OCR Further Pure Core AS 2020 November — Question 2 10 marks

Exam BoardOCR
ModuleFurther Pure Core AS (Further Pure Core AS)
Year2020
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeWrite down transformation matrix
DifficultyModerate -0.8 This is a straightforward Further Pure matrices question testing standard transformations and their properties. Parts (a)-(c) require direct recall and basic matrix multiplication, (d) tests invariant lines by substitution, and (e) asks for standard interpretation of determinants. All parts follow textbook procedures with no novel problem-solving required, making it easier than average even for Further Maths.
Spec4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines4.03i Determinant: area scale factor and orientation
P, Q and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P.
  1. Write down the matrix A. [1]
Q is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). B is the matrix that represents Q.
  1. Find the matrix B. [2]
T is P followed by Q. C is the matrix that represents T.
  1. Determine the matrix C. [2]
\(L\) is the line whose equation is \(y = x\).
  1. Explain whether or not \(L\) is a line of invariant points under T. [2]
An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
  1. Explain what the value of the determinant of C means about
    [3]
Question 2:
AnswerMarks Guidance
2(a) 1 0 
A= 
AnswerMarks
0 −1B1
[1]1.2
(b). 0
 
. 1
1 .
 
AnswerMarks
2 .B1
B1
AnswerMarks
[2]1.1
1.10 0
 →  ie y-axis invariant
1 1
1 1
 → 
AnswerMarks
0 21 0 . .
  or   insufficient for
. . 2 1
B1
AnswerMarks
(c)1 01 0 
C=  
2 10 −1
1 0 
= 
AnswerMarks
2 −1M1
A1
AnswerMarks
[2]1.1
1.1Their A and B but must be the correct
way round (ie BA, not AB).
AnswerMarks
(d)1 0 x x
  = 
2 −1x x
Since each point gets mapped to itself it is a line of
AnswerMarks
invariant pointsM1
A1
AnswerMarks
[2]3.1a
2.2aMultiplying a correct vector
x y
( or  ) correctly into C.
x y
If M0 then SC1 for use of a particular
point leading to correct conclusion
(This can follow from an incorrect
matrix, possibly to show that y = x is
AnswerMarks
not a line of invariant points)If a particular point (eg (1, 1)) is used
then supporting statement required for
M1 that eg this must therefore apply
on the line through (0, 0) and (1, 1).
Can use to
deduce tha1t x =0 y. 𝑥𝑥 𝑥𝑥
� �� �=� �
2 −1 𝑦𝑦 𝑦𝑦
AnswerMarks
(e)det C = 1×–1 – 2×0 = –1
So area of N is the same as area of M oe
But the orientation of N is the reverse of the
AnswerMarks
orientation of M.B1ft
B1ft
B1ft
AnswerMarks
[3]1.1
2.2a
AnswerMarks
2.2aDo not award if implication of
statement is that area of N is negative.
AnswerMarks
ft for det C < 0.Allow “orientation changed” 
Question 2:
2 | (a) | 1 0 
A= 
0 −1 | B1
[1] | 1.2
(b) | . 0
 
. 1
1 .
 
2 . | B1
B1
[2] | 1.1
1.1 | 0 0
 →  ie y-axis invariant
1 1
1 1
 → 
0 2 | 1 0 . .
  or   insufficient for
. . 2 1
B1
(c) | 1 01 0 
C=  
2 10 −1
1 0 
= 
2 −1 | M1
A1
[2] | 1.1
1.1 | Their A and B but must be the correct
way round (ie BA, not AB).
(d) | 1 0 x x
  = 
2 −1x x
Since each point gets mapped to itself it is a line of
invariant points | M1
A1
[2] | 3.1a
2.2a | Multiplying a correct vector
x y
( or  ) correctly into C.
x y
If M0 then SC1 for use of a particular
point leading to correct conclusion
(This can follow from an incorrect
matrix, possibly to show that y = x is
not a line of invariant points) | If a particular point (eg (1, 1)) is used
then supporting statement required for
M1 that eg this must therefore apply
on the line through (0, 0) and (1, 1).
Can use to
deduce tha1t x =0 y. 𝑥𝑥 𝑥𝑥
� �� �=� �
2 −1 𝑦𝑦 𝑦𝑦
(e) | det C = 1×–1 – 2×0 = –1
So area of N is the same as area of M oe
But the orientation of N is the reverse of the
orientation of M. | B1ft
B1ft
B1ft
[3] | 1.1
2.2a
2.2a | Do not award if implication of
statement is that area of N is negative.
ft for det C < 0. | Allow “orientation changed”
P, Q and T are three transformations in 2-D.

P is a reflection in the $x$-axis. A is the matrix that represents P.

\begin{enumerate}[label=(\alph*)]
\item Write down the matrix A. [1]
\end{enumerate}

Q is a shear in which the $y$-axis is invariant and the point $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is transformed to the point $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$. B is the matrix that represents Q.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the matrix B. [2]
\end{enumerate}

T is P followed by Q. C is the matrix that represents T.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Determine the matrix C. [2]
\end{enumerate}

$L$ is the line whose equation is $y = x$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Explain whether or not $L$ is a line of invariant points under T. [2]
\end{enumerate}

An object parallelogram, $M$, is transformed under T to an image parallelogram, $N$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Explain what the value of the determinant of C means about
\begin{itemize}
\item the area of $N$ compared to the area of $M$,
\item the orientation of $N$ compared to the orientation of $M$.
\end{itemize} [3]
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core AS 2020 Q2 [10]}}
This paper (8 questions)
View full paper
Q1 6 Q2 10 Q3 12 Q4 6 Q5 7 Q6 5 Q7 6 Q8 8