Notesale: Turn your study into money

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M11/5/MATSD/SP1/ENG/TZ1/XX/M

MARKSCHEME
May 2011

MATHEMATICAL STUDIES

Standard Level

Paper 1

27 pages

–2–

M11/5/MATSD/SP1/ENG/TZ1/XX/M

This markscheme is confidential and for the exclusive use of
examiners in this examination session
...


–3–

M11/5/MATSD/SP1/ENG/TZ1/XX/M

Paper 1 Markscheme
Instructions to Examiners
Notes:

If in doubt about these instructions or any other marking issues, contact your team leader for
clarification
...

Unless otherwise stated in the question, all numerical answers must be given exactly or
correct to three significant figures
...


(b)

Marks must be noted on candidates’ scripts as in the markscheme
...


(c)

In this paper, the maximum mark is awarded for a correct answer on the answer line
...


(d)

If the answer does not appear on the answer line, but the correct answer is seen in the working box
with no subsequent working, award the maximum mark
...


(f)

Working crossed out by the candidate should not be awarded any marks
...


(h)

If correct working results in a correct answer in the working box but then further working is
developed, full marks should not be awarded
...


–4–

M11/5/MATSD/SP1/ENG/TZ1/XX/M

Please note: Assignment of marks to the answers in all the following examples is for demonstration
purposes only
...

Implementation:
Question: Factorise x 2 − 5 x − 6
Markscheme
(x–6)(x+1)

Candidates’ Scripts

Marking

(i)

Answer line:

(x–6)(x+1)

(C2)

(ii)

Answer line:

(x+6)(x+1)

(A0)(A1)

(iii)

Working box:
(x–6)(x+1)
followed by answer line: x = 6 and –1, or just
6,–1

(A1)
(A0)

(iv)

(A1)(A1)

Working box:
(x–6)(x+1) then x = 6,–1
followed by answer line: x = 6 and –1, or just
6,–1 or factors and roots together

(A1)
(A0)

but
(v)

(vi)

Working box:
(x–6)(x+1) then x = 6,–1
followed by answer line: (x –6) (x+1) only

(C2)

Working box:
(x–6)(x+1) then x = 6,–1
and answer line empty

(A1)(A0)

Question: Using Pythagoras to find a side of a triangle:
Markscheme

9 + 4 = 13

(M1)(A1)

Candidates’ Scripts

Marking

(i)

Answer line:

13 or 3
...
5

9 + 4 = 13 = 6
...
50
Answer line empty

(M1)(A0)

(iv)

Working box:
9 + 4 = 13 = 3
...
16

(M1)(A1)

(3
...


(obvious
transcription
error)

–5–

M11/5/MATSD/SP1/ENG/TZ1/XX/M

Question: Calculate the gradient of the line passing through the points (5,3) and (0,9)
...
To limit the severity of the
penalty, follow through (ft) marks can be awarded
...

• If an answer resulting from follow through is extremely unrealistic (e
...
negative distances or wrong
by large order of magnitude) then the final A mark should not be awarded
...

• If a question is transformed by an error into a different, much simpler question then follow through
might not apply or might be reduced
...

• To award follow through marks for a question part, there must be working present for that part and
not just an answer based on the follow through
...

• Inadvertent use of radians will be penalised the first time it occurs
...
Cases of
this kind will be addressed on an individual basis
...

Question: An investment problem with two different rates of interest and a total amount of $600 split
across the rates in consecutive periods:
Markscheme

Candidate’s Script

(a) $ 600 × 1
...
02) + (
× 1
...
36
(A1)(ft)
Note: The (M1) is for splitting the value
from (a) and forming a sum of products
...


Case (i)
(a) Final amount after 1st period
= $ 600 × 1
...
02 + 301 × 1
...
06

Marking

(M1)
(A0)
(M1)
(A1)(ft)

but note
Case (ii)
an (M0) almost always prohibits the
associated (ft) so
(a)

$ 600 × 1
...
04 = $626
...
02 = $ 602
(b)

No working
...
06 on answer line
...
0

(a)

sin A sin 30
=
4
3

(A1)
A = 41
...
83

Marking
(M1) (A0) (use of sine rule but with wrong
values)
(A0)
(Note: the 2nd (A1) here was not marked (ft)
and cannot be awarded because there was an
earlier error in the same question part
...
26
but case (ii)
6
...

(a)

As A marks are normally dependent on the preceding M mark being awarded, it is not possible to
award (M0)(A1)
...
(See the first example above)
...


