Question 17:
17 | (a) | (i) | After t minutes volume of liquid = 1000 − 5t [litres]
x
so concentration =
1 0 0 0 − 5 t | B1
[1] | 3.1b | AG Volume of liquid (allow “volume”, “amount of
liquid”, “capacity” or “liquid”) in the container must be
given in first step. Do not allow “volume leaving”.
`17 | (a) | (ii) | d𝑥
= 10…
d𝑡
𝑥
…[−]5( )
1000−5𝑡
d x x
 + = 1 0
d t 2 0 0 − t | B1*
B1*
B1dep
[3] | 3.3
3.3
3.3 | Not oe
AG successful completion
17 | (b) |  1 d t
IF is e 2 0 0 − t
1
= e − ln ( 2 0 0 − t ) =
2 0 0 − t
1 d x x 1 0
 + =
2 0 0 − t d t ( 2 0 0 − t ) 2 2 0 0 − t
d  x  1 0 𝑥 10
 = or = ∫ d𝑡
d t 2 0 0 − t 2 0 0 − t 200−𝑡 200−𝑡
𝑥
= −10ln(200−𝑡)[+𝑐]
200−𝑡
t =0,x=0c=10ln200
x  2 0 0 
= − 1 0 ln ( 2 0 0 − t ) + 1 0 ln 2 0 0 = 1 0 ln
2 0 0 − t 2 0 0 − t
 200 
x=10(200−t)ln

200−t | M1
A1
M1
M1*
A1
B1FT
M1dep
A1
[8] | 3.1a
2.1
2.1
2.1
2.1
3.1b
2.1
2.1 | Finding integrating factor
1
Allow 𝐴 where 𝐴 is clearly an unevaluated constant
200−𝑡
of integration and not from incorrect integration
Multiplying equation by their IF, soi by next line
Either, ft their IF (i.e. their IF × 𝑥)
First A1 must have been scored. Condone missing +𝑐 and
bracketing errors.
Correct 𝑐 for their integral
Combining their log terms correctly, do not condone
bracketing errors without clear recovery
AG www
17 | (c) | (i) | Half drawn off when t = 1 0 0
⇒ 𝑥 = 10×100×ln2 = 693 [g] | M1
A1
[2] | 3.4
3.4 | May be embedded but must be seen
Accept 690 or better, or 1000ln2
17 | (c) | (ii) | dx
[x is maximised when] =0
dt
200
10(200−𝑡)ln( ) = 10(200−𝑡)
200−𝑡
 2 0 0 
 l n = 1
2 0 0 − t
2 0 0
 = e
2 0 0 − t
⇒ 𝑡 = 126.42… | M1
M1*
A1
M1dep
A1
[5] | 3.1a
1.1
1.1
1.1
3.4 | soi. Implied by 𝑥 = 10(200−𝑡).
d𝑥 200
Or finding = 10(1−ln( ) ) correctly
d𝑡 200−𝑡
Eliminating logarithms correctly
130 (2sf) or better. Do not accept exact value. Ignore
units.
17 | (d) | In reality the salt would not
dissolve/mix/spread/diffuse/disperse/distribute instantly | B1
[1] | 3.5b | Or “the salt does not
dissolve/mix/spread/diffuse/disperse/distribute
uniformly”. Must be explicit that modelling assumption is
unrealistic.
Ignore subsequent comments
Do not accept reference to volume or exogenous factors,
e.g. evaporation/temperature
Response | Mark
1
is undefined when 𝑥 = 2
3 x − 2 | B1
There is an asymptote at 𝑥 = 2 | B1
The function is not continuous at 𝑥 = 2 | B1
1
When 𝑥 = 2 the denominator will be 0 and is undefined/impossible/invalid/improper/indeterminate/an error/cannot be computed
0 | B1
When 𝑥 = 2 we get 1 divided by 0 which is undefined/impossible/invalid/improper/indeterminate/an error/cannot be computed | B1
1
When 𝑥 = 2 we get which makes the integral undefined/impossible/invalid/improper/indeterminate/an error/incomputable
0 | B1
When 𝑥 = 2 the denominator of the function is 0 and you cannot divide by 0 | B1
1
When 𝑥 = 2 it becomes = undefined
0 | B1
1
When 𝑥 = 2 we get which tends to infinity
0 | B0
1
When 𝑥 = 2 we get which is divergent
0 | B0
1
When 𝑥 = 2 we get which has no solutions
0 | B0
When 𝑥 = 2 we get 1 divided by 0 which is an error function | B0
When 𝑥 < 2 the value is complex/not a real number | B0
It is undefined when 𝑥 → 2 | B0
It is undefined when 𝑥 = 2 | B1
Response | Mark
[The series] converges/works only when −1 < 𝑥 ≤ 1 | A1
The series requires/works/converges when −1 < 𝑥 ≤ 1 | A1
The series converges only when −1 < 𝑥 < 1 | A0
The series converges when −1 ≤ 𝑥 ≤ 1 | A0
The series does not converge when 𝑥 > 1 | A1
The series does not converge when 𝑥 is greater than 1 | A1
The series does not converge when 𝑥 ≥ 1 | A0
The series converges only when |𝑥| < 1 | A0
The series does not converge when |𝑥| > 1 | A1
The series does not converge when |𝑥| ≥ 1 | A0
The series does not converge when |𝑥| ≥ 1 unless 𝑥 = 1 | A1
𝑥 = 2 is greater than 1 so outside the range for convergence | A1
2 ≥ 1 | A0
For series to converge 𝑥 < 1 | A0
2 > 1 so the series does not converge | A1
The series converges when −1 < 𝑥 < 1
The series converges when 𝑥 is between −1 and 1
For the series to converge 𝑥 must be between −1 and 1
The series converges when |𝑥| < 1
2 > 1
The series does not converge when 𝑥 = 2
PMT
Need to get in touch?
If you ever have any questions about OCR qualifications or services (including administration, logistics and teaching) please feel free to get in
touch with our customer support centre.
Call us on
01223 553998
Alternatively, you can email us on
support@ocr.org.uk
For more information visit
ocr.org.uk/qualifications/resource-finder
ocr.org.uk
Twitter/ocrexams
/ocrexams
/company/ocr
/ocrexams
OCR is part of Cambridge University Press & Assessment, a department of the University of Cambridge.
For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored. © OCR
2024 Oxford Cambridge and RSA Examinations is a Company Limited by Guarantee. Registered in England. Registered office
The Triangle Building, Shaftesbury Road, Cambridge, CB2 8EA.
Registered company number 3484466. OCR is an exempt charity.
OCR operates academic and vocational qualifications regulated by Ofqual, Qualifications Wales and CCEA as listed in their
qualifications registers including A Levels, GCSEs, Cambridge Technicals and Cambridge Nationals.
OCR provides resources to help you deliver our qualifications. These resources do not represent any particular teaching method
we expect you to use. We update our resources regularly and aim to make sure content is accurate but please check the OCR
website so that you have the most up-to-date version. OCR cannot be held responsible for any errors or omissions in these
resources.
Though we make every effort to check our resources, there may be contradictions between published support and the
specification, so it is important that you always use information in the latest specification. We indicate any specification changes
within the document itself, change the version number and provide a summary of the changes. If you do notice a discrepancy
between the specification and a resource, please contact us.
Whether you already offer OCR qualifications, are new to OCR or are thinking about switching, you can request more
information using our Expression of Interest form.
Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.