OCR Further Pure Core 1 2023 June — Question 4 11 marks

Exam BoardOCR
ModuleFurther Pure Core 1 (Further Pure Core 1)
Year2023
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeCombined transformation matrix product
DifficultyStandard +0.3 This is a standard Further Maths question on linear transformations requiring routine techniques: identifying geometric transformations (rotation and reflection), using matrix powers with periodicity (A^4 = I), and applying area scaling under transformations. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average even for Further Maths.
Spec4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation
4 The transformations \(T _ { A }\) and \(T _ { B }\) are represented by the matrices \(\mathbf { A }\) and \(\mathbf { B }\) respectively, where \(\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\)
  1. Describe geometrically the single transformation consisting of \(T _ { A }\) followed by \(T _ { B }\).
  2. By considering the transformation \(\mathrm { T } _ { \mathrm { A } }\), determine the matrix \(\mathrm { A } ^ { 423 }\). The transformation \(\mathrm { T } _ { \mathrm { C } }\) is represented by the matrix \(\mathbf { C }\), where \(\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0 \\ 0 & \frac { 1 } { 3 } \end{array} \right)\). The region \(R\) is defined by the set of points \(( x , y )\) satisfying the inequality \(x ^ { 2 } + y ^ { 2 } \leqslant 36\). The region \(R ^ { \prime }\) is defined as the image of \(R\) under \(\mathrm { T } _ { \mathrm { C } }\).
    1. Find the exact area of the region \(R ^ { \prime }\).
    2. Sketch the region \(R ^ { \prime }\), specifying all the points where the boundary of \(R ^ { \prime }\) intersects the coordinate axes.
Question 4:
AnswerMarks Guidance
4(a)  0 1   0 − 1   1 0 
B A = =
1 0 1 0 0 − 1
AnswerMarks
i.e. a reflection in the x-axisM1
A1Multiplication in correct order
Or reflection in the line y = 0
Answer with no working B2
Alternative method:
A represents a rotation anti-clockwise of 900
B represents a reflection in the line y = x
Taken one after the other gives a reflection in the
AnswerMarks
x- axisM1
A1
[2]
AnswerMarks
(b)T is a rotation 90 degrees (anti-clockwise about
A
O)
(423 has remainder 3 when divided by 4) so A423
= A3
 0 1 
So A 4 2 3 = A 3 =
AnswerMarks
− 1 0B1
M1
AnswerMarks
A1For A4 = I or T repeated four times is a 360 degree rotation. Condone
A
clockwise instead of anticlockwise for T so notes that 423 is divisible by 4
A
with remainder 3 so A423 = A3
As A3 represents a 270 degrees rotation anti-clockwise (or 90 degrees
clockwise) or by direct calculation of A3.
[3]
AnswerMarks Guidance
(c)(i) 1
DetC=
6
1
Area of R' = 36=6
AnswerMarks
6M1
A11 1 1
× or seen.
3 2 6
Or area of ellipse = πab = 𝜋×2×3
cao
[2]
Alternative method:
A represents a rotation anti-clockwise of 900
B represents a reflection in the line y = x
Taken one after the other gives a reflection in the
x- axis
M1
A1
AnswerMarks Guidance
4(c) (ii)
(−3,
(3,0)
0)
(0,−2)
AnswerMarks
.M1
A1
A1
AnswerMarks
A1Correct shape, (approximately elliptical, possibly identified by scales, closed,
radius on horizontal axis > radius on vertical axis )
Intercepts labelled at x = ± 3
and y = ± 2(Give A1only rather than A2 if only positive intercepts are labelled
on both)
Give A1 if x and y axes interchanged.
Correctly shaded or labelled R’ and everything else correct.
[4]
Question 4:
4 | (a) |  0 1   0 − 1   1 0 
B A = =
1 0 1 0 0 − 1
i.e. a reflection in the x-axis | M1
A1 | Multiplication in correct order
Or reflection in the line y = 0
Answer with no working B2
Alternative method:
A represents a rotation anti-clockwise of 900
B represents a reflection in the line y = x
Taken one after the other gives a reflection in the
x- axis | M1
A1
[2]
(b) | T is a rotation 90 degrees (anti-clockwise about
A
O)
(423 has remainder 3 when divided by 4) so A423
= A3
 0 1 
So A 4 2 3 = A 3 =
− 1 0 | B1
M1
A1 | For A4 = I or T repeated four times is a 360 degree rotation. Condone
A
clockwise instead of anticlockwise for T so notes that 423 is divisible by 4
A
with remainder 3 so A423 = A3
As A3 represents a 270 degrees rotation anti-clockwise (or 90 degrees
clockwise) or by direct calculation of A3.
[3]
(c) | (i) | 1
DetC=
6
1
Area of R' = 36=6
6 | M1
A1 | 1 1 1
× or seen.
3 2 6
Or area of ellipse = πab = 𝜋×2×3
cao
[2]
Alternative method:
A represents a rotation anti-clockwise of 900
B represents a reflection in the line y = x
Taken one after the other gives a reflection in the
x- axis
M1
A1
4 | (c) | (ii) | (0,2)
(−3,
(3,0)
0)
(0,−2)
. | M1
A1
A1
A1 | Correct shape, (approximately elliptical, possibly identified by scales, closed,
radius on horizontal axis > radius on vertical axis )
Intercepts labelled at x = ± 3
and y = ± 2(Give A1only rather than A2 if only positive intercepts are labelled
on both)
Give A1 if x and y axes interchanged.
Correctly shaded or labelled R’ and everything else correct.
[4]
4 The transformations $T _ { A }$ and $T _ { B }$ are represented by the matrices $\mathbf { A }$ and $\mathbf { B }$ respectively, where $\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)$
\begin{enumerate}[label=(\alph*)]
\item Describe geometrically the single transformation consisting of $T _ { A }$ followed by $T _ { B }$.
\item By considering the transformation $\mathrm { T } _ { \mathrm { A } }$, determine the matrix $\mathrm { A } ^ { 423 }$.

The transformation $\mathrm { T } _ { \mathrm { C } }$ is represented by the matrix $\mathbf { C }$, where $\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0 \\ 0 & \frac { 1 } { 3 } \end{array} \right)$.

The region $R$ is defined by the set of points $( x , y )$ satisfying the inequality $x ^ { 2 } + y ^ { 2 } \leqslant 36$. The region $R ^ { \prime }$ is defined as the image of $R$ under $\mathrm { T } _ { \mathrm { C } }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the exact area of the region $R ^ { \prime }$.
\item Sketch the region $R ^ { \prime }$, specifying all the points where the boundary of $R ^ { \prime }$ intersects the coordinate axes.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1 2023 Q4 [11]}}
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