plumiErrors and convergence: OCR MEI A‑Level Further Maths questions
OCR MEI A-Level Further Maths past-paper questions on 6.5 Errors and convergence, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 53 questions on Errors and convergence, as a list
- June 2025, Numerical Methods, Q2 (6 marks): The numbers p and q are approximated by P 323 and Q 162 . = = P has been found by rounding p to the nearest whole number. Q has been found by chopping q to the
- June 2025, Numerical Methods, Q3 (5 marks): 3 . y 3 3 Approximations to 1 x d x using the midpoint rule, the trapezium rule and Simpson’s rule + 0 5 . with n = and n 1 2 are shown in the table. The table
- June 2025, Numerical Methods, Q6 (7 marks): 1y 6 The midpoint rule is used to find a sequence of approximations to sin x x d . 0 The associated spreadsheet output, together with some further analysis, is
- June 2025, AS Numerical Methods, Q3 (5 marks): (a) Find the relative error when r is chopped to 3 decimal places. [2] (b) Find the relative error when r is rounded to 3 decimal places. [2] 2 5 2 You are give
- June 2025, AS Numerical Methods, Q4 (8 marks): A student is using a spreadsheet to investigate the curve y f ( ) x at the point where x 2 . Some = = of the spreadsheet output is shown below. Table 4.1 shows
- June 2025, AS Numerical Methods, Q5 (7 marks): 5 . y 5 The spreadsheet output shows a sequence of approximations to f ( )x d x using the trapezium 0 5 . rule and Simpson’s rule, together with some further an
- June 2025, AS Numerical Methods, Q8 (7 marks): Table 8.1 shows some values of x and the associated values of ( )x f . Table 8.1 x 0.5 0.9 1.3 f ( )x 1.414 21 1.099 47 0.711 01 (a) Use the central difference
- June 2024, Numerical Methods, Q1 (4 marks): The table shows some values of x, together with the associated values of a function, ( )x f . x 1.9 2 2.1 0.5842 0.6309 0.6753 f ( )x (a) Use the information in
- June 2024, Numerical Methods, Q2 (6 marks): You are given that a tanh ( ) 1 and b tanh ( ) 2 . = = A is the approximation to a formed by rounding tanh 1 to 1 decimal place. ( ) B is the approximation to b
- June 2024, Numerical Methods, Q6 (10 marks): Table 6.1 shows some values of x and the associated values of a function, y f ( ) x . = Table 6.1 x 1.5 1 2 0.840 89 1 1.189 21 f ( )x (a) Explain why it is not
- June 2024, Numerical Methods, Q7 (14 marks): 0y 7 ( )x x A student is using a spreadsheet to find approximations to f d using the midpoint rule, the trapezium rule and Simpson’s rule. Some of the associate
- June 2024, AS Numerical Methods, Q2 (5 marks): A spreadsheet is used to find the approximate value of r + . The output is shown in the table. e A B 1 3.141 593 r 2 e 2.718 282 3 e 5.859 874 r + The formula i
- June 2024, AS Numerical Methods, Q5 (10 marks): Table 5.1 shows some values of x and the associated values of ( )x f . Table 5.1 x 5 5.0001 5.001 5.01 5.1 f(x) 12.182 494 12.183 103 12.188 587 12.243 559 12.8
- June 2024, AS Numerical Methods, Q7 (12 marks): A student uses the midpoint rule, the trapezium rule and Simpson’s Rule to find a series of 1y approximations to f ( )x d x . The associated spreadsheet output,
- June 2023, Numerical Methods, Q1 (7 marks): You are given that ( x , y ) ( . , . ) 0 9 2 3 and ( x , y ) ( . , . ) 1 1 2 7 . 1 = 2 = 1 2 The values of x1 and x2 have been rounded to 1 decimal place. (a) D
- June 2023, Numerical Methods, Q6 (6 marks): (a) (i) Calculate the relative error when r is chopped to 2 decimal places in approximating 2 r + . 2 [2] (ii) Without doing any calculation, explain whether th
- June 2023, Numerical Methods, Q9 (9 marks): 1y 3 9 The trapezium rule is used to calculate 3 approximations to sinh x ( ) d x with 1, 2 and 4 strips 0 respectively. The results are shown in Table 9.1. Tab
- June 2023, AS Numerical Methods, Q1 (6 marks): The table shows some spreadsheet output. The values displayed in cells K21 and K22 are exact, but the values displayed in rows 21 and 22 in columns L, M and N h
- June 2023, AS Numerical Methods, Q3 (5 marks): You are given that 43 2 . 43 2 . P + and Q = = x y x y - and that x 2 17 . and y 2 14 . . = = The values of x and y have been rounded to 2 decimal places. (a) C
