OCR MEI Further Pure with Technology 2024 June — Question 3 20 marks

Exam BoardOCR MEI
ModuleFurther Pure with Technology (Further Pure with Technology)
Year2024
SessionJune
Marks20
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicFirst order differential equations (integrating factor)
TypeIterative/numerical methods
DifficultyStandard +0.8 This is a multi-part Further Maths question combining analytical solution of a first-order linear ODE using integrating factor, isocline analysis requiring algebraic manipulation, qualitative analysis of direction fields, and numerical methods (Euler). While each individual technique is standard for Further Maths, the question requires synthesis across multiple areas and qualitative reasoning about solution behavior, placing it moderately above average difficulty.
Spec4.10a General/particular solutions: of differential equations4.10b Model with differential equations: kinematics and other contexts
3 This question concerns the family of differential equations $$\frac { d y } { d x } = x ^ { 2 } - y + \operatorname { acos } ( x ) \cos ( y ) \quad ( * * )$$ where \(a\) is a constant, \(x \geqslant 0\) and \(y > 0\).
  1. In this part of the question \(a = 0\).
    1. Find the solution to (\textbf{) in which \(y = 1\) when \(x = 0\).
    2. In this part of the question \(m\) is a real number. Show that the equation of the isocline \(\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { m }\) is a parabola.
    3. Using the result given in part (a)(ii), or otherwise, sketch the tangent field for (}) on the axes in the Printed Answer Booklet.
  2. Fig. 3.1 and Fig. 3.2 show the tangent fields for two distinct and unspecified values of \(a\). In each case, a sketch of the solution curve \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) which passes through the point \(( 0,2 )\) is shown for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 3.1} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_399_666_1324_317}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 3.2} \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_397_661_1324_1192}
    \end{figure}
    1. In each case, continue the sketch of the solution curve for \(\frac { 1 } { 2 } \leqslant x \leqslant 3\) on the axes in the Printed Answer Booklet.
    2. State one feature which is present in the continued solution curve for Fig. 3.1 that is not a feature of the continued solution curve for Fig. 3.2.
    3. Using a slider for \(a\), or otherwise, estimate the value of \(a\) for the solution curve shown in Fig. 3.2.
  3. The Euler method for the solution of the differential equation \(\frac { d y } { d x } = f ( x , y )\) is as follows. $$\begin{aligned} & y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right) \\ & x _ { n + 1 } = x _ { n } + h \end{aligned}$$
    1. Construct a spreadsheet to solve (), so that the value of \(a\) and the value of \(h\) can be varied, in the case \(x _ { 0 = 0\) and \(y _ { 0 } = 1\). State the formulae you have used in your spreadsheet.
    2. In this part of the question \(a = 0\). Use your spreadsheet with \(h = 0.1\) to approximate the value of \(y\) when \(x = 0.5\) for the solution to (}) in which \(y = 1\) when \(x = 0\).
    3. Using part (a)(i), state the accuracy of the approximate value of \(y\) given in part (c)(ii).
    4. State one change to your spreadsheet that could improve the accuracy of the approximate value of \(y\) found in part (c)(ii).
  4. The modified Euler method for the solution of the differential equation \(\frac { d y } { d x } = f ( x , y )\) is as follows. \(k _ { 1 } = h f \left( x _ { n } , y _ { n } \right)\) \(k _ { 2 } = h f \left( x _ { n } + h , y _ { n } + k _ { 1 } \right)\) \(y _ { n + 1 } = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)\) \(\mathrm { x } _ { \mathrm { n } + 1 } = \mathrm { x } _ { \mathrm { n } } + \mathrm { h }\)
    1. Adapt your spreadsheet from part (c)(i) to a spreadsheet to solve (**), so that the value of \(a\) and the value of \(h\) can be varied, in the case \(x _ { 0 } = 0\) and \(y _ { 0 } = 1\). State the formulae you have used in your spreadsheet.
    2. In this part of the question \(a = - 0.5\). Use the spreadsheet from part (d)(i) with \(h = 0.1\) to approximate the value of \(y\) when \(x = 0.5\) for the solution to \(( * * )\) in which \(y = 1\) when \(x = 0\). In this part of the question \(a = - 0.5\). The solution to (**) in which \(y = 1\) when \(x = 0\) has a turning point with coordinates \(( c , d )\) where \(0 < c < 1\).
    3. Use the spreadsheet in part (d)(i) to determine the value of \(c\) correct to \(\mathbf { 1 }\) decimal place.
    4. Use the spreadsheet in part (d)(i) to determine the value of \(d\) correct to \(\mathbf { 3 }\) decimal places.
Question 3:
AnswerMarks Guidance
3(a) (i)
[1]1.1 Accept solutions by CAS.
