025 OCR
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
1
2025 OCR AS Level Further Mathematics B (MEI)
Y414/01 Numerical Methods
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Friday 6 June 2025 – Afternoon AS Level
Further Mathematics B (MEI) Y414/01 Numerical
Methods
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(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 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [R/508/5557] OCR is an exempt Charity
DC (PQ/SG) 351558/4 Turn over
Page 1
,025 OCR
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
2
2
1 The method of interval bisection is used to find a sequence of approximations to one of the
roots of the equation ex - x2 - 3x = 0.
The table shows the associated spreadsheet output.
◢ A B C D E F
1 a f(a) b f(b) c f(c)
2 2 -2.61094 3 2.085537 2.5 -1.56751
3 2.5 -1.56751 3 2.085537 2.75 -0.16987
4 2.75 -0.16987 3 2.085537 2.875 0.834799
5 2.75 -0.16987 2.875 0.834799 2.8125 0.303839
6
(a) Write down a suitable formula for cell E2. [1]
The formula in cell A3 is = IF(F2 1 0, E2, A2) .
(b) Write down a similar formula for cell C3. [1]
(c) Complete row 6 of the table in the Printed Answer Booklet. [2]
(d) Without doing any more calculations, write down the value of the root correct to the
maximum number of decimal places which seems justified. You must explain the precision
quoted. [1]
2 The table gives three values of x and the associated values of y.
x -1 2 3
y -3.39 0.18 0.45
Use Lagrange’s form of the interpolating polynomial to construct a polynomial of degree 2 for the
values in the table. Give your answer in the form
y = ax2 + bx + c,
where a, b and c are constants to be determined. [4]
© OCR 2025 Y414/01 Jun25
Page 2
,025 OCR
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
3
3
3 (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]
You are given that y = r2 - 5 and z = (r - 5) 2. You
are also given the following information.
• Y is an approximation to y.
• Z is an approximation to z.
• Y and Z are found by using r = 3.14.
A student states that the relative error in using Y to approximate y is exactly the same as the
relative error in using Z to approximate z, because in each case the calculation involves squaring
and subtracting 5.
(c) Without doing any calculations, explain whether the student is correct. [1]
© OCR 2025 Y414/01 Jun25 Turn over
Page 3
,025 OCR
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
4
4
4 A student is using a spreadsheet to investigate the y = f(x) at the point where x = 2. Some
curve of the spreadsheet output is shown below.
Table 4.1 shows some values of x and the associated values of f(x).
Table 4.2 shows two approximations to f l(2) found using the same method with h = 0.001 and
h = 0.01.
◢ E F G H I J
2
3 Table 4.1
4 x 2 2.001 2.01
5 f(x) 0.30103 0.30125 0.3032
6
7
8 Table 4.2
9 h 0.001 0.01
10 f l(2) 0.21709 0.21661
11
(a) The formula in cell G10 is = (H5 - G5) /G9 .
The formula in cell H10 is = (I5 - G5) /H9 .
State the method being used to find these approximations to f l(2). [1]
(b) • Calculate 0.30125 - 0.30103 .
0.001
• Explain why your answer is not the same as the value displayed in cell G10. [3]
(c) State the value of f l(2) as accurately as you can. You must justify the precision quoted. [1]
(d) Hence determine an estimate for the error when f(2) is used as an approximation to f(2.1). [2]
(e) Explain whether your answer to part (d) is likely to be an over-estimate or an under-estimate.[1]
© OCR 2025 Y414/01 Jun25
Page 4
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
1
2025 OCR AS Level Further Mathematics B (MEI)
Y414/01 Numerical Methods
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Friday 6 June 2025 – Afternoon AS Level
Further Mathematics B (MEI) Y414/01 Numerical
Methods
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(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 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [R/508/5557] OCR is an exempt Charity
DC (PQ/SG) 351558/4 Turn over
Page 1
,025 OCR
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
2
2
1 The method of interval bisection is used to find a sequence of approximations to one of the
roots of the equation ex - x2 - 3x = 0.
The table shows the associated spreadsheet output.
◢ A B C D E F
1 a f(a) b f(b) c f(c)
2 2 -2.61094 3 2.085537 2.5 -1.56751
3 2.5 -1.56751 3 2.085537 2.75 -0.16987
4 2.75 -0.16987 3 2.085537 2.875 0.834799
5 2.75 -0.16987 2.875 0.834799 2.8125 0.303839
6
(a) Write down a suitable formula for cell E2. [1]
The formula in cell A3 is = IF(F2 1 0, E2, A2) .
(b) Write down a similar formula for cell C3. [1]
(c) Complete row 6 of the table in the Printed Answer Booklet. [2]
(d) Without doing any more calculations, write down the value of the root correct to the
maximum number of decimal places which seems justified. You must explain the precision
quoted. [1]
2 The table gives three values of x and the associated values of y.
x -1 2 3
y -3.39 0.18 0.45
Use Lagrange’s form of the interpolating polynomial to construct a polynomial of degree 2 for the
values in the table. Give your answer in the form
y = ax2 + bx + c,
where a, b and c are constants to be determined. [4]
© OCR 2025 Y414/01 Jun25
Page 2
,025 OCR
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
3
3
3 (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]
You are given that y = r2 - 5 and z = (r - 5) 2. You
are also given the following information.
• Y is an approximation to y.
• Z is an approximation to z.
• Y and Z are found by using r = 3.14.
A student states that the relative error in using Y to approximate y is exactly the same as the
relative error in using Z to approximate z, because in each case the calculation involves squaring
and subtracting 5.
(c) Without doing any calculations, explain whether the student is correct. [1]
© OCR 2025 Y414/01 Jun25 Turn over
Page 3
,025 OCR
-2025
AS Level
OCR Further
A/AS Level
Mathematics
Guide.. B (MEI) Y414-01 Numerical Methods Verified Question paper with Marking Scheme Attached.p
4
4
4 A student is using a spreadsheet to investigate the y = f(x) at the point where x = 2. Some
curve of the spreadsheet output is shown below.
Table 4.1 shows some values of x and the associated values of f(x).
Table 4.2 shows two approximations to f l(2) found using the same method with h = 0.001 and
h = 0.01.
◢ E F G H I J
2
3 Table 4.1
4 x 2 2.001 2.01
5 f(x) 0.30103 0.30125 0.3032
6
7
8 Table 4.2
9 h 0.001 0.01
10 f l(2) 0.21709 0.21661
11
(a) The formula in cell G10 is = (H5 - G5) /G9 .
The formula in cell H10 is = (I5 - G5) /H9 .
State the method being used to find these approximations to f l(2). [1]
(b) • Calculate 0.30125 - 0.30103 .
0.001
• Explain why your answer is not the same as the value displayed in cell G10. [3]
(c) State the value of f l(2) as accurately as you can. You must justify the precision quoted. [1]
(d) Hence determine an estimate for the error when f(2) is used as an approximation to f(2.1). [2]
(e) Explain whether your answer to part (d) is likely to be an over-estimate or an under-estimate.[1]
© OCR 2025 Y414/01 Jun25
Page 4
