Oxford Cambridge and RSA Examinations
GCE
Further Mathematics B (MEI)
Y414/01: Numerical Methods
Advanced Subsidiary GCE
Mark Scheme for Autumn 2021Oxford Cambridge and RSA Examinations
OCR (Oxford Cambridge and RS
...
Oxford Cambridge and RSA Examinations
GCE
Further Mathematics B (MEI)
Y414/01: Numerical Methods
Advanced Subsidiary GCE
Mark Scheme for Autumn 2021Oxford Cambridge and RSA Examinations
OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of
qualifications to meet the needs of candidates of all ages and abilities. OCR qualifications
include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals,
Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in
areas such as IT, business, languages, teaching/training, administration and secretarial skills.
It is also responsible for developing new specifications to meet national requirements and the
needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is
invested back into the establishment to help towards the development of qualifications and
support, which keep pace with the changing needs of today’s society.
This mark scheme is published as an aid to teachers and students, to indicate the requirements
of the examination. It shows the basis on which marks were awarded by examiners. It does not
indicate the details of the discussions which took place at an examiners’ meeting before marking
commenced.
All examiners are instructed that alternative correct answers and unexpected approaches in
candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills
demonstrated.
Mark schemes should be read in conjunction with the published question papers and the report
on the examination.
© OCR 2021Y414/01 Mark Scheme November 2021
Text Instructions
1. Annotations and abbreviations
Annotation in scoris Meaning
and
BOD Benefit of doubt
FT Follow through
ISW Ignore subsequent working
M0, M1 Method mark awarded 0, 1
A0, A1 Accuracy mark awarded 0, 1
B0, B1 Independent mark awarded 0, 1
E Explanation mark 1
SC Special case
^ Omission sign
MR Misread
BP Blank page
Highlighting
Other abbreviations in
mark scheme
Meaning
E1 Mark for explaining a result or establishing a given result
dep* Mark dependent on a previous mark, indicated by *. The * may be omitted if only previous M mark.
cao Correct answer only
oe Or equivalent
rot Rounded or truncated
soi Seen or implied
www Without wrong working
AG Answer given
awrt Anything which rounds to
BC By Calculator
DR This indicates that the instruction In this question you must show detailed reasoning appears in the question.Y414/01 Mark Scheme November 2021
2. Subject-specific Marking Instructions for AS/A Level Further Mathematics B (MEI)
a a Annotations must be used during your marking. For a response awarded zero (or full) marks a single appropriate annotation (cross, tick, M0 or ^)
is sufficient, but not required.
For responses that are not awarded either 0 or full marks, you must make it clear how you have arrived at the mark you have awarded and all responses
must have enough annotation for a reviewer to decide if the mark awarded is correct without having to mark it independently.
It is vital that you annotate standardisation scripts fully to show how the marks have been awarded.
Award NR (No Response)
- if there is nothing written at all in the answer space and no attempt elsewhere in the script
- OR if there is a comment which does not in any way relate to the question (e.g. ‘can’t do’, ‘don’t know’)
- OR if there is a mark (e.g. a dash, a question mark, a picture) which isn’t an attempt at the question.
Note: Award 0 marks only for an attempt that earns no credit (including copying out the question).
If a candidate uses the answer space for one question to answer another, for example using the space for 8(b) to answer 8(a), then give benefit of doubt
unless it is ambiguous for which part it is intended.
b An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking
incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not always be judged on the answer alone, and
answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must
be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method.
Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your
Team Leader who will decide on a course of action with the Principal Examiner.
If you are in any doubt whatsoever you should contact your Team Leader.Y414/01 Mark Scheme November 2021
c The following types of marks are available.
M A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not
usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of
using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the
relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified.
A method mark may usually be implied by a correct answer unless the question includes the DR statement, the command words “Determine” or
“Show that”, or some other indication that the method must be given explicitly.
A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated
Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded.
B Mark for a correct result or statement independent of Method marks.
E A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an
unknown result.
Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is
reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a
wrong argument.
d When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and
similarly where there are several B marks allocated. (The notation ‘dep*’ is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in
the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that
no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied
and full credit must be given.
e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B
marks are given for correct work only – differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect
working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such
cases, what is acceptable will be detailed in the mark scheme. If this is not the case, please escalate the question to your Team Leader who will decide on a
course of action with the Principal Examiner.
Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be ‘follow through’. In such cases you
must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow
through questions candidate-by-candidate rather than question-by-question.Y414/01 Mark Scheme November 2021
f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in
SI or in the units of the question. (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km, when this would be
assumed to be the unspecified unit.)
We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly
says so.
When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value.
When a value is not given in the paper accept any answer that agrees with the correct value to 2 s.f. unless a different level of accuracy has been asked for in
the question, or the mark scheme specifies an acceptable range.
NB for Specification A the rubric specifies 3 s.f. as standard, so this statement reads “3 s.f”
Follow through should be used so that only one mark in any question is lost for each distinct accuracy error.
Candidates using a value of 9.80, 9.81 or 10 for g should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is
given in the rubric.
g Rules for replaced work and multiple attempts:
If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others.
If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete.
if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately.
h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the
scheme but following through from the candidate’s data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is
achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors. If a candidate corrects
the misread in a later part, do not continue to follow through. E marks are lost unless, by chance, the given results are established by equivalent working. Note
that a miscopy of the candidate’s own working is not a misread but an accuracy error.
i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers provided that there is nothing in
the wording of the question specifying that analytical methods are required such as the bold “In this question you must show detailed reasoning”, or the command
words “Show” and “Determine. Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no
supporting method score zero. If in doubt, consult your Team Leader.
j If in any case the scheme operates with considerable unfairness consult your Team Leader.Y414/01 Mark Scheme November 2021
Question Answer Marks AOs Guidance
1 (a) (�−1)(�−2)
(−1−1)(−1−2)
× 0.7 + (�−−1)(�−2)
(1−−1)(1−2)
× (−3.56) +
(�−−1)(�−1)
(2−−1)(2−1)
× (−4.94)
M1 1.1 allow sign errors and bracket
errors and one substitution error;
must be 3 terms
M0 if x and y values
interchanged
(P(x)=) 0.25x² ‒ 2.13x ‒ 1.68 oe isw A1
A1
A1
1.1
1.1
1.1
A1 for each correct term
[4]
1 (b)
the x-values must be evenly spaced if
Newton’s forward difference interpolation
formula is to be used and this is not the case
for these data oe
B1 2.4 do not allow eg “these values are
not evenly spaced”
[1]
2 (a) 0.2
0.4955−0.4915
soi M1 3.1a allow 0.495499 or better
0.2
0.4945−0.4925
soi M1 2.1 allow 0.492499 or better
50 or 100 seen A1 1.1 must be from correct, not
approximate upper bounds
50 < f(x) < 100 A1 1.1 condone eg between 50 and 100
but not eg 50 to 100
allow 50 from 100 ‒ 50 seen
inequality must be strictY414/01 Mark Scheme November 2021
Question Answer Marks AOs Guidance
[4]
2 (b)
the calculation involves the subtraction of
nearly equal numbers in the denominator oe
B1 2.2a just subtraction of nearly equal
numbers is insufficient
[1]
2 (c) Amir is wrong because the order of the
sequence of operations has been changed oe
B1 2.4 allow eg it’s a different
calculation
just it’s not subtraction of
nearly equal numbers is
insufficient
[1]
3 (a) 1.558068−1.386294
0.1
or
1.471572−1.386294
0.05
soi M1 1.1
1.71774 A1 1.1
1.70556 A1 1.1
