At a fast-food restaurant, team members use headsets to communicate.

There are six different headsets: \(A, B, C, D, E\) and \(F\).

These headsets work on different frequencies, which means some staff members cannot directly communicate with each other.

The matrix below shows which headsets can directly communicate with each other.

\begin{aligned}
& \quad \quad \quad \quad \textit{receiver} \\
& \quad \ \  A \ \ \  B \ \ \  C \ \ \  D \ \ \  E\ \ \ F \\
\textit{sender}\ \ & \begin{array}{l}
A \\
B \\
C \\
D \\
E \\
F
\end{array}\begin{bmatrix}
0 & 1 & 0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0
\end{bmatrix}\\
\end{aligned}

 
In this matrix:

• the '1' in row \(A\), column \(B\) indicates that headset \(A\) can directly communicate with headset \(B\)
• the '0' in row \(A\), column \(E\) indicates that headset \(A\) cannot directly communicate with headset \(E\).

Which one of the following sequences allows a message to be communicated from headset \(A\) to headset \(E\) ?

1. \(A-C-E\)
2. \(A-D-B-E\)
3. \(A-B-D-E\)
4. \(A-F-D-E\)

The communication matrix below shows the communication links between five people: Steph \((S)\), Tran \((T)\),Ursula \((U)\), Vinh \((V)\) and Wanda \((W)\).

\(\begin{aligned}
&\quad \quad \quad \quad \quad \quad \ \ \ receiver \\
& \begin{array}{lllll}
\quad \quad \quad \quad \quad \ S \ \ \ T \ \ \ U \ \ \  V \ \ \  W\end{array} \\ &
sender \ \ \ \begin{array}{cc} S  \\ T\\ U\\  V\\ W \end{array}
\begin{bmatrix}
0 & 1 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 0
\end{bmatrix} \\
& \end{aligned}\)

In this matrix:

• the '1' in row \(S\), column \(T\) indicates that Steph can communicate directly with Tran
• the '0' in row \(V\), column \(W\) indicates that Vinh cannot communicate directly with Wanda.

Ursula needs to communicate with Steph.

The sequence of communication links that will successfully allow Ursula to communicate with Steph is

1. \(U-T-S\)
2. \(U-W-S\)
3. \(U-T-W-S\)
4. \(U-V-T-S\)
5. \(U-W-T-S\)

The diagram below shows the direct communication links that exist between Sam (S), Tai (T), Umi (U) and Vera (V). For example, the arrow from Umi to Vera indicates that Umi can communicate directly with Vera.
 

A communication matrix can be used to convey the same information.

In this matrix:

• a ‘1’ indicates that a direct communication link exists between a sender and a receiver
• a ‘0’ indicates that a direct communication link does not exist between a sender and a receiver.

The communication matrix could be

The communication matrix below shows the direct paths by which messages can be sent between two people in a group of six people, `U` to `Z`.
 

`{:(qquad qquad qquad qquad qquad qquad qquad qquad text(receiver)),(qquad qquad qquad qquad qquad quad U quad V quad W \ \ \ X \ \ \ Y quad \ Z), (text(sender) qquad {:(U),(V),(W),(X),(Y), (Z):}[(0, 1, 1, 0, 1, 1), (1, 0, 1, 0, 1, 0), (1, 1, 0, 1, 0, 1), (0, 1, 0, 0, 1, 1), (0, 0, 1, 1, 0, 1), (1, 1, 0, 1, 1, 0)]):}`
 

A 1 in the matrix shows that the person named in that row can send a message directly to the person named in that column.

For example, the 1 in row 4, column 2 shows that `X` can send a message directly to `V`.

In how many ways can `Y` get a message to `W` by sending it directly to one other person?

1. 0
2. 1
3. 2
4. 3
5. 4

The matrix below shows how five people, Alan (`A`), Bevan (`B`), Charlie (`C`), Drew (`D`) and Esther (`E`), can communicate with each other.
 

`{:(),(),(text(sender)\ ):}{:(qquadqquadqquad\ text(receiver)),(qquadqquadAquadBquadCquadDquadE),({:(A),(B),(C),(D),(E):}[(0,1,0,1,0),(1,0,0,0,0),(0,0,0,1,1),(1,0,1,0,0),(0,0,1,0,0)]):}`
 

A 1 in the matrix shows that the person named in that row can send a message directly to the person named in that column.

For example, the 1 in row 3 and column 4 shows that Charlie can send a message directly to Drew.

Esther wants to send a message to Bevan.

Which one of the following shows the order of people through which the message is sent?

1. Esther – Bevan
2. Esther – Charlie – Bevan
3. Esther – Charlie – Alan – Bevan
4. Esther – Charlie – Drew – Bevan
5. Esther – Charlie – Drew – Alan – Bevan

A travel company has five employees, Amara (`A`), Ben (`B`), Cheng (`C`), Dana (`D`) and Elka (`E`).

The company allows each employee to send a direct message to another employee only as shown in the communication matrix `G` below.

The matrix `G^2` is also shown below.
 

`{:(),(),(G = text(sender )):}{:(qquadqquadqquad\ text(receiver)),(qquadqquadAquadBquadCquadDquadE),({:(A),(B),(C),(D),(E):}[(0,1,1,1,1),(1,0,1,0,0),(1,1,0,1,0),(0,1,0,0,1),(0,0,0,1,0)]):}qquad{:(),(),(G^2 = text(sender )):}{:(qquadqquadqquad\ text(receiver)),(qquadqquadAquadBquadCquadDquadE),({:(A),(B),(C),(D),(E):}[(2,2,1,2,1),(1,2,1,2,1),(1,2,2,1,2),(1,0,1,1,0),(0,1,0,0,1)]):}`
 

The 1 in row `E`, column `D` of matrix `G` indicates that Elka (sender) can send a direct message to Dana (receiver).

The 0 in row `E`, column `C` of matrix `G` indicates that Elka cannot send a direct message to Cheng.

1. To whom can Dana send a direct message?   (1 mark)
   

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2. Cheng needs to send a message to Elka, but cannot do this directly.
3. Write down the names of the employees who can send the message from Cheng directly to Elka.   (1 mark) 
   

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