Lesson 1.09a

1.09a Locating and approximating roots Quiz: OCR Maths, Unit 9

20 questions

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Lesson 1.09a, Locating and approximating roots: 20 multiple choice questions for the OCR Maths (H240), Unit 9: Numerical Methods, written with Revision Ninja.

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The 20 questions

  1. If a continuous function changes sign over an interval, what must exist within it?

    • A vertical asymptote
    • Exactly one root
    • At least one root
    • No roots

  2. Why does the sign change method fail to detect a root where a curve touches the x-axis?

    • No sign change
    • Undefined function
    • Infinite gradient
    • Discontinuity

  3. For f(x) = x^3 - 2x - 5, what are f(2) and f(3)?

    • f(2) = -1 and f(3) = -16
    • f(2) = 1 and f(3) = 16
    • f(2) = 3 and f(3) = 22
    • f(2) = -1 and f(3) = 16

  4. What action is repeated in each step of the bisection method?

    • Doubling the interval
    • Halving the interval
    • Finding the gradient
    • Differentiating the function

  5. Which property must a function have on an interval for a sign change to guarantee a root?

    • Continuity
    • Periodicity
    • Symmetry
    • Differentiability

  6. For f(x) = x^3 - 4x + 1, which interval gives a sign change?

    • [0, 1]
    • [-1, 0]
    • [2, 3]
    • [3, 4]

  7. For f(x) = cos x - x in radians, what is the sign of f(1)?

    • Zero
    • Undefined
    • Negative
    • Positive

  8. Which interval contains a root of x^2 - 3 = 0, found by a sign change?

    • [2, 3]
    • [0, 1]
    • [1, 2]
    • [-1, 0]

  9. For f(x) = e^x - 4x, which interval shows a sign change?

    • [1, 2]
    • [-1, 0]
    • [0, 1]
    • [3, 4]

  10. For x^3 + x - 1 = 0, the sign test gives f(0) = -1 and f(1) = 1. After testing x = 0.5, which interval contains the root?

    • (0, 0.5)
    • (-1, 0)
    • (1, 1.5)
    • (0.5, 1)

  11. A root of f(x) = 0 lies in the interval 2.342 < x < 2.346. To 2 decimal places, what is the root?

    • 2.35
    • 2.36
    • 2.30
    • 2.34

  12. A root lies in the interval 1.31 < x < 1.34. To 1 decimal place, what is the root?

    • 1.31
    • 1.4
    • 1.2
    • 1.3

  13. What is the value of f(2) for the function f(x) = x^2 - 2x - 2?

    • 2
    • 0
    • -2
    • -1

  14. Which interval of x^2 - 5 gives a sign change?

    • [-1, 1]
    • [0, 1]
    • [2, 3]
    • [3, 4]

  15. Bisection is applied to f(x) = x^2 - 2 on [1, 2]. After testing the midpoint, which interval contains sqrt(2)?

    • [1.5, 2]
    • [1, 2.5]
    • [1.25, 1.5]
    • [1, 1.5]

  16. Bisection on x^2 - 2 has bracket [1, 1.5], with f(1) = -1 and f(1.5) = 0.25. The midpoint 1.25 gives f(1.25) = -0.4375. Which interval now contains the root?

    • [1.5, 1.75]
    • [1, 1.25]
    • [1.25, 1.5]
    • [1.25, 1.75]

  17. Why does f(x) = 1/x change sign between x = -1 and x = 1 without having a root?

    • Local minimum
    • Repeated root
    • Vertical asymptote
    • Stationary point

  18. Why does a sign change test fail for f(x) = (x - 1)^2 on the interval [0, 2]?

    • Discontinuous interval
    • Vertical asymptote
    • No sign change
    • Infinite root

  19. If f is continuous and f(a) f(b) < 0, what is the minimum number of roots in [a, b]?

    • No roots
    • Exactly one
    • At least one
    • Exactly two

  20. For f(x) = x^3 - x - 1 on [1, 1.5], f(1.25) is negative. Which bracket now contains the root?

    • [1, 1.25]
    • [1.25, 1.5]
    • [1.125, 1.25]
    • [1.375, 1.5]

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