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Roots of Equations · CAPE Pure Mathematics Unit 2

73 past-paper questions on Roots of Equations, part of Sequences, Series and Approximations, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise Roots of Equations
  1. 4(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Show that the equation f(x) = 0 has a real root, alpha, in the closed interval [1, 2].
  2. 4(a)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Show that alpha is the only real root of the equation f(x) = 0.
  3. 4(b)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2If x_n is the nth approximation to alpha, use the Newton-Raphson method to show that the (n + 1)th approximation x_{n+1} is given by x_{n+1} = (2x_n^3 + 6) / (3x_n^2 + 4).
  4. 4(a)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Sketch the functions y = sin x and y = x^2 on the SAME axes.
  5. 4(b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Deduce that the function f(x) = sin x - x^2 has EXACTLY two real roots.
  6. 4(c)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Find the interval in which the non-zero root alpha of f(x) lies.
  7. 4(d)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Starting with a first approximation of alpha at x_1 = 0.7, use one iteration of the Newton-Raphson method to obtain a better approximation of alpha to 3 decimal places.
  8. 1(b)(iii)a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Use your graphs to find the value of x satisfying 2^x - e^{-x} = 0.
  9. 1(b)(iii)b)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Use your graphs to find the range of values of x for which 2^x - e^{-x} < 0.
  10. 4(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Show that f(x) = 0 has a root \alpha in the interval (0, 1).
  11. 4(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2If x_1 is a first approximation to \alpha of f(x) = 0 in (0, 1), show that the Newton-Raphson method gives a second approximation x_2 in (0, 1) satisfying x_2 = \frac{3x_1^4 - 1}{4(x_1^3 - 1)}.
  12. 4(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Show that the function f(x) = x^3 - 3x + 1 has a root \alpha in the closed interval [1, 2].
  13. 4(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Use the Newton-Raphson method to show that if x_1 is a first approximation to \alpha in the interval [1, 2], then a second approximation to \alpha in the interval [1, 2] is given by x_2 = \frac{2x_1^3 - 1}{3x_1^2 - 3}.
  14. 3(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Show that the equation x^3 + 3x^2 + 6x - 3 = 0 has a root \alpha between 0 and 1.
  15. 3(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Prove that \alpha is the only real root.
  16. 3(b)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Using TWO iterations of the Newton-Raphson method, find \alpha correct to 2 decimal places.
  17. 4(b)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Show that the function f(x) = x^3 - 6x + 4 has a root x in the closed interval [0, 1].
  18. 4(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2By taking 0.6 as a first approximation of x_1 in the interval [0, 1], use the Newton-Raphson method to obtain a second approximation x_2 in the interval [0, 1].
  19. 4(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Use the intermediate value theorem to determine whether the equation f(x) has any roots in the interval [0.2, 2].
  20. 4(b)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Using x_1 = 0.6 as a first approximation of a root T of f(x), execute FOUR iterations of the Newton–Raphson method to obtain a second approximation, x_2, of T.
  21. 4(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Use the Intermediate Value Theorem to prove that x^3 = 25 has at least one root in the interval [2, 3].
  22. 4(c)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Complete the table to obtain an approximation of the root of the equation x^3 = 25 correct to 2 decimal places.
  23. 4(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Show that the function f(x) = -x^3 + 3x + 4 has a root in the interval [1, 3].
  24. 4(c)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2By taking x_1 = 2.1 as a first approximation of the root in the interval [1, 3], use the Newton–Raphson method to obtain a second approximation, x_2, in the interval [1, 3].
  25. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The equation e^x - x^4 = 0 has a root between
  26. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Let f be a continuous function with f(0) = 1 and f(0.8) = -0.76. The first approximation to the root in [0, 0.8], to three decimal places, using linear interpolation is
  27. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Given that the n^{\text{th}} approximation of the root of the equation x^5 = x^3 + 25 based on the Newton-Raphson method is x_n, then x_{n+1} may be expressed as
  28. 4(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Use the intermediate value theorem to show that f has at least one root in the interval [-2, 0].
  29. 4(b)(ii)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Use at least three iterations of the method of interval bisection to show that f(-0.538) \approx 0 in the interval [-0.7, -0.3].
  30. 4(c)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Use the Newton–Raphson method with initial estimate x_1 = 5.5 to approximate the root of g(x) = \sin 3x in the interval [5, 6], correct to two decimal places.
