    • x = (3 - x)^(1/3)
    • x = (3 + x)^(1/3)

  12. A sequence is defined by x(n+1) = 1 + 1/x(n) with x0 = 1. What is x2?

    • 1.5
    • 1.25
    • 2
    • 1.6667

  13. Newton-Raphson is applied to f(x) = x^2 - 7 starting at x0 = 3. What is x1 to 3 decimal places?

    • 2.667
    • 2.5
    • 2.333
    • 3.5

  14. Applying Newton-Raphson to f(x) = x^2 - 1 with x0 = 0 fails because f'(0) equals what?

    • 1
    • 0
    • Undefined
    • -1

  15. Which iteration arises from Newton-Raphson applied to x^2 - 2 = 0?

    • x(n+1) = 2x(n) - 2
    • x(n+1) = (x(n) + 2/x(n))/2
    • x(n+1) = (x(n) - 2)/x(n)
    • x(n+1) = x(n)^2 + 2

  16. Newton-Raphson is applied to f(x) = x^3 - x - 1 starting at x0 = 1. What is x1?

    • 1.5
    • 1.25
    • 0.5
    • 2

  17. For fixed-point iteration x(n+1) = g(x(n)) to converge near a root alpha, which condition is needed?

    • g'(alpha) > 1
    • g(alpha) = 1
    • |g'(alpha)| < 1
    • g'(alpha) = 0 exactly

  18. For x(n+1) = sqrt(x(n) + 2) with x0 = 1, what is x2 to 3 decimal places?

    • 1.732
    • 2.000
    • 1.414
    • 1.932

  19. Newton-Raphson is applied to f(x) = x^3 - 2x - 5 starting at x0 = 2. What is x1?

    • 2.5
    • 1.9
    • 2.05
    • 2.1

  20. What order of convergence does the Newton-Raphson method typically have near a simple root?

    • Cubic
    • Linear
    • Exponential
    • Quadratic

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