OCR A Level Further Mathematics B Y434/01
Numerical Methods Combined Question paper
& Marking Scheme A+ June 2025
Oxford Cambridge and RSA
Wednesday 18 June 2025 – Afternoon
A Level Further Mathematics B (MEI)
Y434/01 Numerical Methods
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer Booklet. If
you need extra space use the lined page at the end of the Printed Answer Booklet. The
question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be given
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 12 pages.
ADVICE
• Read each question carefully before you start your answer
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 [3]
(b) Determine the range of possible values of R = p - 2q .
(c) Explain why your answer to part (b) is so large. [1]
y0.5
1.3
3 Approximations to 1+x3dx using the midpoint rule, the trapezium rule and Simps on’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 Turn over
, 32
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 y = tanh x - x2 + 4.
of
y
5
0 x
–3 –2 –1 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
0 x
–1 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 Turn over
Numerical Methods Combined Question paper
& Marking Scheme A+ June 2025
Oxford Cambridge and RSA
Wednesday 18 June 2025 – Afternoon
A Level Further Mathematics B (MEI)
Y434/01 Numerical Methods
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer Booklet. If
you need extra space use the lined page at the end of the Printed Answer Booklet. The
question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be given
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 12 pages.
ADVICE
• Read each question carefully before you start your answer
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 [3]
(b) Determine the range of possible values of R = p - 2q .
(c) Explain why your answer to part (b) is so large. [1]
y0.5
1.3
3 Approximations to 1+x3dx using the midpoint rule, the trapezium rule and Simps on’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 Turn over
, 32
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 y = tanh x - x2 + 4.
of
y
5
0 x
–3 –2 –1 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
0 x
–1 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 Turn over
