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CAIE A Level Mathematics 9709

Numerical solution of equations questions

Exam-style questions on Numerical solution of equations, ordered from foundation to challenge. Pick one, write your answer the way you would in the exam, and it comes back marked against the mark scheme, point by point.

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Written by Markquill to follow the Cambridge International syllabus. Markquill is independent and is not affiliated with or endorsed by Cambridge International.

  1. 1 markFoundationSolve x^3 = 1331 for the real value of x.
  2. 2 marksFoundationThe positive root of x^2 = 26 lies between two consecutive integers. Find the lower integer N.
  3. 2 marksFoundationThe equation f(x) = x^3 - 28 = 0 has a single root between two consecutive integers.
  4. 2 marksFoundationGiven f(x) = x^3 - 1001, state the two consecutive integers between which the real root of f(x) = 0 lies.
  5. 2 marksFoundationThe equation x^3 = 68 has one real root. Find the integer N such that the root lies in [N, N+1].
  6. 2 marksFoundationGiven f(x) = x^2 - 17, state the two consecutive integers between which the positive root of f(x) = 0 lies.
  7. 2 marksStandardThe iteration xn+1 = tfrac12 (xn + (36)/(xn) ) converges to √(36). State the value it converges to.
  8. 2 marksStandardThe positive root of x^4 = 1301 lies between two consecutive integers. Find the lower integer N.
  9. 3 marksStandardUse the iterative formula xn+1 = (2xn^3 + 8)/(3xn^2) with x0 = 1 to find the cube root of 8 correct to 3 decimal places.
  10. 2 marksStandardThe iteration xn+1 = (2xn^3 + 512)/(3xn^2) converges to √[3]512. State the value it converges to.
  11. 2 marksStandardThe equation x^2 - 11x + 10 = 0 has two positive roots. State the larger root.
  12. 2 marksStandardThe root of 2^x = 7 lies between two consecutive integers. Find the lower integer N.
  13. 2 marksStandardThe equation ln x = 6 - x has exactly one root. Find the integer N such that the root lies in [N, N+1].
  14. 2 marksStandardThe equation x = 5 + (12)/(x) has a positive root. Find the integer N such that the root lies in [N, N+1].
  15. 2 marksStandardThe equation x^3 + 4x - 25 = 0 has one real root. Find the integer N such that the root lies in [N, N+1].
  16. 2 marksStandardShow that x^3 - 3x - 3 = 0 has a root between two consecutive integers, and find the lower integer N.
  17. 2 marksStandardUsing the iteration xn+1 = (1)/(2) (xn + (169)/(xn) ) with a suitable x0, the sequence converges to a limit L. Find L.
  18. 3 marksStandardThe equation of a curve is y = x^2 - 6x + 8. The x-coordinate of the point where the curve crosses the positive x-axis is given by the iterative formula xn+1 = √(6xn - 8). Use this formula, starting with x0 = 3.8, to…
  19. 2 marksStandardThe equation e^x = 2x + 2 has a root greater than 1. Find the integer N such that this root lies in [N, N+1].
  20. 2 marksStandardThe iteration xn+1 = (xn + 4)/(5) converges to a limit L. Find L.
  21. 1 markStandardThe iteration xn+1 = √(81) trivially, and more generally xn+1 = (81)/(xn) oscillates about √(81). State √(81).
  22. 4 marksChallengeThe equation x^2 - 64 = 0 has a positive root. To find this root, it is rearranged as x = (1)/(2) (x + (64)/(x) ). Using the iteration xn+1 = (1)/(2) (xn + (64)/(xn) ) with x0 = 2, find the root correct to 3 decimal…
  23. 3 marksChallengeThe iteration xn+1 = √(3xn + 4) converges to an integer limit L. Find L.