3(a) (ii)
d𝑥
AnswerMarks Guidance
Which is the equation of a parabola.B1
[1]2.1
3(a) (iii)
A1
AnswerMarks
[2]1.1
1.1Sketch of isoclines (dotted lines) not required.
Correct slopes close to both axes and gradients approximately
constant along parabolic curves.
Fully correct sketch allowing reasonable BOD for transcription.
AnswerMarks Guidance
3(b) (i)
B1
AnswerMarks
[2]1.1
1.1General shape with a minimum.
Curve should be tangent to the tangent field.
Do not award mark if curve crosses a tangent field line in a
manner that place the curve in conflict with the tangent field. That
is the presented curve is qualitatively impossible.
General shape with a maximum.
Curve should be tangent to the tangent field.
Do not award mark if curve crosses a tangent field line in a
manner that place the curve in conflict with the tangent field. That
is the presented curve is qualitatively impossible.
AnswerMarks Guidance
3(b) (ii)
[1]1.1 Alternatives.
Fig 3.1. has a single turning point close to 𝑥 = 1, while Fig. 3.2
has two turning points.
In Fig. 3.2 has a maximum close to 𝑥 = 1 and Fig. 3.1 does not.
In Fig. 3.1. the function decreases from 𝑦 = 2 [before increasing].
Fig. 3.1. has a point of inflection.
AnswerMarks Guidance
3(b) (iii)
[1]2.2b Accept −6 ≤ 𝑎 ≤ −4.5.
3(c) (i)
B2 contains 1
I1 contains 0.1 (value of ℎ)
K1 contains 0 (value of 𝑎)
C2 =A2^2-B2+\(K\)1*COS(A2)*COS(B2)
D2 =\(I\)1*C2
A3 =A2+\(I\)1
AnswerMarks
B3=B2+D2B1
B1
AnswerMarks
[2]3.1a
2.4Give reasonable BOD for possible transcription errors and
consider correct answer to 3(c)(ii) as evidence of correct formulae
in the spreadsheet.
Allow for 𝑎 and ℎ to be varied.
Must reference a cell, a slider or equivalent formulation.
Formulae for 𝑥 and 𝑦 .
𝑛+1 𝑛+1
Allow the specification of “C2” using function notation.
AnswerMarks Guidance
3(c) (ii)
[1]1.1 Answer must be given to at least 4 decimal places 0.6186 given
the context of spreadsheet working.
AnswerMarks Guidance
3(c) (iii)
So accurate to 1 decimal place.B1
[1]3.2b Must see comparison to actual solution value in part (a)(i).
Accept relative accuracy of approximately 4%.
Accept absolute value of difference of approximately 0.0249 (4
decimal places).
AnswerMarks Guidance
3(c) (iv)
[1]3.5c Also accept a named alternative method such as modified Euler.
3(d) (i)
B2 contains 1
I1 contains 0.1 (value of ℎ)
K1 contains −0.5 (value of 𝑎)
C2=A2^2 - B2 + \(K\)1*COS(A2)*COS(B2)
D2 =\(I\)1*C2 (value of 𝑘 )
1
E2= A2+\(I\)1
F2=B2+D2
G2=\(I\)1*(E2^2-
F2+\(K\)1*COS(E2)*COS(F2))
(value of 𝑘 )
2
A3=A2+\(I\)1
AnswerMarks
B3=B2+(1/2)*(D2+G2)B1
B1
AnswerMarks
[2]3.1a
2.4Give reasonable BOD for possible transcription errors and
consider correct answer to 3(d)(ii) as evidence of correct formulae
in the spreadsheet.
Allow for transcription of only changes necessary to 3(c)(i).
Columns for 𝑘 and 𝑘 .
1 2
May see 𝑘 and 𝑘 included explicitly within other formulae.
1 2
Formulae for 𝑥 and 𝑦 .
𝑛+1 𝑛+1
Allow the specification of “C2” and “G2” using function notation.
Allow follow through from part (c)(i).
AnswerMarks Guidance
3(d) (ii)
[1]1.1 Answer must be given to at least 4 decimal places 0.5042 given
the context of spreadsheet working.
AnswerMarks Guidance
3(d) (iv)
0.39218291 < 𝑑 < 0.392334706
AnswerMarks
So 𝑑 = 0.392 to 3 decimal places.M1
A1
AnswerMarks
[2]3.4
2.2bNote ℎ = 0.1 is insufficient to allow an estimate of 𝑑 to 2 decimal
places.
SC B1 for correct value of 𝑐 with no working.
