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 does each Newton-Raphson step use?

    • The chord joining two fixed endpoints
    • The horizontal line y = 0 only
    • A vertical line through the root
    • The tangent to the curve at the current approximation

  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. Which is a way that iterative methods can fail?

    • The starting value is poor or f'(x) is zero, so the iteration diverges or does not settle
    • The method never requires a starting value
    • The iteration needs f to be linear
    • The iteration always converges to the same root whatever the start

  5. What is a cobweb diagram used for?

    • To calculate a derivative
    • To find the area under a curve
    • To locate turning points of f
    • To visualise the convergence of an iteration x(n+1) = g(x(n))

  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. Newton-Raphson is applied to f(x) = x^2 - 1 starting at x0 = 0. Why does the method fail?

    • f(0) = 0, so the method stops at once
    • f'(0) = 0, so the tangent is horizontal and no next point is defined
    • x0 lies exactly on a root
    • f(x) is negative at x = 0

  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. Why is Newton-Raphson often faster than simple fixed-point iteration near a simple root?

    • It uses fewer function evaluations because it is linear
    • It needs no derivative at all
    • It always converges from any starting value
    • It converges quadratically, roughly doubling the number of correct digits each step

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