MATHEMATICS Standard Level Essay

MATHEMATICS Standard Level

16 pages
May 2013
Standard Level
Paper 2– 2 – M13/5/MATME/SP2/ENG/TZ1/XX/M
This markscheme is confidential and for the exclusive use of
examiners in this examination session.
It is the property of the International Baccalaureate and
must not be reproduced or distributed to any other person
without the authorization of the IB Assessment Centre.– 3 – M13/5/MATME/SP2/ENG/TZ1/XX/M
Instructions to Examiners
M Marks awarded for attempting to use a correct Method; working must be seen.
(M) Marks awarded for Method; may be implied by correct subsequent working.
A Marks awarded for an Answer or for Accuracy; often dependent on preceding M marks.
(A) Marks awarded for an Answer or for Accuracy; may be implied by correct subsequent working.
R Marks awarded for clear Reasoning.
N Marks awarded for correct answers if no working shown.
AG Answer given in the question and so no marks are awarded.

Using the markscheme
1 General
Mark according to scoris instructions and the document “Mathematics SL: Guidance for e-marking
May 2013”. It is essential that you read this document before you start marking. In particular,
please note the following. Marks must be recorded using the annotation stamps, using the new scoris
tool. Please check that you are entering marks for the right question.
• If a part is completely correct, (and gains all the “must be seen” marks), use the ticks with
numbers to stamp full marks.
• If a part is completely wrong, stamp A0 by the final answer.
• If a part gains anything else, it must be recorded using all the annotations.
All the marks will be added and recorded by scoris.
2 Method and Answer/Accuracy marks
• Do not automatically award full marks for a correct answer; all working must be checked,
and marks awarded according to the markscheme.
• It is generally not possible to award M0 followed by A1, as A mark(s) depend on the preceding
M mark(s), if any. An exception to this rule is when work for M1 is missing, as opposed to
incorrect (see point 4).
• Where M and A marks are noted on the same line, eg M1A1, this usually means M1 for an attempt
to use an appropriate method (eg substitution into a formula) and A1 for using the correct values.
• Where there are two or more A marks on the same line, they may be awarded independently; so if
the first value is incorrect, but the next two are correct, award A0A1A1.
• Where the markscheme specifies (M2), N3, etc., do not split the marks, unless there is a note.

• Once a correct answer to a question or part-question is seen, ignore further working.
• Most M marks are for a valid method, ie a method which can lead to the answer: it must indicate
some form of progress towards the answer.
• A marks are often dependent on the R mark being awarded for justification for the A mark, in
which case it is not possible to award A1R0. Hence the A1 is not awarded for a correct answer if
no justification or the wrong justification is given.– 4 – M13/5/MATME/SP2/ENG/TZ1/XX/M
3 N marks
If no working shown, award N marks for correct answers. In this case, ignore mark breakdown
(M, A, R).
• Do not award a mixture of N and other marks.
• There may be fewer N marks available than the total of M, A and R marks; this is deliberate as it
penalizes candidates for not following the instruction to show their working.
• There may not be a direct relationship between the N marks and the implied marks. There are
times when all the marks are implied, but the N marks are not the full marks: this indicates that we
want to see some of the working, without specifying what.
• For consistency within the markscheme, N marks are noted for every part, even when these match
the mark breakdown.
• If a candidate has incorrect working, which somehow results in a correct answer, do not award the
N marks for this correct answer. However, if the candidate has indicated (usually by crossing out)
that the working is to be ignored, award the N marks for the correct answer.
4 Implied and must be seen marks
Implied marks appear in brackets eg (M1).

• Implied marks can only be awarded if correct work is seen or if implied in subsequent working
(a correct final answer does not necessarily mean that the implied marks are all awarded). There
are questions where some working is required, but as it is accepted that not everyone will write the
same steps, all the marks are implied, but the N marks are not the full marks for the question.
• Normally the correct work is seen or implied in the next line.
• Where there is an (M1) followed by A1 for each correct answer, if no working shown, one correct
answer is sufficient evidence to award the (M1). An exception to this is where at least one
numerical final answer is not given to the correct three significant figures (see the accuracy
Must be seen marks appear without brackets eg M1.
• Must be seen marks can only be awarded if correct work is seen.
• If a must be seen mark is not awarded because work is missing (as opposed to M0 or A0 for
incorrect work) all subsequent marks may be awarded if appropriate.
5 Follow through marks (only applied after an error is made)
Follow through (FT) marks are awarded where an incorrect answer from one part of a question is
used correctly in subsequent part(s) or subpart(s). Usually, to award FT marks, there must be
working present and not just a final answer based on an incorrect answer to a previous part.
However, if the only marks awarded in a subpart are for the final answer, then FT marks should be
awarded if appropriate. Examiners are expected to check student work in order to award FT marks
where appropriate.
• Within a question part, once an error is made, no further A marks can be awarded for work which
uses the error, but M marks may be awarded if appropriate. (However, as noted above, if an

