OCR H240/02 2017 Specimen — Question 3 9 marks

Exam BoardOCR
ModuleH240/02 (Pure Mathematics and Statistics)
Year2017
SessionSpecimen
Marks9
TopicCompleting the square and sketching
TypeApplied quadratic optimization
DifficultyModerate -0.8 This is a straightforward applied quadratic modeling question requiring basic algebraic manipulation (factorizing to find k), solving a quadratic inequality, and simple interpretation of a model's limitations. All techniques are routine for Stats 1, with no novel problem-solving or conceptual depth required.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02n Sketch curves: simple equations including polynomials1.02z Models in context: use functions in modelling
A publisher has to choose the price at which to sell a certain new book. The total profit, \(£t\), that the publisher will make depends on the price, \(£p\). He decides to use a model that includes the following assumptions. • If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small. • If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small. The graphs below show two possible models. \includegraphics{figure_3}
  1. Explain how model A is inconsistent with one of the assumptions given above. [1]
  2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k(12p - p^2)\), and find the value of the constant \(k\). [2]
  3. The publisher needs to make a total profit of at least £6400. Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie. [4]
  4. Comment briefly on how realistic model B may be in the following cases. • \(p = 0\) • \(p = 12.1\) [2]
A publisher has to choose the price at which to sell a certain new book.
The total profit, $£t$, that the publisher will make depends on the price, $£p$.
He decides to use a model that includes the following assumptions.

• If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
• If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.

The graphs below show two possible models.

\includegraphics{figure_3}

\begin{enumerate}[label=(\alph*)]
\item Explain how model A is inconsistent with one of the assumptions given above. [1]

\item Given that the equation of the curve in model B is quadratic, show that this equation is of the form $t = k(12p - p^2)$, and find the value of the constant $k$. [2]

\item The publisher needs to make a total profit of at least £6400. Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie. [4]

\item Comment briefly on how realistic model B may be in the following cases.
• $p = 0$
• $p = 12.1$ [2]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/02 2017 Q3 [9]}}
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