A LEVEL FURTHER MATHEMATICS B (MEI): Actual June 2025 PAST PAPER Y434/01:
Numerical Methods. All Assessment Questions & Mark Scheme [OCR Y434/01]
Exam Resource Summary
The A Level Further Mathematics B (MEI) June 2025 Numerical Methods Paper (OCR Y434/01)
integrates the full official examination paper with its complete mark scheme to provide a focused and
practical revision resource. This paper evaluates students’ understanding of numerical approximation
techniques, iterative methods, numerical solution of equations, numerical integration, and error analysis,
alongside the application of algorithms to real mathematical problems. By presenting every question next
to its corresponding marking criteria, the resource offers clear guidance on method accuracy, logical
reasoning, and the structured working required to achieve higher marks. This combined format supports
efficient revision, builds precision and confidence in computational approaches, and strengthens the
analytical skills essential for success in the May/June 2026 OCR A Level Further Mathematics B (MEI)
Numerical Methods examination.
OCR is an exempt Charity
Turn over
, 2
1 The table shows some values of x and the associated values of f(x).
x 1.9 2.0 2.1
f(x) 0.216 92 0.2 0.184 84
(a) Determine an estimate of f l(2.0) using the forward difference method. [2]
(b) Determine an estimate of f l(2.0) using the central difference method. [2]
2 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 nearest whole number.
(a) (i) Find the maximum possible relative error in using P to approximate p. [1]
(ii) Find the maximum possible relative error in using Q to approximate q. [1]
200
(b) Determine the range of possible values of R = p -2q
. [3]
(c) Explain why your answer to part (b) is so large. [1]
y0.5
1.3
3 Approximations to 1 + x3 dx using the midpoint rule, the trapezium rule and Simpson’s rule
with n = 1 and n = 2 are shown in the table. The table is incomplete.
n Mn Tn S2n
1 1.081 111
2 1.074 256
(a) Complete the copy of the table in the Printed Answer Booklet. Give your answers to
6 decimal places. [4]
(b) Without doing any further calculations, state the value of y 1.3 1 + x3 dx as accurately as
0.5
possible. You must justify the precision quoted. [1]
© OCR 2025 Y434/01 Jun25
, 3
4 The Newton-Raphson method is to be used to find the positive root of the equation
tanh x - x2 + 4 = 0 .
The diagram shows part of the graph of y = tanh x - x2 + 4.
y
5
x
–3 –2 –1 0 1 2 3 4
–5
–10
(a) On the copy of the diagram in the Printed Answer Booklet, show how the Newton-Raphson
method works to find x1 using the starting value x0 = 1. [1]
(b) Use the Newton-Raphson method using the starting value x0 = 1 to determine the values of
x1 and x2 correct to 8 decimal places. [4]
(c) Continue the iteration to determine the value of the positive root of the equation
tanh x - x2 + 4 = 0 correct to 7 decimal places. [2]
© OCR 2025 Y434/01 Jun25 Turn over
, 4
5 The diagram shows part of the graph of y = sinh x - 3x + 1.
y
10
5
x
–1 0 1 2 3 4
–5
The largest positive root, a, of the equation sinh x - 3x + 1 = 0 is to be found using the secant
method with x0 = 2 and x1 = 3.
Table 5.1 shows the associated spreadsheet output.