A mark is not awarded because work is missing, all subsequent marks may be awarded
if appropriate)
• Exceptions to this rule will be explicitly noted on the markscheme.
• If the question becomes much simpler because of an error then use discretion to award fewer
FT marks.
• If the error leads to an inappropriate value (eg probability greater than 1, use of r >1 for the sum
of an infinite GP, sin 1.5 θ = , non integer value where integer required), do not award the mark(s)
for the final answer(s).– 5 – M13/5/MATME/SP2/ENG/TZ1/XX/M
• The markscheme may use the word “their” in a description, to indicate that candidates may be
using an incorrect value.
• If a candidate makes an error in one part, but gets the correct answer(s) to subsequent part(s),
award marks as appropriate, unless the question says hence. It is often possible to use a different
approach in subsequent parts that does not depend on the answer to previous parts.
• In a “show that” question, if an error in a previous subpart leads to not showing the required
answer, do not award the final A1. Note that if the error occurs within the same subpart, the FT
rules may result in further loss of marks.
• Where there are anticipated common errors, the FT answers are often noted on the markscheme,
to help examiners. It should be stressed that these are not the only FT answers accepted, neither
should N marks be awarded for these answers.
6 Mis-read
If a candidate incorrectly copies information from the question, this is a mis-read (MR). A candidate
should be penalized only once for a particular mis-read. Use the MR stamp to indicate that this is a

misread. Do not award the first mark in the question, even if this is an M mark, but award all others
(if appropriate) so that the candidate only loses one mark for the misread.
• If the question becomes much simpler because of the MR, then use discretion to award
fewer marks.
• If the MR leads to an inappropriate value (eg probability greater than 1, use of r >1 for the sum of
an infinite GP, sin 1.5 θ = , non integer value where integer required), do not award the mark(s) for
the final answer(s).
• Miscopying of candidates’ own work does not constitute a misread, it is an error.
7 Discretionary marks (d)
An examiner uses discretion to award a mark on the rare occasions when the markscheme does not
cover the work seen. In such cases the annotation DM should be used and a brief note written next to
the mark explaining this decision.
8 Alternative methods
Candidates will sometimes use methods other than those in the markscheme. Unless the question
specifies a method, other correct methods should be marked in line with the markscheme. If in doubt,
contact your team leader for advice.
• Alternative methods for complete questions are indicated by METHOD 1, METHOD 2, etc.
• Alternative solutions for part-questions are indicated by EITHER . . . OR.
• Where possible, alignment will also be used to assist examiners in identifying where these
alternatives start and finish.
9 Alternative forms
Unless the question specifies otherwise, accept equivalent forms.

• As this is an international examination, accept all alternative forms of notation.
• In the markscheme, equivalent numerical and algebraic forms will generally be written in
brackets immediately following the answer.
• In the markscheme, simplified answers, (which candidates often do not write in examinations),
will generally appear in brackets. Marks should be awarded for either the form preceding the
bracket or the form in brackets (if it is seen). – 6 – M13/5/MATME/SP2/ENG/TZ1/XX/M
10 Accuracy of Answers
If the level of accuracy is specified in the question, a mark will be allocated for giving the final answer
to the required accuracy. When this is not specified in the question, all numerical answers should be
given exactly or correct to three significant figures
Candidates should NO LONGER be penalized for an accuracy error (AP). Examiners should award
marks according to the rules given in these instructions and the markscheme. Accuracy is not the
same as correctness – an incorrect value does not achieve relevant A marks. It is only final answers
which may lose marks for accuracy errors, not intermediate values. Please check work carefully for
FT. Further information on which answers are accepted is given in a separate booklet, along with
examples. It is essential that you read this carefully, as there are a number of changes.
Do not accept unfinished numerical final answers such as 3/0.1 (unless otherwise stated). As a rule,
numerical answers with more than one part (such as fractions) should be given using integers
(eg 6/8). Calculations which lead to integers should be completed, with the exception of fractions
which are not whole numbers.
Clarification of intermediate values accuracy instructions
Intermediate values do not need to be given to the correct three significant figures. If candidates work

with any rounded values, this could lead to an incorrect answer, in which case award A0 for the
final answer. However, do not penalise inaccurate intermediate values that lead to an acceptable
final answer.
11 Calculators
A GDC is required for paper 2, but calculators with symbolic manipulation features (eg TI-89) are not
Calculator notation
The Mathematics SL guide says:
Students must always use correct mathematical notation, not calculator notation.
Do not accept final answers written using calculator notation. However, do not penalize the use of
calculator notation in the working.
12 Style
The markscheme aims to present answers using good communication, eg if the question asks to find
the value of k, the markscheme will say k = 3, but the marks will be for the correct value 3 – there is
usually no need for the “ k = ”. In these cases, it is also usually acceptable to have another variable,
as long as there is no ambiguity in the question, eg if the question asks to find the value of p and of q,
then the student answer needs to be clear. Generally, the only situation where the full answer is
required is in a question which asks for equations – in this case the markscheme will say “must be
an equation”.
The markscheme often uses words to describe what the marks are for, followed by examples, using the
eg notation. These examples are not exhaustive, and examiners should check what candidates have
written, to see if they satisfy the description. Where these marks are M marks, the examples may