[3]
(b)
the data must be (equally spaced) either side
of 2 oe
B1 2.4
[1]
(c)
central difference method is 2nd order method
whereas forward difference is 1st order method
oe
B1 2.4 allow eg central difference
method usually more accurate oe
[1]Y414/01 Mark Scheme November 2021
Question Answer Marks AOs Guidance
4 (a) B1 2.4 tangent from (1,1) cutting the
x-axis at (3,0)
allow tangent cutting xaxis between (but not
touching) the curve and x
= 3.2
[1]
4 (b)
f(2.8179205) and f(2.8179215) evaluated M1 1.1 must be fully evaluated, not just
substituted
0.000 000 34(336) and ‒ 0.000 000 195(334)
sign change so correct to 6 dp
A1 2.2a
[2]
4 (c) the ratios of differences are decreasing
(rapidly) so
the convergence is faster than 1st order
B1
B1
[2]
2.2a
2.2b
4 (d) #NUM! is displayed when the spreadsheet is
unable to compute a value oe
in this case it is because it is not possible to
find the natural logarithm or square root of a
negative number
B1
B1
1.1
1.1
[2]Y414/01 Mark Scheme November 2021
Question Answer Marks AOs Guidance
4 (e) N O P Q
0.01 ‒ 0.90517 0.03112 0.00098
B1 1.1
0.01×0.00098−−0.90517×0.03112
0.00098−−0.90517
soi M1 1.1
R S
0.03110 0.00033 – 0.00040
A1
A1
1.1
1.1
[4]
4 (f) 0.0311 since last two estimates agree to this
precision
B1 2.2b
[1]
4 (g) =IF(S8<0,R8,N8) or =IF(S8>0,N8,R8) B1 1.1 must see =
[1]
5 (a) S2n = 2.296296296 B1 1.1 D2
difference = ‒0.04003(1213) B1 1.1 E3
ratio = 0.1317(47194…) B1 1.1 awrt 0.1375 NB F4
2.264070598+2�8
3
= 2.250517186 M1 3.1a or 2.264070598+�8
2
= 2.253905539Y414/01 Mark Scheme November 2021
Question Answer Marks AOs Guidance
�8 = 2.24374048 A1 1.1 C5
[5]
5 (b)
extrapolation with S2n and difference from
table
M1 2.1 eg 2.250483378 and ‒3.38×10‒5 M1M1 only available for
extrapolation to infintity
from T32 or M32 with
r = 0.25
2.250481181 and ‒2.2×10-6 used A1 3.1a may be in partial extrapolation
eg to S128
2.250481181 ‒2.2×10-6 × 1
1−�
A1 1.1 r = 0.06499 – 0.065 or 1
16
NB 2.25048103
comparison of their answer from extrapolation
with S64
M1 1.1 dependent on award of M1
2.250481 is certain or 2.2504810 is possible, A1 2.2b allow 2.25048103 must follow extrapolation
to infinity from S64
[5]
6 (a) =(D5 ‒ D4)/D4 or = D5/D4 ‒ 1 B1 1.1 must see =
[1]
6 (b) y = nx B1 3.3
[1]
6 (c) 2.7185−�5
�5 oe M1 3.4
‒0.000518(286998) A1 1.1
5 × (‒0.0001037) = ‒0.0005185
so model is a good fit for n = 5
A1 3.5aY414/01 Mark Scheme November 2021
Question Answer Marks AOs Guidance
[3]
6 (d) the model becomes less good as the
approximation moves further away from e
B1 3.5b
[1]
6 (e) ‒0.000 0259 2 or ‒2.592×10‒5 B1 1.1
[1]
6 (f) so that (the values in cells) F18 and G18 are
also multiplied by (the value in cell) D21
B1 2.4 allow “absolute cell reference”
[1]
6 (g) z = �
�
oe B1 3.5c
[1]
7 (a) g(1.875) ‒ g(1.865)
0.01
soi M1 1.1 NB 0.010408−−0.00728
0.01
awrt 0.424 A1 1.1 0.424(18146683)
[2]
7 (b) this suggests the iteration will converge to the
root because ‒1 < g′(1.87) < 1 so she will use
the iteration
B1 2.2b FT their value
[1]
7 (c) P12
Q13
= O12 ‒ O11
=P13/P12
B1
B1
1.1
1.1
if B0B0 allow SC1 for
O12 ‒ O11 and P13/P12
[2]
7 (d) ratio is tending towards constant (0.424)
which suggests 1st order convergence
B1
B1
1.1
2.2b
[2]Y414/01 Mark Scheme November 2021
Question Answer Marks AOs Guidance
7 (e) comparison of last two estimates
1.869 is certain or 1.8691 is possible
M1
A1
2.4
2.2b
if M0 allow SC1 for either 1.869
is certain or 1.8691 is possible
with insufficient reasoning
[2]
7 (f) 2
1.86893356
1.86912211
1.86912218
1.86912218
1.86912218 [since �4 = �3 ] cao
M1
A1
A1
1.1
1.1
1.1
at least one iterate found using
relaxed iteration
�1, �2 correct to 8 dp
must see 8th dp justified by repeat
allow justification by
calculation of sign change
[3]OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
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