Table 5.1
◢ A B C D E
2 r xr f(xr) xr +1 f(xr +1)
3 0 2 –1.37314 3 2.01787
4 1 3 2.017875 2.404935 –0.7211
5 2 2.404935 –0.72109 2.561598 –0.2451
6 3 2.561598 –0.24513 2.642285 0.06018
7 4 2.642285 0.060175 2.626382 –0.0035
8 5 2.626382 –0.00348 2.627252 –5E-05
9 6 2.627252 –4.5E-05 2.627264 3.5E-08
(a) Write down a suitable formula for cell D4. [2]
(b) State the most accurate approximation to a in the spreadsheet output. [1]
(c) Determine whether this approximation is accurate to 6 decimal places. [2]
(d) Explain what would happen if the initial entry in cell D3 were to be changed from 3 to 1. [1]
© OCR 2025 Y434/01 Jun25
Numerical Methods. All Assessment Questions & Mark Scheme [OCR Y434/01]
Exam Resource Summary
The A Level Further Mathematics B (MEI) June 2025 Numerical Methods Paper (OCR Y434/01)
integrates the full official examination paper with its complete mark scheme to provide a focused and
practical revision resource. This paper evaluates students’ understanding of numerical approximation
techniques, iterative methods, numerical solution of equations, numerical integration, and error analysis,
alongside the application of algorithms to real mathematical problems. By presenting every question next
to its corresponding marking criteria, the resource offers clear guidance on method accuracy, logical
reasoning, and the structured working required to achieve higher marks. This combined format supports
efficient revision, builds precision and confidence in computational approaches, and strengthens the
analytical skills essential for success in the May/June 2026 OCR A Level Further Mathematics B (MEI)
Numerical Methods examination.
OCR is an exempt Charity
Turn over
, 2
1 The table shows some values of x and the associated values of f(x).
x 1.9 2.0 2.1
f(x) 0.216 92 0.2 0.184 84
(a) Determine an estimate of f l(2.0) using the forward difference method. [2]
(b) Determine an estimate of f l(2.0) using the central difference method. [2]
2 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 nearest whole number.
(a) (i) Find the maximum possible relative error in using P to approximate p. [1]
(ii) Find the maximum possible relative error in using Q to approximate q. [1]
200
(b) Determine the range of possible values of R = p -2q
. [3]
(c) Explain why your answer to part (b) is so large. [1]
y0.5
1.3
3 Approximations to 1 + x3 dx using the midpoint rule, the trapezium rule and Simpson’s rule
with n = 1 and n = 2 are shown in the table. The table is incomplete.
n Mn Tn S2n
1 1.081 111
2 1.074 256
(a) Complete the copy of the table in the Printed Answer Booklet. Give your answers to
6 decimal places. [4]
(b) Without doing any further calculations, state the value of y 1.3 1 + x3 dx as accurately as
0.5
possible. You must justify the precision quoted. [1]
© OCR 2025 Y434/01 Jun25
, 3
4 The Newton-Raphson method is to be used to find the positive root of the equation
tanh x - x2 + 4 = 0 .
The diagram shows part of the graph of y = tanh x - x2 + 4.
y
5
x
–3 –2 –1 0 1 2 3 4
–5
–10
(a) On the copy of the diagram in the Printed Answer Booklet, show how the Newton-Raphson
method works to find x1 using the starting value x0 = 1. [1]
(b) Use the Newton-Raphson method using the starting value x0 = 1 to determine the values of
x1 and x2 correct to 8 decimal places. [4]
(c) Continue the iteration to determine the value of the positive root of the equation
tanh x - x2 + 4 = 0 correct to 7 decimal places. [2]
© OCR 2025 Y434/01 Jun25 Turn over
, 4
5 The diagram shows part of the graph of y = sinh x - 3x + 1.
y
10
5
x
–1 0 1 2 3 4
–5
The largest positive root, a, of the equation sinh x - 3x + 1 = 0 is to be found using the secant
method with x0 = 2 and x1 = 3.
Table 5.1 shows the associated spreadsheet output.
Table 5.1
◢ A B C D E
2 r xr f(xr) xr +1 f(xr +1)
3 0 2 –1.37314 3 2.01787
4 1 3 2.017875 2.404935 –0.7211
5 2 2.404935 –0.72109 2.561598 –0.2451
6 3 2.561598 –0.24513 2.642285 0.06018
7 4 2.642285 0.060175 2.626382 –0.0035
8 5 2.626382 –0.00348 2.627252 –5E-05
9 6 2.627252 –4.5E-05 2.627264 3.5E-08
(a) Write down a suitable formula for cell D4. [2]
(b) State the most accurate approximation to a in the spreadsheet output. [1]
(c) Determine whether this approximation is accurate to 6 decimal places. [2]
(d) Explain what would happen if the initial entry in cell D3 were to be changed from 3 to 1. [1]
© OCR 2025 Y434/01 Jun25
