Question 6 - A-Level Maths

OCR FD1 AS 2018 March — Question 6 16 marks

Exam Board	OCR
Module	FD1 AS (Further Decision 1 AS)
Year	2018
Session	March
Marks	16
Topic	Linear Programming
Type	Constraint derivation verification
Difficulty	Moderate -0.3 This is a standard linear programming question requiring explanation of constraint derivation and graphical optimization. Part (i) involves straightforward algebraic manipulation to verify given constraints, part (ii) is routine 2D graphing with z fixed, and part (iii) requires standard linear programming optimization. While it has multiple parts and requires careful algebraic work, it follows predictable patterns for AS-level Decision Maths with no novel problem-solving insight required.
Spec	7.06a LP formulation: variables, constraints, objective function 7.06b Slack variables: converting inequalities to equations 7.06d Graphical solution: feasible region, two variables

6 An online magazine consists of an editorial, articles, reviews and advertisements.
The magazine must have a total of at least 12 pages. The editorial always takes up exactly half a page. There must be at least 3 pages of articles and at most 1.5 pages of reviews. At least a quarter but fewer than half of the pages in the magazine must be used for advertisements. Let $x$ be the number of pages used for articles, $y$ be the number of pages used for reviews and $z$ be the number of pages used for advertisements. The constraints on the values of $x , y$ and $z$ are: $$\begin{aligned} & x + y + z \geqslant 11.5 \\ & x \geqslant 3 \\ & y \leqslant 1.5 \\ & 2 x + 2 y - 2 z + 1 \geqslant 0 \\ & 2 x + 2 y - 6 z + 1 \leqslant 0 \\ & y \geqslant 0 \end{aligned}$$

(a) Explain why $x + y + z \geqslant 11.5$.
(b) Explain why only one non-negativity constraint is needed.
(c) Show that the requirement that at least one quarter of the pages in the magazine must be used for advertisements leads to the constraint $2 x + 2 y - 6 z + 1 \leqslant 0$. Advertisements bring in money but are not popular with the subscribers. The editor decides to limit the number of pages of advertisements to at most four.
Graph the feasible region in the case when $z = 4$ using the axes in the Printed Answer Booklet. To be successful the magazine needs to maximise the number of subscribers.
The editor has found that when $z \leqslant 4$ the expected number of subscribers is given by $P = 300 x + 400 y$.
(a) What is the maximum expected number of subscribers when $z = 4$ ?
(b) By first considering the feasible region for $z = k$, where $k \leqslant 4$, find an expression for the maximum number of subscribers in terms of $k$. \section*{END OF QUESTION PAPER} \section*{OCR} \section*{Oxford Cambridge and RSA}

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6 An online magazine consists of an editorial, articles, reviews and advertisements.\\
The magazine must have a total of at least 12 pages. The editorial always takes up exactly half a page. There must be at least 3 pages of articles and at most 1.5 pages of reviews. At least a quarter but fewer than half of the pages in the magazine must be used for advertisements.

Let $x$ be the number of pages used for articles, $y$ be the number of pages used for reviews and $z$ be the number of pages used for advertisements.

The constraints on the values of $x , y$ and $z$ are:

$$\begin{aligned}
& x + y + z \geqslant 11.5 \\
& x \geqslant 3 \\
& y \leqslant 1.5 \\
& 2 x + 2 y - 2 z + 1 \geqslant 0 \\
& 2 x + 2 y - 6 z + 1 \leqslant 0 \\
& y \geqslant 0
\end{aligned}$$
\begin{enumerate}[label=(\roman*)]
\item (a) Explain why $x + y + z \geqslant 11.5$.\\
(b) Explain why only one non-negativity constraint is needed.\\
(c) Show that the requirement that at least one quarter of the pages in the magazine must be used for advertisements leads to the constraint $2 x + 2 y - 6 z + 1 \leqslant 0$.

Advertisements bring in money but are not popular with the subscribers. The editor decides to limit the number of pages of advertisements to at most four.
\item Graph the feasible region in the case when $z = 4$ using the axes in the Printed Answer Booklet.

To be successful the magazine needs to maximise the number of subscribers.\\
The editor has found that when $z \leqslant 4$ the expected number of subscribers is given by $P = 300 x + 400 y$.
\item (a) What is the maximum expected number of subscribers when $z = 4$ ?\\
(b) By first considering the feasible region for $z = k$, where $k \leqslant 4$, find an expression for the maximum number of subscribers in terms of $k$.

\section*{END OF QUESTION PAPER}

\section*{OCR}
\section*{Oxford Cambridge and RSA}
\end{enumerate}

\hfill \mbox{\textit{OCR FD1 AS 2018 Q6 [16]}}

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