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Numerical methods: roots, iteration and Newton–Raphson

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 46

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 21 parts, 46 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q9(c) · 3 marks
Using the iteration formula ; tn+1 = 5 ln with t1 = 8 tn 1 : (c) (i) find the value of t3 to 3 decimal places, (ii) find, by repeated iteration, the time taken for the car to reach maximum speed. (3)
2. June 2024 · Paper 1 · Q3(a) · 2 marks
pi : 3. f(x) = x x pi x < < tan 2 Given that the equation f(x) = 0 has a single root alpha (a) show that alpha lies in the interval [3.6, 3.7] (2)
3. June 2024 · Paper 1 · Q3(c) · 2 marks
(c) Using 3.7 as a first approximation for alpha , apply the Newton–Raphson method once to obtain a second approximation for alpha . Give your answer to 3 decimal places. (2)
4. June 2024 · Paper 2 · Q6(c) · 3 marks
The iterative formula x ln x n 1 O n is used with x1 = 0.6 to find an approximate value for alpha (c) Calculate, giving each answer to 4 decimal places, (i) the value of x2 (ii) the value of alpha (3)
5. June 2023 · Paper 1 · Q15(c) · 1 marks
x e The equation x has two positive roots alpha and beta where beta > alpha x e A student uses the iteration formula x n e xn 1 O x n e in an attempt to find approximations for alpha and beta x e Diagram 1 shows a plot of part of the curve with equation y and part of the x e line with equation y = x Using Diagram 1 on page 42 (c) draw a staircase diagram to show that the iteration formula starting with x1 = 1 can be used to find an approximation for beta (1)
6. June 2023 · Paper 1 · Q15(d) · 3 marks
Use the iteration formula with x1 = 1, to find, to 3 decimal places, (d) (i) the value of x2 (ii) the value of beta (3)
7. June 2023 · Paper 1 · Q15(e) · 2 marks
Using a suitable interval and a suitable function that should be stated (e) show that alpha = 0.432 to 3 decimal places. (2)
8. June 2022 · Paper 1 · Q8(c) · 3 marks
Using the iteration formula tn +1 = - 1 + ln( tn + ) with t1 = 7 (c) (i) find the value of t3 to 3 decimal places, (ii) find, by repeated iteration, the time taken for the car to reach maximum speed. (3)
9. June 2022 · Paper 2 · Q6(b) · 1 marks
The curve crosses the x-axis at x = alpha , as shown in Figure 2. Given that, to 3 decimal places, f(4) = 4.274 and f(5) = -1.212 (b) explain why alpha must lie in the interval [4, 5] (1)
10. June 2022 · Paper 2 · Q6(c) · 2 marks
(c) Taking x0 = 5 as a first approximation to alpha , apply the Newton-Raphson method once to f(x) to obtain a second approximation to alpha . Show your method and give your answer to 3 significant figures. (2)
11. Oct 2021 · Paper 1 · Q4(b) · 3 marks
The iterative formula xn + 1 = 1 7 (2 + 4xn 2 - 2xn 3) is used to find an approximate value for alpha . Starting with x1 = 0.3 (b) calculate, giving each answer to 4 decimal places, (i) the value of x2 (ii) the value of x4 (3)
12. Oct 2021 · Paper 1 · Q4(c) · 2 marks
Using a suitable interval and a suitable function that should be stated, (c) show that alpha is 0.341 to 3 decimal places. (2)
13. Oct 2020 · Paper 2 · Q7(c) · 3 marks
(c) Use the iteration formula xn xn+1 = 4 - with x1 = 2 to find (i) the value of x2 to 5 decimal places, (ii) the x coordinate of P to 5 decimal places. (3)
14. June 2019 · Paper 2 · Q11(b) · 2 marks
The point P(Į, 2) lies on C. (b) Show that 1.5 < Į < 1.6 (2)
15. June 2019 · Paper 2 · Q11(c) · 2 marks
A possible iteration formula that could be used in an attempt to find Į is x x xn = - n + n Using this formula with x1 = 1.5 (c) find x4 to 3 decimal places, (2)
16. June 2019 · Paper 2 · Q11(d) · 2 marks
(d) describe the long-term behaviour of xn (2)
17. June 2018 · Paper 1 · Q4(a) · 2 marks
4. The curve with equation y = 2 ln(8 - x) meets the line y = x at a single point, x = alpha . (a) Show that 3 < alpha < 4 (2)
18. June 2018 · Paper 1 · Q4(b) · 2 marks
y y = x y = 2 ln(8 - x) x Figure 2 Figure 2 shows the graph of y = 2 ln(8 - x) and the graph of y = x. A student uses the iteration formula xn + 1 = 2 ln(8 - xn), n ∈ in an attempt to find an approximation for alpha . Using the graph and starting with x1 = 4 (b) determine whether or not this iteration formula can be used to find an approximation for alpha , justifying your answer. (2)
19. June 2018 · Paper 2 · Q5(a) · 3 marks
5. The equation 2x3 + x2 - 1 = 0 has exactly one real root. (a) Show that, for this equation, the Newton-Raphson formula can be written x + x + x + = n n n x + x n n (3)
20. June 2018 · Paper 2 · Q5(b) · 2 marks
Using the formula given in part (a) with x1 = 1 (b) find the values of x2 and x3 (2)
21. June 2018 · Paper 2 · Q5(c) · 1 marks
(c) Explain why, for this question, the Newton-Raphson method cannot be used with x1 = 0 (1)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · Paper 1 · Q9
Mark scheme June 2025 Paper 1 Q9(c)
June 2024 · Paper 1 · Q3
Mark scheme June 2024 Paper 1 Q3(c)
June 2024 · Paper 2 · Q6
Mark scheme June 2024 Paper 2 Q6(c)
June 2023 · Paper 1 · Q15
Mark scheme June 2023 Paper 1 Q15(e)
June 2022 · Paper 1 · Q8
Mark scheme June 2022 Paper 1 Q8(c)
June 2022 · Paper 2 · Q6
Mark scheme June 2022 Paper 2 Q6(c)
Oct 2021 · Paper 1 · Q4
Mark scheme Oct 2021 Paper 1 Q4(c)
Oct 2020 · Paper 2 · Q7
Mark scheme Oct 2020 Paper 2 Q7(c)
June 2019 · Paper 2 · Q11
Mark scheme June 2019 Paper 2 Q11(d)
June 2018 · Paper 1 · Q4
Mark scheme June 2018 Paper 1 Q4(b)
June 2018 · Paper 2 · Q5
Mark scheme June 2018 Paper 2 Q5(c)