OCR MEI Further Pure Core 2024 June — Question 8 10 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2024
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeMatrix powers and repeated transformations
DifficultyStandard +0.3 This is a structured multi-part question on matrix transformations that guides students through standard techniques: identifying a shear transformation, computing a determinant, and proving a matrix power result by induction. While it requires Further Maths knowledge, each part follows predictable patterns with clear scaffolding, making it slightly easier than average for Further Pure content.
Spec4.01a Mathematical induction: construct proofs4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation
8
  1. Specify fully the transformation T of the plane associated with the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)\) and \(\lambda\) is a non-zero constant.
    1. Find detM.
    2. Deduce two properties of the transformation T from the value of detM.
  3. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)\), where \(n\) is a positive integer.
  4. Hence specify fully a single transformation which is equivalent to \(n\) applications of the transformation T.
Question 8:
AnswerMarks Guidance
8(a) Shear
with x-axis fixed, mapping (0, 1) to (, 1)M1
A1
AnswerMarks
[2]1.2
1.1Do not accept β€œsheaf”
Accept π‘₯-axis is a line of invariant points (not an invariant
line only). β€œShear parallel to π‘₯-axis” is insufficient.
Accept alternative mappings, e.g. (1,1) to (1+πœ†,1).
Do not accept shear factor.
AnswerMarks Guidance
8(b) (i)
[1]1.1
8(b) (ii)
Preserves orientationB1
B1
AnswerMarks
[2]1.2
1.2FT their determinant. Condone area scale factor = 1.
FT their determinant. Do not accept β€œorientation not
reversed”.
AnswerMarks Guidance
8(c) 1 1Γ—πœ† 1 πœ†
When n = 1 [𝐌1 =]( ) = ( ) so true when 𝑛 = 1
0 1 0 1
[Assume result holds for n = k]
1 π‘˜πœ† 1 πœ†
[πŒπ‘˜+1 =]( )( )
0 1 0 1
1 πœ†+π‘˜πœ† 1 πœ†(1+π‘˜)
[πŒπ‘˜+1 =]( ) = ( )
0 1 0 1
So true for n = 1 and if true for n = k then
AnswerMarks
true for n = k + 1, so true for all nB1
M1
A1*
A1dep
AnswerMarks
[4]2.1
2.1
2.2a
AnswerMarks
2.41Γ—πœ† must be seen. β€œTrue when 𝑛 = 1” could appear
later.
πŒπ‘˜+1 = πŒπ‘˜πŒ or 𝐌k+1 = πŒπŒπ‘˜ could be used.
Required matrix with intermediate step seen
𝑛 = 1 must have been considered
AnswerMarks Guidance
8(d) A shear with x-axis fixed, mapping (0,1) to (n ο€±)
[1]1.1 Accept π‘₯-axis is a line of invariant points (not an invariant
line only).
Accept alternative mappings, e.g. (1,1) to (1+π‘›πœ†,1).
Do not accept shear factor
Question 8:
8 | (a) | Shear
with x-axis fixed, mapping (0, 1) to (, 1) | M1
A1
[2] | 1.2
1.1 | Do not accept β€œsheaf”
Accept π‘₯-axis is a line of invariant points (not an invariant
line only). β€œShear parallel to π‘₯-axis” is insufficient.
Accept alternative mappings, e.g. (1,1) to (1+πœ†,1).
Do not accept shear factor.
8 | (b) | (i) | 1 | B1
[1] | 1.1
8 | (b) | (ii) | Preserves area
Preserves orientation | B1
B1
[2] | 1.2
1.2 | FT their determinant. Condone area scale factor = 1.
FT their determinant. Do not accept β€œorientation not
reversed”.
8 | (c) | 1 1Γ—πœ† 1 πœ†
When n = 1 [𝐌1 =]( ) = ( ) so true when 𝑛 = 1
0 1 0 1
[Assume result holds for n = k]
1 π‘˜πœ† 1 πœ†
[πŒπ‘˜+1 =]( )( )
0 1 0 1
1 πœ†+π‘˜πœ† 1 πœ†(1+π‘˜)
[πŒπ‘˜+1 =]( ) = ( )
0 1 0 1
So true for n = 1 and if true for n = k then
true for n = k + 1, so true for all n | B1
M1
A1*
A1dep
[4] | 2.1
2.1
2.2a
2.4 | 1Γ—πœ† must be seen. β€œTrue when 𝑛 = 1” could appear
later.
πŒπ‘˜+1 = πŒπ‘˜πŒ or 𝐌k+1 = πŒπŒπ‘˜ could be used.
Required matrix with intermediate step seen
𝑛 = 1 must have been considered
8 | (d) | A shear with x-axis fixed, mapping (0,1) to (n ο€±) | B1
[1] | 1.1 | Accept π‘₯-axis is a line of invariant points (not an invariant
line only).
Accept alternative mappings, e.g. (1,1) to (1+π‘›πœ†,1).
Do not accept shear factor
8
\begin{enumerate}[label=(\alph*)]
\item Specify fully the transformation T of the plane associated with the matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)$ and $\lambda$ is a non-zero constant.
\item \begin{enumerate}[label=(\roman*)]
\item Find detM.
\item Deduce two properties of the transformation T from the value of detM.
\end{enumerate}\item Prove that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)$, where $n$ is a positive integer.
\item Hence specify fully a single transformation which is equivalent to $n$ applications of the transformation T.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q8 [10]}}
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