OCR MEI Further Pure Core 2019 June — Question 11 12 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2019
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeDescribe reflection from matrix
DifficultyStandard +0.3 This is a straightforward Further Maths question on recognizing standard transformation matrices (rotation and reflection) and finding mirror lines. The matrices are in standard forms with recognizable patterns (3-4-5 triangle for rotation, diagonal matrix for x-axis reflection). Part (b) requires matrix multiplication and finding the mirror line, while part (c) tests understanding that matrix multiplication order matters. All techniques are routine for Further Pure students with no novel insight required.
Spec4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products
11
  1. Specify fully the transformations represented by the following matrices.
Question 11:
AnswerMarks Guidance
11(a) M rotation
1M1 3.1a
through cos1(3/5) or 53.1 or 0.927 radsA1 1.1b
anti-clockwise about OA1 1.2
M reflection in x-axis
AnswerMarks Guidance
2M reflection in x-axis
2B1 1.2
[4]
AnswerMarks Guidance
11(b) 3 4
5 5
M =
3
4 3
AnswerMarks Guidance
5 5B1 1.1b
B1
3 4
x x
5 5
  
4 3y y
AnswerMarks Guidance
5 5M1 3.1a
attempt to find invariantor inv line y = mx [+c]
points
3 4 4 3
 x yx, x y y
AnswerMarks Guidance
5 5 5 5A2 1.1b
 m = ½ , 2 A12m2+3m2=0 A1
either or both m = ½ , 2 A1
 y = ½ x so y = ½ x is mirror lineA1 2.2a
Alternative solutionM1
1   1m2 2m    5 3 5 4 
1m2   2m m2 1   5 4  5 3 
M1
AnswerMarks Guidance
 2m2  5m + 2 = 0A1
 m = ½ m = ½ A2
[5]
AnswerMarks Guidance
11(c)  3 4
M =  
4 5 5
 
 4 3
  
AnswerMarks Guidance
 5 5B1 1.1b
M ≠ M [so can’t represent same reflection]
AnswerMarks Guidance
4 3M1 3.1a
4
as in part (b)
AnswerMarks Guidance
so mirror line cannot be the same, andA1 2.4
statement is incorrectincorrect
1.1b
M1
3.1a
A2
1.1b
Question 11:
11 | (a) | M rotation
1 | M1 | 3.1a
through cos1(3/5) or 53.1 or 0.927 rads | A1 | 1.1b | oe e.g. sin1(4/5), tan1(4/3) | 53 or 0.93 rads or better
anti-clockwise about O | A1 | 1.2 | or positive rotation about O
M reflection in x-axis
2 | M reflection in x-axis
2 | B1 | 1.2 | 1.2 | or Ox or y = 0 | or Ox or y = 0
[4]
11 | (b) | 3 4
5 5
M =
3
4 3
5 5 | B1 | 1.1b
B1
3 4
x x
5 5
  
4 3y y
5 5 | M1 | 3.1a
attempt to find invariant | or inv line y = mx [+c]
points
3 4 4 3
 x yx, x y y
5 5 5 5 | A2 | 1.1b | either or both | 2m2+3m2=0 A1
 m = ½ , 2 A1 | 2m2+3m2=0 A1
either or both |  m = ½ , 2 A1
 y = ½ x so y = ½ x is mirror line | A1 | 2.2a | accept valid geometric args
Alternative solution | M1
1   1m2 2m    5 3 5 4 
1m2   2m m2 1   5 4  5 3 
M1
 2m2  5m + 2 = 0 | A1
 m = ½ |  m = ½ | A2 | must discount m = 2 | must discount m = 2
[5]
11 | (c) |  3 4

M =  
4 5 5
 
 4 3
  
 5 5 | B1 | 1.1b
M ≠ M [so can’t represent same reflection]
4 3 | M1 | 3.1a | or attempt to find mirror line | M must be different
4
as in part (b)
so mirror line cannot be the same, and | A1 | 2.4 |  y = ½ x , so statement is
statement is incorrect | incorrect
1.1b
M1
3.1a
A2
1.1b
11
\begin{enumerate}[label=(\alph*)]
\item Specify fully the transformations represented by the following matrices.

\begin{itemize}
  \item $\mathbf { M } _ { 1 } = \left( \begin{array} { r r } \frac { 3 } { 5 } & - \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)$
  \item $\mathbf { M } _ { 2 } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)$
\item Find the equation of the mirror line of the reflection R represented by the matrix $\mathbf { M } _ { 3 } = \mathbf { M } _ { 1 } \mathbf { M } _ { 2 }$.
\item It is claimed that the reflection represented by the matrix $\mathbf { M } _ { 4 } = \mathbf { M } _ { 2 } \mathbf { M } _ { 1 }$ has the same mirror line as R . Explain whether or not this claim is correct.
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2019 Q11 [12]}}
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