PMT
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Though we make every effort to check our resources, there may be contradictions between published support and the
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Question 3:
3 | (a) | (i) | 𝑦 = 𝑥2 −2𝑥 +2−𝑒−𝑥. | B1
[1] | 1.1 | Accept solutions by CAS.
3 | (a) | (ii) | d𝑦 = 𝑚 ⟺ 𝑥2 −𝑦 = 𝑚.
d𝑥
Which is the equation of a parabola. | B1
[1] | 2.1
3 | (a) | (iii) | M1
A1
[2] | 1.1
1.1 | Sketch of isoclines (dotted lines) not required.
Correct slopes close to both axes and gradients approximately
constant along parabolic curves.
Fully correct sketch allowing reasonable BOD for transcription.
3 | (b) | (i) | B1
B1
[2] | 1.1
1.1 | General shape with a minimum.
Curve should be tangent to the tangent field.
Do not award mark if curve crosses a tangent field line in a
manner that place the curve in conflict with the tangent field. That
is the presented curve is qualitatively impossible.
General shape with a maximum.
Curve should be tangent to the tangent field.
Do not award mark if curve crosses a tangent field line in a
manner that place the curve in conflict with the tangent field. That
is the presented curve is qualitatively impossible.
3 | (b) | (ii) | e.g. Local minimum close to 𝑥 = 1 | B1
[1] | 1.1 | Alternatives.
Fig 3.1. has a single turning point close to 𝑥 = 1, while Fig. 3.2
has two turning points.
In Fig. 3.2 has a maximum close to 𝑥 = 1 and Fig. 3.1 does not.
In Fig. 3.1. the function decreases from 𝑦 = 2 [before increasing].
Fig. 3.1. has a point of inflection.
3 | (b) | (iii) | 𝑎 = −5.5 | B1
[1] | 2.2b | Accept −6 ≤ 𝑎 ≤ −4.5.
3 | (c) | (i) | A2 contains 0
B2 contains 1
I1 contains 0.1 (value of ℎ)
K1 contains 0 (value of 𝑎)
C2 =A2^2-B2+$K$1*COS(A2)*COS(B2)
D2 =$I$1*C2
A3 =A2+$I$1
B3=B2+D2 | B1
B1
[2] | 3.1a
2.4 | Give reasonable BOD for possible transcription errors and
consider correct answer to 3(c)(ii) as evidence of correct formulae
in the spreadsheet.
Allow for 𝑎 and ℎ to be varied.
Must reference a cell, a slider or equivalent formulation.
Formulae for 𝑥 and 𝑦 .
𝑛+1 𝑛+1
Allow the specification of “C2” using function notation.
3 | (c) | (ii) | Spreadsheet gives 0.618559 | B1
[1] | 1.1 | Answer must be given to at least 4 decimal places 0.6186 given
the context of spreadsheet working.
3 | (c) | (iii) | Actual solution is 0.64346934.
So accurate to 1 decimal place. | B1
[1] | 3.2b | Must see comparison to actual solution value in part (a)(i).
Accept relative accuracy of approximately 4%.
Accept absolute value of difference of approximately 0.0249 (4
decimal places).
3 | (c) | (iv) | Use a smaller value of ℎ. | B1
[1] | 3.5c | Also accept a named alternative method such as modified Euler.
3 | (d) | (i) | A2 contains 0
B2 contains 1
I1 contains 0.1 (value of ℎ)
K1 contains −0.5 (value of 𝑎)
C2=A2^2 - B2 + $K$1*COS(A2)*COS(B2)
D2 =$I$1*C2 (value of 𝑘 )
1
E2= A2+$I$1
F2=B2+D2
G2=$I$1*(E2^2-
F2+$K$1*COS(E2)*COS(F2))
(value of 𝑘 )
2
A3=A2+$I$1
B3=B2+(1/2)*(D2+G2) | B1
B1
[2] | 3.1a
2.4 | Give reasonable BOD for possible transcription errors and
consider correct answer to 3(d)(ii) as evidence of correct formulae
in the spreadsheet.
Allow for transcription of only changes necessary to 3(c)(i).
Columns for 𝑘 and 𝑘 .
1 2
May see 𝑘 and 𝑘 included explicitly within other formulae.
1 2
Formulae for 𝑥 and 𝑦 .
𝑛+1 𝑛+1
Allow the specification of “C2” and “G2” using function notation.
Allow follow through from part (c)(i).
3 | (d) | (ii) | Spreadsheet gives 0.504226904 | B1
[1] | 1.1 | Answer must be given to at least 4 decimal places 0.5042 given
the context of spreadsheet working.