  31. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The equation e^x - x^4 = 0 has a root between
  32. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1By using the Newton–Raphson method with a first approximation x_n, the second approximation x_{n+1} for a root of the equation x^5 = x^3 + 25 may be expressed as
  33. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1Let f be a continuous function with f(0) = 1 and f(0.8) = -0.76. The first approximation to the root in [0, 0.8], using linear interpolation, to 3 decimal places is
  34. 4(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Show that h(x) = 0 has a root on the interval [0, 1].
  35. 4(b)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Use the iteration x_(n+1) = 1 / (x_n^2 + 1) with initial estimate x_1 = 0.7 to estimate the root of h correct to 2 decimal places.
  36. 4(c)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Use two iterations of the Newton-Raphson method with initial estimate x_1 = 1 to approximate the root of the equation g(x) = e^(4x - 3) - 4 in the interval [1, 2]. Give your answer correct to 3 decimal places.
  37. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1By using the Newton–Raphson method with a first approximation x_n, the second approximation x_{n+1} for a root of the equation x^5 = x^3 + 25 may be expressed as
  38. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The equation x^3 - x - 3 = 0 has one real positive root in the interval (n, n + 1). The value of n is
  39. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Let f be a continuous function with f(0) = 1 and f(0.8) = -0.76. The first approximation to the root in [0, 0.8], to three decimal places, using linear interpolation is
  40. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1It is given that the equation e^{0.5x} + x^2 - 3.5x = 0 has exactly one root in the interval [0, 1]. Applying the interval bisection twice, a more accurate determination of the interval containing the root is
  41. 4(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Use the intermediate value theorem to show that f(x) = \sqrt{x} - \cos x has a root in the interval [0, 1].
  42. 4(c)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Use two iterations of the interval bisection method to approximate the root of f in the interval [0, 1].
  43. 4(d)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Show that x_{n+1} = \sqrt[3]{\frac{9 - 3x_n}{2}} is an appropriate iterative formula for finding the root of f(x) = -2x^3 - 3x + 9.
  44. 4(d)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Apply the iterative formula with initial approximation x_1 = 1, to obtain a third approximation, x_3, of the root of the equation.
  45. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The equation e^x - x^4 = 0 has a root between
  46. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1A continuous function is defined by f(0) = 1 and f(0.8) = -0.76. The first approximation to the root in [0, 0.8], to 3 decimal places, using linear interpolation is
  47. 4(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Use the intermediate value theorem to show that the equation 4 cos(x) - x^3 + 2 = 0 has a root in the interval (1, 1.5).
  48. 4(c)(ii)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Use linear interpolation to approximate the value of the root of the equation 4 cos(x) - x^3 + 2 = 0 in the interval (1, 1.5), correct to two decimal places.
  49. 4(d)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2The equation 3e^x = 1 - 2 ln(x) has a root in the interval (0, 1). Taking x_1 = 0.2 as the first approximation, use the Newton-Raphson method to find a second approximation, x_2, of the root in the interval (0, 1).
  50. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The equation e^x - x^4 = 0 has a root between
  51. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The equation x^3 - x - 3 = 0 has one real positive root in the interval (n, n + 1). The value of n is
  52. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1A continuous function is defined by f(0) = 1 and f(0.8) = -0.76. \nThe first approximation to the root in [0, 0.8], to 3 decimal places, using linear interpolation is
  53. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1It is given that the equation e^{0.5x} + x^2 - 3.5x = 0 has exactly one root in the interval [0, 1]. \nApplying the interval bisection twice, a more accurate determination of the interval containing the root is
  54. 3(a)6 marks· Pure Mathematics · Unit 2 Q3 3(a)Use two iterations of the Newton-Raphson method, with the initial estimate x₀ = 0.3, to calculate a new estimate for the root of f(x) = 3x² - 4x + 1.
  55. 3(c)(i)3 marks· Pure Mathematics · Unit 2 Q3 3(c)(i)Show that h(x) = 0 has a root on the interval [0, 1].
  56. 3(c)(ii)6 marks· Pure Mathematics · Unit 2 Q3 3(c)(ii)Use the iteration x_(n+1) = (x_n² + 1) / (2x_n + 1) with initial estimate x₁ = 0.7 to estimate the root of h(x) = 0, correct to 2 decimal places.
