| Date | May 2018 | Marks available | 3 | Reference code | 18M.1.hl.TZ0.4 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Show that | Question number | 4 | Adapted from | N/A |
Question
The transformations T1, T2, T3, T4, in the plane are defined as follows:
T1 : A rotation of 360° about the origin
T2 : An anticlockwise rotation of 270° about the origin
T3 : A rotation of 180° about the origin
T4 : An anticlockwise rotation of 90° about the origin.
The transformation T5 is defined as a reflection in the \(x\)-axis.
The transformation T is defined as the composition of T3 followed by T5 followed by T4.
Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the operation of composition of transformations.
Show that T1, T2, T3, T4 under the operation of composition of transformations form a group. Associativity may be assumed.
Show that this group is cyclic.
Write down the 2 × 2 matrices representing T3, T4 and T5.
Find the 2 × 2 matrix representing T.
Give a geometric description of the transformation T.
Markscheme
A2
[2 marks]
Note: Award A1 for 6, 7 or 8 correct.
the table is closed – no new elements A1
T1 is the identity A1
T3 (and T1) are self-inverse; T2 and T4 are an inverse pair. Hence every element has an inverse A1
hence it is a group AG
[3 marks]
all elements in the group can be generated by T2 (or T4) R1
hence the group is cyclic AG
[1 mark]
T3 is represented by \(\left( {\begin{array}{*{20}{c}}
{ - 1}&0 \\
0&{ - 1}
\end{array}} \right)\) A1
T4 is represented by \(\left( {\begin{array}{*{20}{c}}
0&{ - 1} \\
1&0
\end{array}} \right)\) A1
T5 is represented by \(\left( {\begin{array}{*{20}{c}}
1&0 \\
0&{ - 1}
\end{array}} \right)\) A1
[3 marks]
\(\left( {\begin{array}{*{20}{c}}
0&{ - 1} \\
1&0
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
1&0 \\
0&{ - 1}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{ - 1}&0 \\
0&{ - 1}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
0&{ - 1} \\
{ - 1}&0
\end{array}} \right)\) (M1)A1
Note: Award M1A0 for multiplying the matrices in the wrong order.
[2 marks]
a reflection in the line \(y = - x\) A1
[1 mark]
