OCR MEI Further Numerical Methods 2023 June — Question 1 7 marks

Exam BoardOCR MEI
ModuleFurther Numerical Methods (Further Numerical Methods)
Year2023
SessionJune
Marks7
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Mark schemeDownload PDF ↗
TopicNumerical integration
TypeTrapezium rule applied to real-world data
DifficultyStandard +0.8 This question requires understanding the subtle distinction between rounding and chopping errors, then applying interval arithmetic to a quotient where the denominator can be very small, leading to significant error magnification. Part (d) requires conceptual insight into why division amplifies uncertaintyβ€”going beyond routine calculation to demonstrate understanding of numerical analysis principles.
1 You are given that \(\left( x _ { 1 } , y _ { 1 } \right) = ( 0.9,2.3 )\) and \(\left( x _ { 2 } , y _ { 2 } \right) = ( 1.1,2.7 )\).
The values of \(x _ { 1 }\) and \(x _ { 2 }\) have been rounded to \(\mathbf { 1 }\) decimal place.
  1. Determine the range of possible values of \(x _ { 2 } - x _ { 1 }\). The values of \(y _ { 1 }\) and \(y _ { 2 }\) have been chopped to \(\mathbf { 1 }\) decimal place.
  2. Determine the range of possible values of \(y _ { 2 } - y _ { 1 }\). You are given that \(m = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } }\).
  3. Determine the range of possible values of \(m\).
  4. Explain why your answer to part (c) is much larger than your answer to part (a) and your answer to part (b).
Question 1:
AnswerMarks Guidance
1(a) 1.15βˆ’0.85 or 1.05βˆ’0.95
oe isw
AnswerMarks
2 1M1
A11.1a
1.1both values attempted, one must be correct;
allow non-strict inequality or eg 0.1 to 0.3 oe
allow SC1 for correct answer unsupported
AnswerMarks Guidance
0.1 < π‘₯π‘₯ βˆ’π‘₯π‘₯ < 0.3[2]
1(b) 2.8βˆ’2.3 π‘œπ‘œπ‘œπ‘œ 2.7βˆ’2.4
oe isw
AnswerMarks
2 1M1
A11.1a
1.1both values attempted, one must be correct
allow non-strict inequality or eg 0.3 to 0.5 oe
allow SC1 for correct answer unsupported
AnswerMarks Guidance
0.3 < 𝑦𝑦 βˆ’π‘¦π‘¦ < 0.5[2]
1(c) 0.5 0.3
π‘‘π‘‘β„Žπ‘’π‘’π‘’π‘’π‘œπ‘œ 0.1 𝐚𝐚𝐚𝐚𝐚𝐚 π‘‘π‘‘β„Žπ‘’π‘’π‘’π‘’π‘œπ‘œ 0.3
AnswerMarks
cao iswM1
A13.1a
3.2a
[2]
AnswerMarks Guidance
1(d) 1 < π‘šπ‘š < 5
the denominator involves the subtraction
AnswerMarks Guidance
of nearly equal quantities oeB1 2.4
between two nearly equal numbers
[1]
Question 1:
1 | (a) | 1.15βˆ’0.85 or 1.05βˆ’0.95
oe isw
2 1 | M1
A1 | 1.1a
1.1 | both values attempted, one must be correct;
allow non-strict inequality or eg 0.1 to 0.3 oe
allow SC1 for correct answer unsupported
0.1 < π‘₯π‘₯ βˆ’π‘₯π‘₯ < 0.3 | [2]
1 | (b) | 2.8βˆ’2.3 π‘œπ‘œπ‘œπ‘œ 2.7βˆ’2.4
oe isw
2 1 | M1
A1 | 1.1a
1.1 | both values attempted, one must be correct
allow non-strict inequality or eg 0.3 to 0.5 oe
allow SC1 for correct answer unsupported
0.3 < 𝑦𝑦 βˆ’π‘¦π‘¦ < 0.5 | [2]
1 | (c) | 0.5 0.3
π‘‘π‘‘β„Žπ‘’π‘’π‘’π‘’π‘œπ‘œ 0.1 𝐚𝐚𝐚𝐚𝐚𝐚 π‘‘π‘‘β„Žπ‘’π‘’π‘’π‘’π‘œπ‘œ 0.3
cao isw | M1
A1 | 3.1a
3.2a
[2]
1 | (d) | 1 < π‘šπ‘š < 5
the denominator involves the subtraction
of nearly equal quantities oe | B1 | 2.4 | must refer to denominator or to division by difference
between two nearly equal numbers
[1]
1 You are given that $\left( x _ { 1 } , y _ { 1 } \right) = ( 0.9,2.3 )$ and $\left( x _ { 2 } , y _ { 2 } \right) = ( 1.1,2.7 )$.\\
The values of $x _ { 1 }$ and $x _ { 2 }$ have been rounded to $\mathbf { 1 }$ decimal place.
\begin{enumerate}[label=(\alph*)]
\item Determine the range of possible values of $x _ { 2 } - x _ { 1 }$.

The values of $y _ { 1 }$ and $y _ { 2 }$ have been chopped to $\mathbf { 1 }$ decimal place.
\item Determine the range of possible values of $y _ { 2 } - y _ { 1 }$.

You are given that $m = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } }$.
\item Determine the range of possible values of $m$.
\item Explain why your answer to part (c) is much larger than your answer to part (a) and your answer to part (b).
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2023 Q1 [7]}}
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