  57. 3(d)6 marks· Pure Mathematics · Unit 2 Q3 3(d)Use the Newton-Raphson method with initial estimate x₀ = 5.5 to approximate the root of g(x) = sin 3x in the interval [5, 6], correct to 2 decimal places.
  58. 4(a)(i)3 marks· Pure Mathematics · Unit 2 Q4 4(a)(i)Use the intermediate value theorem to prove that x⁴ - 2x³ + x² + 2 = 0 has a root in the interval [-2, -1.5].
  59. 4(a)(i)3 marks· Pure Mathematics · Unit 2 Q4 4(a)(i)Use the intermediate value theorem to prove that f(x) = 2x³ + x² + 9x - 5 has at least one root in the interval [0.2, 1].
  60. 4(a)(ii)5 marks· Pure Mathematics · Unit 2 Q4 4(a)(ii)Hence, use three iterations of the interval bisection method to estimate the root of f in the interval [0.2, 1].
  61. 4(a)(ii)6 marks· Pure Mathematics · Unit 2 Q4 4(a)(ii)Hence, use two iterations of the Newton–Raphson method, with initial estimate x₁ = -1.5, to calculate a new estimate of the root of x⁴ - 2x³ + x² + 2 = 0 in the interval [-2, -1.5].
  62. 4(b)(i)4 marks· Pure Mathematics · Unit 2 Q4 4(b)(i)Show that θₙ₊₁ = sin⁻¹(2 / (8 - θₙ²)) is a suitable iteration for the approximation of the roots of the equation cosec θ = 4 - (1/2)θ².
  63. 4(b)(i)3 marks· Pure Mathematics · Unit 2 Q4 4(b)(i)Use the Intermediate Value Theorem to prove that 4eˣ + 2x² - 5 = 0 has a root in the interval [0, 1].
  64. 4(b)(ii)5 marks· Pure Mathematics · Unit 2 Q4 4(b)(ii)Hence, using θ₁ = 0 as the first approximation, calculate a new estimate for the root of cosec θ = 4 - (1/2)θ², correct to three decimal places.
  65. 4(b)(ii)6 marks· Pure Mathematics · Unit 2 Q4 4(b)(ii)Use four iterations of the interval bisection method to calculate an approximation of the root of 4eˣ + 2x² - 5 = 0 in the interval [0, 1].
  66. 4(b)(iii)5 marks· Pure Mathematics · Unit 2 Q4 4(b)(iii)Use three iterations of the linear interpolation method to approximate the root of 4eˣ + 2x² - 5 = 0 in the interval (-2, -1).
  67. 4(c)(i)3 marks· Pure Mathematics · Unit 2 Q4 4(c)(i)Using the intermediate value theorem, show that cos x = xe^(-x) has a root between x = 1 and x = 1.5.
  68. 4(c)(i)4 marks· Pure Mathematics · Unit 2 Q4 4(c)(i)Use the Intermediate Value Theorem to show that the equation 4 sin 2x + x³ – 3 = 0 has a root in the interval [0, 1].
  69. 4(c)(ii)6 marks· Pure Mathematics · Unit 2 Q4 4(c)(ii)Use three iterations of the interval bisection method to obtain an approximation of the root.
  70. 4(c)(ii)6 marks· Pure Mathematics · Unit 2 Q4 4(c)(ii)Hence, use the method of interval bisection to determine, correct to 1 decimal place, the approximate root of cos x = xe^(-x) which lies in the interval (1, 1.5).
  71. 4(d)5 marks· Pure Mathematics · Unit 2 Q4 4(d)Use the iteration x_(n+1) = (sin x_n + 2) / 3 and the initial approximation x = 1 to calculate an approximate value of the root of f(x) = sin x – 3x + 2, correct to 2 decimal places.
  72. 4(d)(i)4 marks· Pure Mathematics · Unit 2 Q4 4(d)(i)Use the Newton-Raphson formula to show that for n ≥ 1, with a given initial estimate x_n, x_(n+1) = (3x_n^4 + 13)/(4x_n^3 + 1).
  73. 4(d)(ii)3 marks· Pure Mathematics · Unit 2 Q4 4(d)(ii)Hence, or otherwise, use the Newton-Raphson method with the initial estimate x_1 = 2 to calculate, to 2 decimal places, a new estimate, x_2, of the root.

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