9.1.1 Locating roots by change of sign Quiz: Pearson Edexcel Maths, Unit 9

The 20 questions

1. What mathematical principle underpins the location of roots using a change of sign?

   • Differentiable functions
   • Continuous functions
   • Periodic functions
   • Linear functions

2. If f(a) and f(b) have opposite signs for continuous f, what exists between a and b?

   • At least one root
   • An asymptote
   • Exactly two roots
   • No roots

3. What does f(a) * f(b) < 0 indicate about a root between a and b?

   • Function is undefined
   • Root exists
   • Root is zero
   • No root exists

4. Which theorem formally justifies the location of roots by sign change?

   • Intermediate value theorem
   • Rolle's theorem
   • Mean value theorem
   • Fundamental theorem

5. If f(2) = -3 and f(3) = 4, what is the most likely conclusion?

   • Two roots exist
   • No root exists
   • Root at 2.5
   • Root between 2 and 3

6. Why might a sign change interval fail to detect roots?

   • Continuous function
   • Odd root count
   • Steep gradient
   • Even root count

7. Which type of root cannot be located using a simple sign change method?

   • Negative root
   • Rational root
   • Tangent root
   • Simple root

8. What type of root cannot be located using a simple sign change test?

   • Integer root
   • Repeated root
   • Negative root
   • Irrational root

9. Evaluate f(x) = x^3 - 5x - 1 at x = 2 to test for a root.

   • 1
   • -3
   • -5
   • 3

10. For f(x) = x^2 - 3, test x = 1 to find the function value.

    • -4
    • 2
    • -2
    • 0

11. Given f(1) = -2 and f(2) = 1 for f(x) = x^3 - 2x - 1, locate the root.

    • Between 1 and 2
    • Between 0 and 1
    • At exactly 1.5
    • Between 2 and 3

12. If f(0) = -1 and f(1) = 2, what can be stated about the interval [0, 1]?

    • Contains no roots
    • Contains a root
    • Is undefined
    • Contains two roots

13. Test f(x) = e^x - 3x at x = 1 to check the sign.

    • Negative
    • Positive
    • Zero
    • Undefined

14. Using f(1) = -2 and f(2) = 3, what is the first midpoint to test?

    • 1.2
    • 1.5
    • 1.8
    • 2.5

15. If f(1) = -1 and f(1.5) = 0.5, where is the root now bracketed?

    • At 1.5
    • Between 1 and 1.5
    • Between 1.5 and 2
    • Between 1 and 2

16. Why might f(x) = 1/x appear to change sign across x = 0 without a root?

    • Local maximum
    • Inflection point
    • Stationary point
    • Vertical asymptote

17. Locate the integer interval for the root of x^3 - 3x - 1 = 0 given f(1)=-3 and f(2)=1.

    • Between 0 and 1
    • Between 2 and 3
    • Between -1 and 0
    • Between 1 and 2

18. What does a sign change method help establish before using iterative formulas?

    • Starting interval
    • Derivative value
    • Maximum turning point
    • Exact root

19. If f(x) = ln(x) - 2, evaluate f(e^2).

    • 1
    • 0
    • e^2
    • 2

20. What interval contains the root for f(x) = x^2 - 5 given f(2) = -1 and f(3) = 4?

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

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