Implementation: Question: (a) χ 2 calculated followed by (b) degrees of freedom found and (c) and (d)
comparison to critical value
...
)
Markscheme
(a)

χ calc 2 = 3
...
92

Marking

(b)

n=4

(A1)

(b)

n=4

(c)

χ crit 2 = 9
...
92

(A1)

(b)

n=4

(A1)

(c)

χ crit 2 = 4
...
92

(A1)

(b) n = 1

(A0)

(c) χ crit 2 = 3
...
Thus, if an answer is wrong then the
working must be carefully analysed in order that marks are awarded for a different method in a
manner that is consistent with the markscheme
...
This includes alternatives obtained with a graphic display calculator
...

x=−

(M1)(A1)

(M1)(A1)(ft)

Coordinates are (–7/4, –73/8)

(A1)(ft)

OR

OR

f '( x ) = 4 x + 7 , 4x+7 = 0
so x = –7/4
(M1) for attempting to take a derivative and setting it to 0
(A1) for answer

(M1)
(A1)

146
73
f (− 7 ) = −
=−
4
16
8
(M1) for using f(–7/4), (A1) for answer
...
For example:

sin θ
for tan θ
...

(d)

As this is an international examination, all valid alternative forms of notation should be accepted
...
7; 1’7; 1 ⋅ 7 ; 1,7
...


(e)

Different forms of notation for set properties (e
...
complement): A′ ; A ; Ac ; U − A ; ( A U/A
...

p⇒ q; p→q; q ⇐ p
...
In such cases, (d) should be used to indicate where an examiner has used discretion
...


–9–
5

M11/5/MATSD/SP1/ENG/TZ1/XX/M

Accuracy of Answers
Unless otherwise stated in the question, all numerical answers must be given exactly or correct to 3
significant figures
...


This penalty is applied to the final answer of a question part only
...


THE ACCURACY PENALTY IS APPLIED AT MOST ONCE PER PAPER! Subsequent accuracy
errors can be ignored and full marks awarded if all else is correct
...

An accuracy penalty must be recorded in proximity to the incorrect answer as (A1)(AP)
...
This mark is not deducted in the item box but from
the final whole paper total automatically by scoris
...
In all such cases the final mark is not awarded if the rounding does not follow
the instructions given in the question
...
A mark for specified accuracy
can be regarded as a (ft) mark regardless of an immediately preceding (M0)
...
If the
1
rounding is incorrect, an accuracy penalty should be applied as detailed above
...
Reduction of a
fraction to its lowest terms is not essential
...
g
...
All other powers
(e
...
of non-integers) and values of transcendental functions such as sine and cosine must be evaluated
...
When this happens, multiple C
marks can be split e
...
(A1)(A1)(AP)
...

However, do not accept answers to one significant figure without working
...


– 10 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

Special cases
An answer taken directly from the IB chi squared statistical table can be given and used to the same level
of accuracy as appears in the table (3 decimal places) or correct to 3 significant figures
...
This problem will be dealt with on an individual basis as the
need arises
...
61 3sf)

Candidates’ Scripts
(i)
(ii)

Marking

Working box: nothing
but answer line: 4

(C0)

Working box: nothing
but answer line: 3
...
6

(C1)(C1)(AP)

(iii)

Working box:
Answer line:

9 + 4 = 13
3
...
60555

(v)

Working box:

(vi)

Working box:
9 + 4 = 14 = 3
...
60

(M1)
(A1)(AP)
(M1)
(A1)(AP)

(M1)(A1)(AP)
(M1)(A0)

– 11 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

If the accuracy is specified in the question e
...
give your answer correct to 4 decimal places, then there
would be one extra mark available as follows:
Markscheme

9 + 4 = 13 (M1)(A1)
= 3
...
606

(C0)

Working box: nothing
but answer line: 3
...
6

(iv)

Working box:
Answer line:

9 + 4 = 13
3
...
7417
whether transferred to answer line or not
...
2361
whether transferred to answer line or not
...
61 or 3
...


(M1)(A1)
(A0)

(C0)

Premature Rounding
Accuracy errors in a final answer, which result from premature rounding earlier in the same question part,
should not receive an accuracy penalty
...

If there is a mark available for a prematurely rounded answer and the rounding occurs at this stage, then
the inappropriate rounding should be penalised with (A0) but the answer can then be allowed to follow
through to the end of the question
...
Some discretion should be used
to deny a (ft) mark if the rounding is very bad and the answer far from its required value
...

