\begin{enumerate}[label=(\roman*)]
\item Rachel thinks that the answer given in the newspaper article is not sensible. Give a verbal argument why Rachel might think that the batsman should be given out.

Rachel tries to formalise her argument. She defines four simple propositions.\\
o: "The batsman is given out."\\
lb: "The batsman is given out (LBW)."\\
c: "The batsman is given out (caught)."\\
b: "The ball hit the bat."
\item An implication of the batsman not being out (LBW) is that the ball has hit the bat. Write this down in terms of Rachel's propositions.
\item Similarly, write down the implication of the batsman not being out (caught).
\item Using your answers to parts (ii) and (iii) write down the implication of a batsman being not out, in terms of $b$ and $\sim b$.\\[0pt]
[You may assume that if $\mathrm { w } \Rightarrow \mathrm { y }$ and $\mathrm { x } \Rightarrow \mathrm { z }$, then $( \mathrm { w } \wedge \mathrm { x } ) \Rightarrow ( \mathrm { y } \wedge \mathrm { z } )$. ]
\item By writing down the contrapositive of your implication from part (iv), produce an implication which supports Rachel's argument.\\
(b) A classroom rule has been broken by either Anja, Bobby, Catherine or Dimitria, or by a subset of those four. The teacher knows that Dimitria could not have done it on her own.

Let $a$ be the proposition "Anja is guilty", and similarly for $b , c$ and $d$.
\item Express the teacher's knowledge as a compound proposition.

Evidence emerges that Bobby and Catherine were elsewhere at the time, so they cannot be guilty. This can be expressed as the compound proposition $\sim ( b \vee c )$.
\item Construct a truth table to show the truth values of the compound proposition given by the conjunction of the two compound propositions, one from part (i) and one given above.
\item What does your truth table tell you about who is guilty?

3 Three products, A, B and C are to be made.\\
Three supplements are included in each product. Product A has 10 g per kg of supplement $\mathrm { X } , 5 \mathrm {~g}$ per kg of supplement Y and 5 g per kg of supplement Z .

Product B has 5 g per kg of supplement $\mathrm { X } , 5 \mathrm {~g}$ per kg of supplement Y and 3 g per kg of supplement Z .\\
Product C has 12 g per kg of supplement $\mathrm { X } , 7 \mathrm {~g}$ per kg of supplement Y and 5 g per kg of supplement Z .\\
There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .\\
Product A will sell at $\pounds 7$ per kg and costs $\pounds 3$ per kg to produce. Product B will sell at $\pounds 5$ per kg and costs $\pounds 2$ per kg to produce. Product C will sell at $\pounds 4$ per kg and costs $\pounds 3$ per kg to produce.

The profit is to be maximised.
\item Explain how the initial feasible tableau shown in Fig. 3 models this problem.

\begin{table}[h]

\begin{center}
\begin{tabular}{|l|l|}
\hline
1(v) &  \\
\hline
1(vi) &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
1
\item &  \\
\hline
 &  \\
\hline
 &  \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|l|l|}
\hline
2(a)(i) &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{enumerate}

\hfill \mbox{\textit{OCR MEI D2 2014 Q2 [16]}}