include ones using poor notation, to indicate what is acceptable. A valid method is one which will
allow candidate to proceed to the next step eg if a quadratic function is given in factorised form, and
the question asks for the zeroes, then multiplying the factors does not necessarily help to find the
zeros, and would not on its own count as a valid method.– 7 – M13/5/MATME/SP2/ENG/TZ1/XX/M
13 Candidate work
If a candidate has drawn a line through work on their examination script, or in some other way crossed
out their work, do not award any marks for that work.
14. Diagrams
The notes on how to allocate marks for sketches usually refer to passing through particular points or
having certain features. These marks can only be awarded if the sketch is approximately the correct
shape. All values given will be an approximate guide to where these points/features occur. In some
questions, the first A1 is for the shape, in others, the marks are only for the points and/or features. In
both cases, unless the shape is approximately correct, no marks can be awarded (unless otherwise
stated). However, if the graph is based on previous calculations, FT marks should be awarded if
appropriate.– 8 – M13/5/MATME/SP2/ENG/TZ1/XX/M
1. (a) d = 3 A1 N1
[1 mark]
(b) (i) correct substitution into term formula (A1)
eg 100 u = +5 3(99) , 5 3(100 1) + −
100 u = 302 A1 N2
(ii) correct substitution into sum formula (A1)

eg 100 ( ) 100 2(5) 99(3) 2
S = + , 100
100 (5 302) 2
S = +
100 S =15350 A1 N2
[4 marks]
(c) correct substitution into term formula (A1)
eg 1502 5 3( 1), 1502 3 2 =+ − = + n n
n = 500 A1 N2
[2 marks]
Total [7 marks]
2. (a) valid approach (M1)
eg 35 26 − , 26 35 + = p
p = 9 A1 N2
[2 marks]
(b) (i) mean = 26.7 A2 N2
(ii) recognizing that variance is 2 (sd) (M1)
eg 2 11.021… , σ = var , 2 11.158…
2 σ =121 A1 N2
[4 marks]
Total [6 marks]– 9 – M13/5/MATME/SP2/ENG/TZ1/XX/M
3. (a) p = 5 , q = 7, r = 7 (accept r = 5 ) A1A1A1 N3

[3 marks]
(b) correct working (A1)
eg 5 7 12
(3 ) ( 2) 7
   Ч Ч−
 
, 792 , 243 , 7 −2 , 24634368
coefficient of term in 5 x is −24634368 A1 N2
Note: Do not award the final A1 for an answer that contains x.
[2 marks]
Total [5 marks]
4. (a) (i)
1 11
1 10
2 12
  − −
  =       − −
A A1 N1
(ii) 1
21 1
20 1


  −   = −     
A A2 N2
Note: Award A1 for 6, 7 or 8 correct elements.
[3 marks]
(b) evidence of multiplying by −1 A (in any order) (M1)
eg −1 X AB = , −1 BA , one correct element
    = −     
X (accept x = 9 , y = −8 , z = 3.5 ) A2 N3
Note: Award A1 for two correct elements.
[3 marks]
Total [6 marks]– 10 – M13/5/MATME/SP2/ENG/TZ1/XX/M
5. (a)
A1A1A1 N3
Note: Award A1 for approximately correct shape crossing x-axis with 3 3.5 < <x .
Only if this A1 is awarded, award the following:
A1 for maximum in circle, A1 for endpoints in circle.
[3 marks]

(b) (i) t = π(exact), 3.14 A1 N1
(ii) recognizing distance is area under velocity curve (M1)
eg s v = ∫ , shading on diagram, attempt to integrate v
valid approach to find the total area (M1)
eg area A + area B, vt vt d d − ∫ ∫ , 3.14 5
0 3.14 vt vt d d + ∫ ∫ , v ∫
correct working with integration and limits (accept dx or missing dt) (A1)
eg 3.14 3.14
0 5 vt vt d d + ∫ ∫ , 3.067…+ 0.878…, 5 sin
0 e 1 t − ∫
distance = 3.95(m) A1 N3
[5 marks]
Total [8 marks]
6. (a) (i) k = 2 A1 N1
(ii) p = −1 A1 N1
(iii) q = 5 A1 N1
[3 marks]
(b) recognizing one transformation (M1)
eg horizontal stretch by 1
, reflection in x-axis,
A′ is (2, 5) − A1A1 N3
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