Markscheme
a = 25 + 36 = 61
sin( A) sin(32)
=
5
61

Candidate’s Script

(M1)(A1)

(i)

(M1)(A1)(ft)

a = 25 + 36 = 61
= 7
...
8

(A1)(ft)

a = 25 + 36 = 61

(M1)(A1)

sin( A) sin(32)
=
7
...
8

A = 55
...
8
5

(A1)(ft)

a = 25 + 36 = 61

(M1)(A1)

(A1)(ft)

(ii)

(iii)

sin( A) sin(32)
=
7
...
83) = 56
...
0

6

(A0)(ft)
(The rounding is
severe and the
answer quite far
from correct)
...
This will
usually be either whole units or two decimal places, but could differ in rare instances depending on the
currency in question
...
This penalty is applied to the final answer of a question part
only
...

THE FINANCIAL ACCURACY PENALTY IS APPLIED AT MOST ONCE PER PAPER! Subsequent
financial accuracy errors can be ignored and full marks awarded, if all else is correct
...
This is
different to what we have done previously awarding (A0)(FP)
...

The financial accuracy penalty is imposed only for rounding to the wrong level of accuracy and NOT for
incorrect rounding to the required number of places
...

No single answer can receive two penalties
...

Please see the examples below
...
If this
instruction is not present, then do not apply the penalty
...


Example: A financial question demands accuracy correct to 2dp
...
6189
Markscheme
Financial accuracy penalty (FP) is
applicable in this question
$231
...
62 or 231
...
)

231
...
)

231
...
)

231

(A1)(FP)
(Both types of
error occurred but
(FP) takes
priority
...
00

(A1)(AP)
(It’s not clear
whether nearest
dollar or 2dp was
really intended but
we interpret as 2dp
rounded
incorrectly
...

This applies both to missing units and to incorrect units
...
Please see section G in the guidance document which clearly explains, with the use of
screenshots how this works in scoris
...

A unit penalty must be recorded in proximity to the incorrect answer as (A1)(UP)
...
This mark is not deducted in the item box but from
the final whole paper total automatically by scoris
...
If this instruction is not present, then do not apply
the penalty
...

NOTE: In this context, symbols for currency such as $ or GBP etc are not considered units
...

Missing degree symbols and percentage symbols are also not eligible for a unit penalty
...
If an answer is rounded incorrectly and also has wrong
or missing units, apply the accuracy penalty (AP) only
...

Example: A question has answer to part (i) of 66
...
The answer before rounding is 66
...

Part (ii) involves dividing by 60 with units of cms-1
...


– 15 –

Markscheme
Unit penalty (UP) applies in parts (i) and
(ii)
(i)
66
...
2cm

(A1)

(ii)

1
...
2

(A1)(UP)

(ii)

1
...
2 cm

(A1)

(ii)

1
...


(ii)

1
...


(ii)

1
...
10 cms
(A1)

8

(A1)(ft)

Graphic Display Calculators
Candidates will often be obtaining solutions directly from their calculators
...
No method marks can be awarded for incorrect answers supported only
by calculator notation
...


– 16 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

BEFORE MARKING ANY SCRIPT YOU SHOULD CHECK WHETHER OR NOT
THERE ARE ANY SUPPLEMENTARY SHEETS ADDED TO THE SCRIPT
QUESTION 1
(a)

p = 1
...
44
48

(M1)

Note: Award (M1) for correctly substituted equation for p
...
75

7

1
...
98  
 50 

(A1)

(C2)

(A1)
(A1)(ft)

(C2)

Note: Follow through from part (b)(i), irrespective of whether working is
shown
...
f
...
78 or 1
...

(c)

1
...
75
× 100
1
...


Note: Follow through from parts (a) and (b)
...
1 %

Notes: % sign not required
...
1%
If 100 missing and incorrect answer, award (M0)(A0)
...


(A1)(ft)

(C2)
[6 marks]

– 17 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

QUESTION 2
(a)

q = 25 − (4 + 3 + 8 + 4 + 1)

(M1)

Note: Award (M1) for subtraction from 25 of all values from the table
...
2

(C2)

(A2)(ft)

(C2)

(A1)

(C1)

(A1)(ft)

(C1)

Note: Award (M1) for use of mean formula with correct substitution
...

(ii)

2

(iii)

1
...

Award (A1)(AP) for 1
...


[6 marks]

– 18 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

QUESTION 3
(a)

−4, − 3, − 2, − 1, 0, 1, 2

(A1)

(C1)

(A1)(ft)(A1)(ft)

(C2)

(A1)(ft)

(C1)

(A1)(ft)(A1)(ft)

(C2)

Note: Award (A1) for correct numbers, do not penalise if braces, brackets
or parentheses seen
...
571, 57
...

Follow through from part (a)
...

(c)

1
(0
...
3%)
7

Note: Follow through from part (a)
...
143 , 14
...