- June 2023, AS Numerical Methods, Q5 (9 marks): y 5 2 5 The table shows some approximations to x 3 x 2 d x , correct to 8 decimal places, which + + 0 have been found using the midpoint rule and the trapezium
- June 2023, AS Numerical Methods, Q8 (9 marks): Table 8.1 shows some values of x and the associated values of a function ( )x f . Table 8.1 x 1.2 2 2.8 ( )x 0.9638087 1.3258177 1.4439304 f (a) Use the forward
- June 2022, Numerical Methods, Q1 (4 marks): C 3 7622 . and S 3 6269 . are used to approximate cosh2 and sinh2 respectively. = = (a) Determine whether these approximations are the result of chopping or rou
- June 2022, Numerical Methods, Q2 (7 marks): The table shows some values of x and the associated values of y f ( ) x . = x 2.75 3 3.25 f(x) 0.920799 1 1.072858 d y at x (a) Calculate an estimate of d = usi
- June 2022, Numerical Methods, Q5 (9 marks): Kai uses the midpoint rule, trapezium rule and Simpson’s rule to find approximations to by f ( ) x d x , where a and b are constants. The associated spreadsheet
- June 2022, Numerical Methods, Q6 (11 marks): Charlie uses fixed point iteration to find a sequence of approximations to the root of the equation 1 2 x x sin - ( ) 1 0 . - + = a k. 2 x x x x Charlie uses th
- June 2022, AS Numerical Methods, Q1 (7 marks): You are given the following simultaneous equations. 2 x 1 8 . y 5 + = 3 x 2 8 . y 3 + = The constants and the coefficients of x are exact, but the coefficients
- June 2022, AS Numerical Methods, Q2 (6 marks): The table shows some values of x and the associated values of y = f (x). x 1.98 1.99 2 2.01 2.02 f (x) 1.10311648 1.10514069 1.10714872 1.10914075 1.11111695 d
- June 2022, AS Numerical Methods, Q3 (5 marks): Ali uses the trapezium rule with h = 1, h = 0.5 and h = 0.25 to calculate three approximations to 2y f ( )x d x . Ali’s results are shown in the table. 1 h n Tn
- June 2022, AS Numerical Methods, Q6 (10 marks): Fig. 6.1 shows some values of x and the associated values of f (x). x 1 1.25 1.5 1.75 2 f (x) 0 0.1609640 0.2924813 0.4036775 0.5 Fig. 6.1 2y Sam is using the m
- November 2021, Numerical Methods, Q1 (5 marks): (a) (i) Determine the relative error when • 1.414214 is used to approximate 2, • 1.4142142 is used to approximate 2. [3] (ii) Write down the relationship betwee
- November 2021, Numerical Methods, Q5 (9 marks): When Nina does the weekly grocery shopping she models the total cost by adding up the cost of each item in her head as she goes along. To simplify matters she r
- November 2021, Numerical Methods, Q7 (15 marks): y 7 tanh x x Sarah uses the trapezium rule to find a sequence of approximations to ( ) d . 0 Her spreadsheet output is shown in Fig. 7.1. n Tn difference ratio
- November 2021, AS Numerical Methods, Q2 (6 marks): The table shows 2 values of x and the associated values of (fx . The values of x are exact and the ) values of ( ) f x have been rounded to 3 decimal places. x
- November 2021, AS Numerical Methods, Q5 (10 marks): y 5 f x x A student is trying to find an accurate approximation to ( ) d using the trapezium rule, the 1 midpoint rule and Simpson’s rule. Some of the spreadshe
- November 2021, AS Numerical Methods, Q6 (9 marks): En n to approximate en, and calculates the 6 James is investigating relative error. He uses .2 718 = relative error for various values of n. Some of his results
- November 2020, Numerical Methods, Q1 (4 marks): Fig. 1 shows some spreadsheet output. A 1 1E-17 2 1E-17 3 1E-29 Fig. 1 (a) Write the value displayed in cell A3 in standard mathematical notation. [1] The for
- November 2020, Numerical Methods, Q3 (7 marks): At Heathwick airport each passenger’s luggage is weighed before being loaded into the hold of the aeroplane. Each weight is displayed digitally in kg to 1 decim
- November 2020, Numerical Methods, Q4 (10 marks): y 3 4 (a) Use the trapezium rule with 1 strip to calculate an estimate of x x 1 d , giving your + [2] answer correct to six decimal places. 1 Fig. 4 shows some
- November 2020, AS Numerical Methods, Q2 (8 marks): ex2 2 Fig. 2.1 shows the graph of y f ( ) x where ( )x f for 0 x 1 3 . . = = G G y 7 6 5 4 3 2 1 0 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Fig.