3 | (d) | (iv) | From the spreadsheet with ℎ = 0.01
0.39218291 < 𝑑 < 0.392334706
So 𝑑 = 0.392 to 3 decimal places. | M1
A1
[2] | 3.4
2.2b | Note ℎ = 0.1 is insufficient to allow an estimate of 𝑑 to 2 decimal
places.
SC B1 for correct value of 𝑐 with no working.
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.
3 This question concerns the family of differential equations

$$\frac { d y } { d x } = x ^ { 2 } - y + \operatorname { acos } ( x ) \cos ( y ) \quad ( * * )$$

where $a$ is a constant, $x \geqslant 0$ and $y > 0$.
\begin{enumerate}[label=(\alph*)]
\item In this part of the question $a = 0$.
\begin{enumerate}[label=(\roman*)]
\item Find the solution to (\textbf{) in which $y = 1$ when $x = 0$.
\item In this part of the question $m$ is a real number. Show that the equation of the isocline $\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { m }$ is a parabola.
\item Using the result given in part (a)(ii), or otherwise, sketch the tangent field for (}) on the axes in the Printed Answer Booklet.
\end{enumerate}\item Fig. 3.1 and Fig. 3.2 show the tangent fields for two distinct and unspecified values of $a$. In each case, a sketch of the solution curve $\mathrm { y } = \mathrm { g } ( \mathrm { x } )$ which passes through the point $( 0,2 )$ is shown for $0 \leqslant x \leqslant \frac { 1 } { 2 }$.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
  \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_399_666_1324_317}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
  \includegraphics[alt={},max width=\textwidth]{6d485052-b0db-4c33-b374-4fd7b6f0759c-4_397_661_1324_1192}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item In each case, continue the sketch of the solution curve for $\frac { 1 } { 2 } \leqslant x \leqslant 3$ on the axes in the Printed Answer Booklet.
\item State one feature which is present in the continued solution curve for Fig. 3.1 that is not a feature of the continued solution curve for Fig. 3.2.
\item Using a slider for $a$, or otherwise, estimate the value of $a$ for the solution curve shown in Fig. 3.2.
\end{enumerate}\item The Euler method for the solution of the differential equation $\frac { d y } { d x } = f ( x , y )$ is as follows.

$$\begin{aligned}
& y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right) \\
& x _ { n + 1 } = x _ { n } + h
\end{aligned}$$
\begin{enumerate}[label=(\roman*)]
\item Construct a spreadsheet to solve (\textbf{), so that the value of $a$ and the value of $h$ can be varied, in the case $x _ { 0 } = 0$ and $y _ { 0 } = 1$. State the formulae you have used in your spreadsheet.
\item In this part of the question $a = 0$. Use your spreadsheet with $h = 0.1$ to approximate the value of $y$ when $x = 0.5$ for the solution to (}) in which $y = 1$ when $x = 0$.
\item Using part (a)(i), state the accuracy of the approximate value of $y$ given in part (c)(ii).
\item State one change to your spreadsheet that could improve the accuracy of the approximate value of $y$ found in part (c)(ii).
\end{enumerate}\item The modified Euler method for the solution of the differential equation $\frac { d y } { d x } = f ( x , y )$ is as follows.\\
$k _ { 1 } = h f \left( x _ { n } , y _ { n } \right)$\\
$k _ { 2 } = h f \left( x _ { n } + h , y _ { n } + k _ { 1 } \right)$\\
$y _ { n + 1 } = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$\\
$\mathrm { x } _ { \mathrm { n } + 1 } = \mathrm { x } _ { \mathrm { n } } + \mathrm { h }$
\begin{enumerate}[label=(\roman*)]
\item Adapt your spreadsheet from part (c)(i) to a spreadsheet to solve (**), so that the value of $a$ and the value of $h$ can be varied, in the case $x _ { 0 } = 0$ and $y _ { 0 } = 1$. State the formulae you have used in your spreadsheet.
\item In this part of the question $a = - 0.5$. Use the spreadsheet from part (d)(i) with $h = 0.1$ to approximate the value of $y$ when $x = 0.5$ for the solution to $( * * )$ in which $y = 1$ when $x = 0$.

In this part of the question $a = - 0.5$. The solution to (**) in which $y = 1$ when $x = 0$ has a turning point with coordinates $( c , d )$ where $0 < c < 1$.
\item Use the spreadsheet in part (d)(i) to determine the value of $c$ correct to $\mathbf { 1 }$ decimal place.
\item Use the spreadsheet in part (d)(i) to determine the value of $d$ correct to $\mathbf { 3 }$ decimal places.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure with Technology 2024 Q3 [20]}}
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