| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Challenging +1.8 This is a multivariable calculus optimization problem requiring partial derivatives, solving a system of nonlinear equations, and second derivative test using the Hessian determinant. While the techniques are standard for Further Maths, the algebraic manipulation is non-trivial and the context requires careful interpretation of the second-order conditions to determine which variable to change. |
| Spec | 8.05d Partial differentiation: first and second order, mixed derivatives8.05e Stationary points: where partial derivatives are zero |
5 A trading company deals in two goods. The formula used to estimate $z$, the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is\\
$z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }$,\\
where $x$ and $y$ are the masses, in thousands of tonnes, of the two goods.
You are given that $x > 0$ and $y > 0$.
\begin{enumerate}[label=(\alph*)]
\item In the first week of trading, it was found that the values of $x$ and $y$ corresponded to the stationary value of $z$.
Determine the total cost to the company for this week.
\item For the second week, the company intends to make a small change in either $x$ or $y$ in order to reduce the total weekly cost.
Determine whether the company should change $x$ or $y$. (You are not expected to say by how much the company should reduce its costs.)
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2021 Q5 [11]}}