A20 find approximate solutions to equations numerically using iteration Quiz: Pearson Edexcel Maths, Unit 2

The 20 questions

1. How does numerical iteration find solutions to an equation?

   • Factorisation
   • Repeated formula
   • Graphical sketching
   • Expansion

2. What indicates that a root of a function lies between two values?

   • Sign change
   • Zero gradient
   • Equal values
   • Maximum point

3. The starting value used in an iterative method is called

   • the gradient
   • the initial estimate
   • the final answer
   • the turning point

4. What does a sign change between f(a) and f(b) indicate?

   • minimum point
   • root exists
   • y-intercept
   • maximum point

5. Before using an iterative method, what form must an equation be rearranged into?

   • ax + b = 0
   • y = mx + c
   • x^2 = c
   • x = g(x)

6. What does an iterative numerical method primarily approximate?

   • turning point
   • gradient
   • root
   • y-intercept

7. How do you improve the accuracy of an iterative approximation?

   • more iterations
   • larger steps
   • rounding earlier
   • fewer iterations

8. Iteration is a method for finding

   • gradients of curves
   • exact algebraic solutions
   • approximate solutions numerically
   • equations of tangents

9. For f(x) = x^2 - 5, f(2) < 0 and f(3) > 0. A root lies

   • between 2 and 3
   • between 3 and 4
   • between 1 and 2
   • at x = 5

10. Using x_(n+1) = 1 + 1/x_n with x_0 = 1, the value of x_2 is

    • 1.5
    • 2
    • 1
    • 1.6667

11. Using x_(n+1) = 2 + 1/x_n with x_0 = 2, the value of x_1 is

    • 3
    • 2.5
    • 2.25
    • 2.4

12. Using x_(n+1) = 6/(x_n + 1) with x_0 = 2, the value of x_1 is

    • 4
    • 1.5
    • 2
    • 3

13. If f(2) = -1 and f(3) = 16, what are the respective signs of f(2) and f(3)?

    • both negative
    • negative then positive
    • positive then negative
    • both positive

14. Rearranging x^3 + 4x - 10 = 0 as x = (10 - x^3) / 4 and starting at x_0 = 1, x_1 is

    • 2.5
    • 2.25
    • 1.5
    • 2.75

15. For a cubic with f(2) < 0 and f(2.5) > 0, the next interval to test is

    • 2 to 2.5
    • 1 to 2
    • 3 to 3.5
    • 2.5 to 3

16. Successive iterates are 1.4, 1.38, 1.38 to 2 decimal places. The root is best quoted as

    • 1.3
    • 1.4
    • 1.5
    • 1.38

17. For f(x) = x^3 + x - 4, f(1) < 0 and f(1.5) > 0. Which interval must contain a root?

    • between 1 and 1.5
    • between 0 and 1
    • between 1.5 and 2
    • between 2 and 3

18. Using x_(n+1) = (x_n + 3/x_n) / 2 with x_0 = 2, the value of x_1 is

    • 2.5
    • 1.75
    • 2
    • 1.5

19. For f(x) = x^3 - 4x - 1, f(2.1) < 0 and f(2.2) > 0. The root to 1 decimal place is

    • 2.0
    • 2.2
    • 2.1
    • 2.3

20. Using x_(n+1) = 10 / (x_n + 1) with x_0 = 3, the value of x_2 to 2 decimal places is

    • 2.9
    • 2.86
    • 2.85
    • 2.5

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