Follow through from part (a)
...

(b)

If I visited (either) Sarah’s Snackbar or Pete’s Eats (then) I did not
visit Alan’s Diner

Note: Award (A1) for If… (then), (A1) for Sarah’s Snackbar or Pete’s
Eats, (A1) for did not visit Alan’s Diner
...
Follow through from part (a)
...
The final (A1) is awarded for
rounding their answer correct to the nearest km
...

(c)

4
...
3354, (A1)(ft) for ×104
...
Accept 4
...

[6 marks]

QUESTION 6
(a)

x=−

4
−2

x=2

(M1)
(A1)

OR
dy
= 4 − 2x
dx
x=2
(2, 7) or x = 2, y = 7

(M1)
(A1)
(A1)

(C3)

Notes: Award (M1)(A1)(A0) for 2 , 7 without parentheses
...

( x =) 4

(A1)

(C2)

OR
Using symmetry of graph x = 2 + 2
...

( x =) 4

(A1)(ft)

(C2)

[6 marks]

– 21 –
QUESTION 7
(a)

M11/5/MATSD/SP1/ENG/TZ1/XX/M

Unit penalty applies in parts (a) and (b)

AC 2 = 7
...
62

(M1)

Note: Award (M1) for correct substitution in Pythagoras Theorem
...
52

(M1)

(C2)

Note: Award (M1) for correct substitution in Pythagoras Theorem
...
5 m

(UP)

(A1)(ft)

(C2)

Note: Follow through from their answer to part (a)
...
5
3
...
5
12
...


θ = 16
...

Award (M1)(A0) for use of radians (0
...

[6 marks]

– 22 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

QUESTION 8
(a)

45000 + (5 − 1)1750

(M1)(A1)

Note: Award (M1) for substituted AP formula, (A1) for correct
substitutions
...


(b)

10
( 2 (45000) + (10 − 1)(1750) )
2

(M1)(A1)

Notes: Award (M1) for substituted AP formula, (A1)(ft) for correct
substitutions
...

= 528750 USD

(A1)(ft)

(C3)

Notes: Accept 529 000
...


[6 marks]

– 23 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

QUESTION 9
(a)

Tomorrow

Today

0
...
2

No Rain

0
...
4

0
...
3

No Rain

(A1)(A1)(A1)

(C3)

Note: Award (A1) for each correct pair
...
4 × 0
...
6 × 0
...
Follow through from their
tree diagram provided all probabilities are between 0 and 1
...
74

(A1)(ft)

(C3)
[6 marks]

QUESTION 10
(a)

0−2
6−0
1  2

= −  − , − 0
...

Accept equivalent forms for the equation of a line
...
5
2

(A1)(M1)

Note: Award (A1) for 1
...

= 4
...
333
...
Accept equivalent forms
...
Follow through from their common ratio found in
part (a)
...
333 used from part (a) award (M1)(A1)(ft) for an
answer of 79 285 or 79 300 irrespective of whether working is
shown
...

Follow through from parts (a) and (b)
...

k = 10

(A1)(ft)

(C3)

Notes: Follow through from parts (a) and (b)
...
Accept alternative methods
...
333 and 79 285 used award (M1)(M1)(A1)(ft) for k = 5
...
333 and 79 300 used award (M1)(M1)(A0)
...

Award (A1) for consistent use of ≤ with t
...

[6 marks]

– 26 –

M11/5/MATSD/SP1/ENG/TZ1/XX/M

QUESTION 13
(a)

y = −2 x + 8

(M1)

Note: Award (M1) for rearrangement of equation or for –2 seen
...

k =0
(ii)

y=

(A1)

1
x + c (can be implied)
2

Note: Award (M1) for substitution of

(C2)

(M1)

1
into equation of the line
...

forms for the equation of a line
...

2

1
y = ( x − 4)
2

(A1)(ft)

(C2)
[6 marks]

Notes: Follow through from parts (a) and (b)(i)
...


Accept equivalent

– 27 –
QUESTION 14
(UP) (a)
(b)

M11/5/MATSD/SP1/ENG/TZ1/XX/M

Unit penalty applies in parts (a) and (d)

61 kg

(A1)

66 − 52

(A1)

= 14

(C1)

(A1)(ft)

(C2)

(C1)

Note: Award (A1) for identifying quartiles, (A1)(ft) for correct
subtraction of their quartiles
...
5 × 20 + 56
...

= 53 kg

(UP)

(A1)

(C2)
[6 marks]

QUESTION 15
(a)

y = 1
...
25 − 1
= 0
...
25)
...
25 − a −2
a=2
y = 1
...
25

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