- November 2020, AS Numerical Methods, Q3 (7 marks): Fig. 3 shows three values of x and the associated values of ( )x f . x 1 2 5 2.53 7.71 47.37 f ( )x Fig. 3 (a) Use Lagrange’s method to construct the interpolat
- November 2020, AS Numerical Methods, Q4 (8 marks): A small company leases a machine which they use for scanning, printing and photocopying. The cost of each job completed is displayed in pounds to the nearest te
- November 2020, AS Numerical Methods, Q5 (11 marks): y 5 ( )x x Arthur uses a spreadsheet to approximate f d using the trapezium rule. Some of the output is shown in Fig. 5.1. 0 C D E F 3 x y 4 T 5 0 1.732051 6 1
- June 2019, Numerical Methods, Q1 (6 marks): Fig. 1 shows some spreadsheet output concerning the values of a function, ( )x . f A B C 1 x f ( )x 2 1 0.367879441 0.367879441 3 2 0.018315639 0.38619508 4 3
- June 2019, Numerical Methods, Q5 (12 marks): Fig. 5 shows spreadsheet output concerning the estimation of the derivative of a function ( )x f at x 2 using the forward difference method. = A B C D 1 h est
- June 2019, Numerical Methods, Q6 (10 marks): y x 6 The spreadsheet output in Fig. 6 shows approximations to x - d x found using the midpoint 0 rule, denoted by Mn, and the trapezium rule, denoted by Tn. A
- June 2019, AS Numerical Methods, Q2 (8 marks): You are given that ( ) f x log ( 2 x 1 ) . = + 10 (a) Calculate the absolute error when ( ) f 5 is used to approximate ( . f 4 98 . ) [2] (b) Hence find an appr
- June 2019, AS Numerical Methods, Q3 (6 marks): Sallysavers Building Society offers a “Bigsaver” account. Interest is payable at the end of each year. The rate of interest is 2.33% per annum. A customer model
- June 2019, AS Numerical Methods, Q5 (3 marks): 1 5 (a) Explain what problem may arise in the evaluation of x for large values of x and y. [2] - 10 10 y 10 10 y x 1 1 - (b) Explain why re-writing x as ( may a
- June 2019, AS Numerical Methods, Q8 (12 marks): ^ h 1 y 8 Fig. 8.1 shows estimates of sin x d x found using the trapezium rule, denoted by Tn, together 0 with some further analysis of these estimates. n Tn di
- June 2018, AS Numerical Methods, Q1 (4 marks): The coordinates of point A are (1.05, 2.71) and the coordinates of point B are (1.07, 3.08). In each case the x and y values have been rounded to two decimal pl
- June 2018, AS Numerical Methods, Q2 (4 marks): The table in Fig. 2 shows some values of log x 3 which are correct to 6 decimal places. x 2 2.25 2.5 2.75 3 log3x 0.630930 0.738140 0.834044 0.920799 1 Fig. 2 3
- June 2018, AS Numerical Methods, Q3 (8 marks): The spreadsheet output in Fig. 3 shows Table 1 and Table 2. A B C D E F G 1 x 2 2.1 2.01 2.001 2.0001 2.00001 2 f(x) 0.64 0.625876972 0.638573473 0.639857204 0.
- June 2018, AS Numerical Methods, Q7 (12 marks): x 2 7 Fig. 7.1 shows part of the curve y 2 x + . 3 = - y 6 5 4 3 2 1 0 x 0 1 2 3 4 5 Fig. 7.1 7 ^ h y 4 x 2 (i) State, with a reason, whether using the trapeziu
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